Thesis for the final dissertation for my MS degree in Computer Science & Engineering.
Subject is Mathematical Optimization in the field of Network Pricing Problems.
The present thesis is the result of a research carried on from September 2011 to August 2012; part of this work has been developed at the Graphes et Optimisation Mathématique (G.O.M.) group of the Université Libre de Bruxelles, under the supervision of Professor Martine Labbé; further development and experimentation have been carried on at the Laboratorio di Ricerca Operativa of the Università degli studi di Trieste with Professor Lorenzo Castelli as supervisor.
This document summarizes a master's thesis that implements a reliable overlay multicast protocol on wireless sensor nodes. The thesis first discusses related work on wireless sensor networks, communication schemes, hardware, and the Contiki operating system. It then presents the design of the Sensor Nodes Overlay Multicast Communication (SNOMC) protocol, including node roles, message types, design models, data structures, and the SNOMC algorithm. The implementation of SNOMC in Contiki is described, along with implementations of UDP and TCP for comparison. An evaluation analyzes the performance of transmitting small and large messages using SNOMC.
This document is the dissertation of Davi Correia submitted in 2006 for the degree of Doctor of Philosophy in Electrical and Computer Engineering. The dissertation proposes a higher-order perfectly matched layer (PML) that combines advantages of regular and complex frequency shifted PMLs. It applies the second-order PML to open-region, waveguide and periodic electromagnetic problems. Results show the second-order PML outperforms regular and CFS-PMLs and its performance is independent of simulation technique used. The dissertation contains 7 chapters presenting formulation, implementation and numerical results of the higher-order PML applied to various problems.
This dissertation proposes variational methods for signal and image processing problems. Part I introduces Variational Mode Decomposition (VMD), a method for decomposing signals into intrinsic mode functions. VMD models the decomposition as an optimization problem that minimizes the bandwidth of individual modes while reconstructing the original signal. Part II extends VMD to multidimensional signals like images, and introduces binary support functions to allow for spatially compact modes. Part III presents a variational model for removing stripes from remote sensing imagery by leveraging sparsity and total variation regularization with L1 fidelity. The dissertation demonstrates these variational methods on a variety of synthetic and real-world signals and images.
This document presents a final year project that aims to analyze musical instrument sounds from a large database (SOL) to classify instruments and playing techniques. The project uses feature extraction techniques like MFCC and scattering transform to represent audio samples as vectors in a space where distance correlates with perceptual similarity. Two ranking metrics, mean average precision and precision at k, evaluate how well the feature spaces can discriminate between classes. The document also discusses human perception of timbre and explores optimizing feature spaces based on perceptual judgments through metric learning. The goal is to develop acoustic descriptions of sounds that align with how humans interpret and differentiate instruments.
This document describes a hybrid method using finite difference time domain (FDTD) and finite element method (FEM) to model electromagnetic wave scattering from a flat surface containing a narrow groove. The authors divided the geometry into an outer region modeled with FDTD and an inner region within the groove modeled with FEM. They implemented this hybrid method in MATLAB but encountered problems with boundary conditions and updating fields between regions that they solved using Mur's absorbing boundary conditions and separating field variables between regions. The final code produced expected results for this 2D cavity problem model.
This document discusses various techniques for solving macroeconometric models. It begins with an overview of direct and iterative methods for solving systems of equations, including LU and QR factorization, sparse matrix methods, stationary iterative methods like Jacobi and Gauss-Seidel, and nonstationary methods like conjugate gradient. It then focuses on techniques for large models, such as block triangular decomposition and parallel computing. Finally, it covers solution methods for rational expectations models, including the stacked-time approach and Newton's method.
This dissertation proposes and analyzes distributed algorithms for solving convex optimization problems in networked systems. Specifically, it considers algorithms for distributed convex optimization that are robust to noise, distributed online convex optimization over time-varying graphs, distributed saddle-point subgradient methods with consensus constraints, and distributed nuclear norm regularization for multi-task learning. It also studies stability properties of stochastic differential equations with persistent noise. The analysis establishes convergence and performance guarantees for the distributed optimization algorithms and characterizes different notions of stability for stochastic systems.
Thesis. A comparison between some generative and discriminative classifiers.Pedro Ernesto Alonso
This thesis comprises Naive Bayes, Full Bayesian Network, Artificial Neural Networks, Support Vector Machines and Logistic Regression. For classification purposes.
This document summarizes a master's thesis that implements a reliable overlay multicast protocol on wireless sensor nodes. The thesis first discusses related work on wireless sensor networks, communication schemes, hardware, and the Contiki operating system. It then presents the design of the Sensor Nodes Overlay Multicast Communication (SNOMC) protocol, including node roles, message types, design models, data structures, and the SNOMC algorithm. The implementation of SNOMC in Contiki is described, along with implementations of UDP and TCP for comparison. An evaluation analyzes the performance of transmitting small and large messages using SNOMC.
This document is the dissertation of Davi Correia submitted in 2006 for the degree of Doctor of Philosophy in Electrical and Computer Engineering. The dissertation proposes a higher-order perfectly matched layer (PML) that combines advantages of regular and complex frequency shifted PMLs. It applies the second-order PML to open-region, waveguide and periodic electromagnetic problems. Results show the second-order PML outperforms regular and CFS-PMLs and its performance is independent of simulation technique used. The dissertation contains 7 chapters presenting formulation, implementation and numerical results of the higher-order PML applied to various problems.
This dissertation proposes variational methods for signal and image processing problems. Part I introduces Variational Mode Decomposition (VMD), a method for decomposing signals into intrinsic mode functions. VMD models the decomposition as an optimization problem that minimizes the bandwidth of individual modes while reconstructing the original signal. Part II extends VMD to multidimensional signals like images, and introduces binary support functions to allow for spatially compact modes. Part III presents a variational model for removing stripes from remote sensing imagery by leveraging sparsity and total variation regularization with L1 fidelity. The dissertation demonstrates these variational methods on a variety of synthetic and real-world signals and images.
This document presents a final year project that aims to analyze musical instrument sounds from a large database (SOL) to classify instruments and playing techniques. The project uses feature extraction techniques like MFCC and scattering transform to represent audio samples as vectors in a space where distance correlates with perceptual similarity. Two ranking metrics, mean average precision and precision at k, evaluate how well the feature spaces can discriminate between classes. The document also discusses human perception of timbre and explores optimizing feature spaces based on perceptual judgments through metric learning. The goal is to develop acoustic descriptions of sounds that align with how humans interpret and differentiate instruments.
This document describes a hybrid method using finite difference time domain (FDTD) and finite element method (FEM) to model electromagnetic wave scattering from a flat surface containing a narrow groove. The authors divided the geometry into an outer region modeled with FDTD and an inner region within the groove modeled with FEM. They implemented this hybrid method in MATLAB but encountered problems with boundary conditions and updating fields between regions that they solved using Mur's absorbing boundary conditions and separating field variables between regions. The final code produced expected results for this 2D cavity problem model.
This document discusses various techniques for solving macroeconometric models. It begins with an overview of direct and iterative methods for solving systems of equations, including LU and QR factorization, sparse matrix methods, stationary iterative methods like Jacobi and Gauss-Seidel, and nonstationary methods like conjugate gradient. It then focuses on techniques for large models, such as block triangular decomposition and parallel computing. Finally, it covers solution methods for rational expectations models, including the stacked-time approach and Newton's method.
This dissertation proposes and analyzes distributed algorithms for solving convex optimization problems in networked systems. Specifically, it considers algorithms for distributed convex optimization that are robust to noise, distributed online convex optimization over time-varying graphs, distributed saddle-point subgradient methods with consensus constraints, and distributed nuclear norm regularization for multi-task learning. It also studies stability properties of stochastic differential equations with persistent noise. The analysis establishes convergence and performance guarantees for the distributed optimization algorithms and characterizes different notions of stability for stochastic systems.
Thesis. A comparison between some generative and discriminative classifiers.Pedro Ernesto Alonso
This thesis comprises Naive Bayes, Full Bayesian Network, Artificial Neural Networks, Support Vector Machines and Logistic Regression. For classification purposes.
Hub location models in public transport planningsanazshn
This document is a dissertation written in German on the topic of hub location models in public transport planning. It begins with an introduction that describes hub location problems, aspects of multi-period planning, and solution procedures. The dissertation then reviews literature on hub location problems and formulations. It presents new mathematical formulations for public transport applications and extensions. Finally, it discusses solution methods like Lagrangian relaxation, Benders decomposition, and heuristic algorithms.
This document provides an introduction to integral calculus and demonstrates how to perform integral calculations using the computer algebra system Sage. It covers key integral calculus concepts such as the definition of the integral, Riemann sums, the Fundamental Theorem of Calculus, and techniques for evaluating integrals such as substitution, integration by parts, and trigonometric substitutions. It also discusses applications of integrals to computing areas, volumes, arc lengths, averages, and centers of mass. The document is intended as a preliminary version of an instructional text on integral calculus using Sage.
Pattern classification via unsupervised learnersNick Palmer
I study classification problems in a standard learning framework, in which an unsupervised learner creates a discriminant function over each class and observations are labeled by the learner returning the highest value associated with that observation. I examine whether this approach gains significant advantage over traditional discriminant techniques.
Probably Approximately Correct learning distributions over class labels under L1 distance or KL-divergence is shown to imply PAC classification in this framework. I determine bounds on the regret associated with the resulting classifier, taking into account the possibility of variable misclassification penalties, and demonstrate the advantage of estimating the a posteriori probability distributions over class labels in the setting of Optical Character Recognition.
Unsupervised learners can be used to learn a class of probabilistic
concepts (stochastic rules denoting the probability that an observation has a positive
label in a 2-class setting). I demonstrate a situation where unsupervised learners
can be used even when it is hard to learn distributions over class labels – in this case
the discriminant functions do not estimate the class probability densities.
If that isn't exciting enough, I then use a standard state-merging technique to PAC-learn a class of probabilistic automata. The results show that by learning the distribution over outputs under the weaker L1 distance rather than KL-divergence we are able to learn without knowledge of the expected length of an output. It is also shown that for a restricted class of these automata learning under L1 distance is equivalent to learning under KL-divergence.
Multiuser detection based on generalized side-lobe canceller plus SOVA algorithmAitor López Hernández
This is my undergraduate thesis project, consisting on comparing the performance of several classical multi user detection strategies with another solution of our choice, based on side-lobe cancellation.
Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zinggRohit Bapat
This document provides an overview of computational fluid dynamics (CFD) and summarizes its key steps and concepts. It discusses the fundamentals of CFD, including conservation laws, governing equations, finite difference approximations, semi-discrete and finite volume methods, and time-marching algorithms. The document is intended to introduce readers to the basic theory and methods in CFD for modeling fluid flow and transport phenomena.
This document is the thesis presented by Sophie Hautphenne to obtain a PhD in computer science from the Université Libre de Bruxelles in October 2009. The thesis investigates algorithmic approaches for computing extinction probabilities and other measures for branching processes and their extensions, including Markovian trees, reducible branching processes, branching processes in random environments, and those subject to catastrophes. It develops both linear and quadratic algorithms and applies matrix analytic methods.
All Minimal and Maximal Open Single Machine Scheduling Problems Are Polynomia...SSA KPI
This thesis presents Boolean linear programming models for three single machine scheduling problems with equal job lengths and release dates. It is proven that the problems are polynomially solvable by showing that the models have the property of total dual integrality. In Chapter 4, a model for minimizing total weighted completion time with preemption and unit job processing times is presented. The model is based on an assignment problem formulation. It is shown that the model has total unimodularity and total dual integrality, implying polynomial time solvability. These results are generalized to arbitrary equal job processing times in Chapter 5. In Chapter 6, models are presented for the preemptive and non-preemptive versions of minimizing total weighted tardiness
This document is the doctoral dissertation of Takayoshi Yoshimura submitted to Nagoya Institute of Technology in January 2002. The dissertation proposes improvements to HMM-based text-to-speech systems including simultaneous modeling of phonetic and prosodic parameters, characteristic conversion using speaker interpolation, and improving synthesized speech quality. The contributions aim to more accurately model speech and allow for voice conversion capabilities in text-to-speech systems.
This document describes a Matlab implementation of neural networks. It begins with an introduction to neural networks and associative memory, explaining how neural networks can be used to create associative memories that recall stored information based on partial cues. It then discusses implementing associative memory using neural networks and provides Matlab functions for storing and recalling information. The document goes on to describe perceptrons, multi-layer networks, and backpropagation networks. It concludes by presenting three applications of backpropagation networks: solving the XOR problem, curve fitting, and time series forecasting.
This document summarizes a master's thesis on optimizing quantum states for phase sensitivity in quantum interference measurements. The thesis investigates the fundamental sensitivity limits of two-mode interferometry in the presence of photon loss. It develops a computer algorithm to find the optimal quantum states for N=2 and 3 photons and compares their performance to the standard quantum limit. While focusing on practical implementation possibilities, it provides a comprehensive discussion of statistical methods and their potential for realizing optimized measurements.
This document is an introduction to plasma physics that covers several key topics:
1. It defines plasma as a gas of charged particles and discusses the conditions needed for a plasma state, including debye shielding and plasma parameters.
2. It describes different models for plasma description including fluid, MHD, and two-fluid models. It also covers continuity, Euler, and state equations.
3. It discusses MHD equilibria and waves, including Alfven and magnetosonic modes.
4. It examines MHD discontinuities and shocks.
5. It presents the two-fluid description and generalized Ohm's law.
6. It explores waves in dispers
Machine learning solutions for transportation networksbutest
This dissertation proposes machine learning solutions for problems in transportation networks. It contains four main contributions:
1. A probabilistic graphical model called a Gaussian Tree Model that describes multivariate traffic patterns using fewer parameters than standard models. This allows learning from less data.
2. A dynamic probabilistic model of traffic flow inspired by macroscopic flow models. It handles uncertainty and incorporates observations using a particle filter for prediction.
3. Two new optimization algorithms for vehicle routing that use the traffic flow model for routing in volatile environments.
4. A method for detecting traffic accidents using supervised learning that outperforms manual methods. It addresses data biases using dynamic Bayesian networks to improve performance with little labeled data.
This document provides an overview of normal moveout (NMO) and seismic data processing. It begins with an introduction to seismic data acquisition, including seismic sources, geophones, hydrophones, and recording systems. Next, it discusses seismic data processing objectives such as improving signal-to-noise ratio and resolution. Then it provides details on NMO, including its purpose of flattening reflections and correcting for offset. Finally, it describes different seismic velocities that are important for NMO, including interval, average and root-mean-square velocities.
Fuzzy and Neural Approaches in Engineering MATLABESCOM
This document provides an introduction to a MATLAB supplement for the book "Fuzzy and Neural Approaches in Engineering". It describes MATLAB as an educational software package for technical computing. The supplement contains MATLAB code examples that demonstrate concepts from the book, such as neural networks, fuzzy logic, and hybrid systems. It is intended to help readers gain a practical understanding of implementing soft computing techniques in MATLAB.
This document is a thesis submitted by M.P.P. (Maran) van Heesch for the degree of Master of Science in Econometrics and Mathematical Economics. It proposes a game theoretic framework to analyze users' incentives to join a technological mechanism called Wi-5, which aims to manage Wi-Fi channel selection and transmission power. The framework combines non-cooperative and cooperative game theory to model scenarios where no users, all users, or some users join Wi-5. It also presents a use case of an apartment building to initialize the framework and provides two examples applying the framework.
The document contains lecture notes on algorithm design techniques, including greedy algorithms, divide and conquer, dynamic programming, approximation algorithms, and local search. It provides examples of problems that can be solved using each technique and outlines algorithms for interval scheduling, scheduling to minimize lateness, finding closest pair of points, weighted interval scheduling, and more. It also introduces matroids as a theoretical foundation for greedy algorithms and proves properties like maximum independent sets having the same size.
Branch and-bound nearest neighbor searching over unbalanced trie-structured o...Michail Argyriou
Master thesis of Mike Argyriou in Technological University of Crete about
Branch and-bound nearest neighbor searching over unbalanced trie-structured overlays.
This document is a thesis submitted by Victor Arulchandran for the degree of Doctor of Philosophy at Brunel University. The thesis investigates free vibrations of thin elastic circular cylindrical panels localized near the edge using the Kirchhoff-Love theory of shells. Specifically, it analyzes:
1) Bending, extensional, and super-low frequency vibrations of a semi-infinite cylindrical shell.
2) The effects of varying panel thickness, wavelength, Poisson's ratio, and circumferential length.
3) Bending, extensional, and super-low frequency vibrations localized at the interface of two joined cylindrical shells.
Asymptotic solutions are derived and computational methods are
Hub location models in public transport planningsanazshn
This document is a dissertation written in German on the topic of hub location models in public transport planning. It begins with an introduction that describes hub location problems, aspects of multi-period planning, and solution procedures. The dissertation then reviews literature on hub location problems and formulations. It presents new mathematical formulations for public transport applications and extensions. Finally, it discusses solution methods like Lagrangian relaxation, Benders decomposition, and heuristic algorithms.
This document provides an introduction to integral calculus and demonstrates how to perform integral calculations using the computer algebra system Sage. It covers key integral calculus concepts such as the definition of the integral, Riemann sums, the Fundamental Theorem of Calculus, and techniques for evaluating integrals such as substitution, integration by parts, and trigonometric substitutions. It also discusses applications of integrals to computing areas, volumes, arc lengths, averages, and centers of mass. The document is intended as a preliminary version of an instructional text on integral calculus using Sage.
Pattern classification via unsupervised learnersNick Palmer
I study classification problems in a standard learning framework, in which an unsupervised learner creates a discriminant function over each class and observations are labeled by the learner returning the highest value associated with that observation. I examine whether this approach gains significant advantage over traditional discriminant techniques.
Probably Approximately Correct learning distributions over class labels under L1 distance or KL-divergence is shown to imply PAC classification in this framework. I determine bounds on the regret associated with the resulting classifier, taking into account the possibility of variable misclassification penalties, and demonstrate the advantage of estimating the a posteriori probability distributions over class labels in the setting of Optical Character Recognition.
Unsupervised learners can be used to learn a class of probabilistic
concepts (stochastic rules denoting the probability that an observation has a positive
label in a 2-class setting). I demonstrate a situation where unsupervised learners
can be used even when it is hard to learn distributions over class labels – in this case
the discriminant functions do not estimate the class probability densities.
If that isn't exciting enough, I then use a standard state-merging technique to PAC-learn a class of probabilistic automata. The results show that by learning the distribution over outputs under the weaker L1 distance rather than KL-divergence we are able to learn without knowledge of the expected length of an output. It is also shown that for a restricted class of these automata learning under L1 distance is equivalent to learning under KL-divergence.
Multiuser detection based on generalized side-lobe canceller plus SOVA algorithmAitor López Hernández
This is my undergraduate thesis project, consisting on comparing the performance of several classical multi user detection strategies with another solution of our choice, based on side-lobe cancellation.
Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zinggRohit Bapat
This document provides an overview of computational fluid dynamics (CFD) and summarizes its key steps and concepts. It discusses the fundamentals of CFD, including conservation laws, governing equations, finite difference approximations, semi-discrete and finite volume methods, and time-marching algorithms. The document is intended to introduce readers to the basic theory and methods in CFD for modeling fluid flow and transport phenomena.
This document is the thesis presented by Sophie Hautphenne to obtain a PhD in computer science from the Université Libre de Bruxelles in October 2009. The thesis investigates algorithmic approaches for computing extinction probabilities and other measures for branching processes and their extensions, including Markovian trees, reducible branching processes, branching processes in random environments, and those subject to catastrophes. It develops both linear and quadratic algorithms and applies matrix analytic methods.
All Minimal and Maximal Open Single Machine Scheduling Problems Are Polynomia...SSA KPI
This thesis presents Boolean linear programming models for three single machine scheduling problems with equal job lengths and release dates. It is proven that the problems are polynomially solvable by showing that the models have the property of total dual integrality. In Chapter 4, a model for minimizing total weighted completion time with preemption and unit job processing times is presented. The model is based on an assignment problem formulation. It is shown that the model has total unimodularity and total dual integrality, implying polynomial time solvability. These results are generalized to arbitrary equal job processing times in Chapter 5. In Chapter 6, models are presented for the preemptive and non-preemptive versions of minimizing total weighted tardiness
This document is the doctoral dissertation of Takayoshi Yoshimura submitted to Nagoya Institute of Technology in January 2002. The dissertation proposes improvements to HMM-based text-to-speech systems including simultaneous modeling of phonetic and prosodic parameters, characteristic conversion using speaker interpolation, and improving synthesized speech quality. The contributions aim to more accurately model speech and allow for voice conversion capabilities in text-to-speech systems.
This document describes a Matlab implementation of neural networks. It begins with an introduction to neural networks and associative memory, explaining how neural networks can be used to create associative memories that recall stored information based on partial cues. It then discusses implementing associative memory using neural networks and provides Matlab functions for storing and recalling information. The document goes on to describe perceptrons, multi-layer networks, and backpropagation networks. It concludes by presenting three applications of backpropagation networks: solving the XOR problem, curve fitting, and time series forecasting.
This document summarizes a master's thesis on optimizing quantum states for phase sensitivity in quantum interference measurements. The thesis investigates the fundamental sensitivity limits of two-mode interferometry in the presence of photon loss. It develops a computer algorithm to find the optimal quantum states for N=2 and 3 photons and compares their performance to the standard quantum limit. While focusing on practical implementation possibilities, it provides a comprehensive discussion of statistical methods and their potential for realizing optimized measurements.
This document is an introduction to plasma physics that covers several key topics:
1. It defines plasma as a gas of charged particles and discusses the conditions needed for a plasma state, including debye shielding and plasma parameters.
2. It describes different models for plasma description including fluid, MHD, and two-fluid models. It also covers continuity, Euler, and state equations.
3. It discusses MHD equilibria and waves, including Alfven and magnetosonic modes.
4. It examines MHD discontinuities and shocks.
5. It presents the two-fluid description and generalized Ohm's law.
6. It explores waves in dispers
Machine learning solutions for transportation networksbutest
This dissertation proposes machine learning solutions for problems in transportation networks. It contains four main contributions:
1. A probabilistic graphical model called a Gaussian Tree Model that describes multivariate traffic patterns using fewer parameters than standard models. This allows learning from less data.
2. A dynamic probabilistic model of traffic flow inspired by macroscopic flow models. It handles uncertainty and incorporates observations using a particle filter for prediction.
3. Two new optimization algorithms for vehicle routing that use the traffic flow model for routing in volatile environments.
4. A method for detecting traffic accidents using supervised learning that outperforms manual methods. It addresses data biases using dynamic Bayesian networks to improve performance with little labeled data.
This document provides an overview of normal moveout (NMO) and seismic data processing. It begins with an introduction to seismic data acquisition, including seismic sources, geophones, hydrophones, and recording systems. Next, it discusses seismic data processing objectives such as improving signal-to-noise ratio and resolution. Then it provides details on NMO, including its purpose of flattening reflections and correcting for offset. Finally, it describes different seismic velocities that are important for NMO, including interval, average and root-mean-square velocities.
Fuzzy and Neural Approaches in Engineering MATLABESCOM
This document provides an introduction to a MATLAB supplement for the book "Fuzzy and Neural Approaches in Engineering". It describes MATLAB as an educational software package for technical computing. The supplement contains MATLAB code examples that demonstrate concepts from the book, such as neural networks, fuzzy logic, and hybrid systems. It is intended to help readers gain a practical understanding of implementing soft computing techniques in MATLAB.
This document is a thesis submitted by M.P.P. (Maran) van Heesch for the degree of Master of Science in Econometrics and Mathematical Economics. It proposes a game theoretic framework to analyze users' incentives to join a technological mechanism called Wi-5, which aims to manage Wi-Fi channel selection and transmission power. The framework combines non-cooperative and cooperative game theory to model scenarios where no users, all users, or some users join Wi-5. It also presents a use case of an apartment building to initialize the framework and provides two examples applying the framework.
The document contains lecture notes on algorithm design techniques, including greedy algorithms, divide and conquer, dynamic programming, approximation algorithms, and local search. It provides examples of problems that can be solved using each technique and outlines algorithms for interval scheduling, scheduling to minimize lateness, finding closest pair of points, weighted interval scheduling, and more. It also introduces matroids as a theoretical foundation for greedy algorithms and proves properties like maximum independent sets having the same size.
Branch and-bound nearest neighbor searching over unbalanced trie-structured o...Michail Argyriou
Master thesis of Mike Argyriou in Technological University of Crete about
Branch and-bound nearest neighbor searching over unbalanced trie-structured overlays.
This document is a thesis submitted by Victor Arulchandran for the degree of Doctor of Philosophy at Brunel University. The thesis investigates free vibrations of thin elastic circular cylindrical panels localized near the edge using the Kirchhoff-Love theory of shells. Specifically, it analyzes:
1) Bending, extensional, and super-low frequency vibrations of a semi-infinite cylindrical shell.
2) The effects of varying panel thickness, wavelength, Poisson's ratio, and circumferential length.
3) Bending, extensional, and super-low frequency vibrations localized at the interface of two joined cylindrical shells.
Asymptotic solutions are derived and computational methods are
Master Thesis - A Distributed Algorithm for Stateless Load BalancingAndrea Tino
The algorithm object of this thesis deals with the problem of balancing data units
across different stations in the context of storing large amounts of information in
data stores or data centres. The approaches being used today are mainly based on
employing a central balancing node which often requires information from the different
stations about their load state.
The algorithm being proposed here follows the opposite strategy for which data is
balanced without the use of any centralized balancing unit, thus fulfilling the distributed
property, and without gathering any information from stations about their
current load state, thus the stateless property.
This document will go through the details of the algorithm by describing the idea
and the mathematical principles behind it. By means of an analytical proof, the equation
of balancing will be devised and introduced. Later on, tests and simulations,
carried on by means of different environments and technologies, will illustrate the
effectiveness of the approach. Results will be introduced and discussed in the second
part of this document together with final notes about current state of art, challenges
and deployment considerations in real scenarios.
MatConvNet is a MATLAB toolbox that implements convolutional neural networks (CNNs) for computer vision tasks. It aims to provide a simple and flexible environment for researchers to prototype and test new CNN architectures. Key features include exposing CNN building blocks as MATLAB functions, optimized CPU and GPU implementations for efficient training of large models on large datasets, and the ability to easily develop new blocks within MATLAB. Pre-trained versions of popular CNN models are also provided.
MatConvNet is a MATLAB toolbox that implements convolutional neural networks for computer vision tasks. It provides functions for common CNN layers like convolution, pooling, and normalization. These can be combined to easily prototype new CNN architectures. MatConvNet supports efficient GPU computation, allowing it to train complex models on large datasets. It aims to be a simple and flexible environment for computer vision researchers to experiment with CNNs within the MATLAB platform.
This document is Roman Zeyde's 2013 master's thesis from the Technion submitted in partial fulfillment of the requirements for a Master of Science degree in Computer Science. The thesis describes research on computational electrokinetics, which involves developing a numerical scheme to solve the governing equations for electrokinetic phenomena such as electrophoresis and ion exchange. The numerical scheme is based on a finite volume method in spherical coordinates. Results are presented comparing the numerical solutions to asymptotic analytical solutions for steady-state velocity profiles.
This document is a thesis submitted by David Liebman to the State University of New York at New Paltz for the degree of Master of Science in Computer Science. The goal of the thesis is to create a chatbot using natural language processing and deep learning models. The thesis provides background on recurrent neural networks, transformers, and pre-trained language models like GPT-2. It then describes the experimental design and setup for installing chatbot models on devices like the Raspberry Pi. Several chatbot experiments are conducted using GRU, transformer, and GPT-2 models with discussion of the results.
This document is the thesis of Alessandro Adamo submitted for a PhD in Mathematics and Statistics for Computational Sciences. The thesis proposes a new algorithm called LIMAPS (Lipschitzian Mappings for Sparse recovery) for solving underdetermined linear systems based on nonconvex Lipschitzian mappings. Chapter 1 provides theoretical foundations on sparse recovery and compressive sensing. Chapter 2 introduces LIMAPS and its iterative scheme for sparse representation and sparsity minimization. Chapters 3 and 4 apply LIMAPS to face recognition and ECG signal compression respectively, demonstrating its effectiveness on real-world applications.
This document is a master's thesis that examines localization techniques in wireless sensor networks. It provides background on wireless sensor networks and how they emerged from military applications but are now used in various civil applications. The thesis focuses on developing and analyzing new localization algorithms. It presents the results of experiments measuring received signal strength indication (RSSI) from wireless sensor nodes, which indicate significant fluctuations that could limit the reliability of localization schemes. Overall, the thesis evaluates localization methods and develops new algorithms to improve positioning accuracy in wireless sensor networks.
This document is a thesis presented by Miguel de Vega Rodrigo to obtain a doctorate in engineering sciences from the Université libre de Bruxelles in 2008. The thesis models future all-optical networks without buffering capabilities, specifically optical burst switching (OBS) and optical packet switching (OPS) networks. It covers the functional and hardware implementation of such networks, characterization of internet traffic that will enter these networks, and mathematical modeling approaches for the networks and traffic.
Trade-off between recognition an reconstruction: Application of Robotics Visi...stainvai
Autonomous and ecient action of robots requires a robust robot vision system that can
cope with variable light and view conditions. These include partial occlusion, blur, and
mainly a large scale dierence of object size due to variable distance to the objects. This
change in scale leads to reduced resolution for objects seen from a distance. One of the
most important tasks for the robot's visual system is object recognition. This task is also
aected by orientation and background changes. These real-world conditions require a
development of specic object recognition methods.
This work is devoted to robotic object recognition. We develop recognition methods
based on training that includes incorporation of prior knowledge about the problem.
The prior knowledge is incorporated via learning constraints during training (parameter
estimation). A signicant part of the work is devoted to the study of reconstruction
constraints. In general, there is a tradeo between the prior-knowledge constraints and
the constraints emerging from the classication or regression task at hand. In order to
avoid the additional estimation of the optimal tradeo between these two constraints, we
consider this tradeo as a hyper parameter (under Bayesian framework) and integrate
over a certain (discrete) distribution. We also study various constraints resulting from
information theory considerations.
Experimental results on two face data-sets are presented. Signicant improvement in
face recognition is achieved for various image degradations such as, various forms of image
blur, partial occlusion, and noise. Additional improvement in recognition performance is
achieved when preprocessing the degraded images via state of the art image restoration
techniques.
This report discusses quantum cryptography and potential attacks on quantum key distribution systems. It provides background on quantum cryptography and describes the BB84 quantum key distribution protocol. It then analyzes several potential attacks on quantum key distribution systems, including photon number attacks, spectral attacks, and random number attacks that are relatively easy to solve. It focuses on the more challenging "faked-state" attack, providing details on how an attacker could implement this attack in practice using superconducting nanowire single-photon detectors. The report evaluates the security of quantum key distribution against these attacks.
This document provides an introduction to wireless sensor networks (WSNs). It discusses WSN architecture and protocol stacks, challenges and constraints of WSNs, and applications of WSNs. It also examines WSN integration with the Internet. The document is divided into chapters that cover topics such as the wireless channel, the physical layer, medium access control, routing, topology control, distributed detection and estimation, distributed learning, positioning and localization, and time synchronization in WSNs.
This document is the preface to the book "Mining of Massive Datasets" by Anand Rajaraman and Jeffrey D. Ullman. It provides an overview of the book, which covers topics related to applying algorithms to very large datasets, including distributed file systems, map-reduce, similarity search, data streams, search engines, frequent patterns, clustering, and applications like advertising and recommendations. The preface outlines the book's content, prerequisites, and acknowledges contributions from others.
Robust link adaptation in HSPA EvolvedDaniel Göker
This master's thesis studies a robust link adaptation technique for HSDPA networks. The technique adds an offset to the estimated channel quality indicator (CQI) to account for errors caused by measurement delay and noise. Simulations compare the robust technique to existing CQI adjustment in terms of block error rate stability and throughput. Results show the robust technique achieves the target block error rate on average across different user speeds, while existing CQI adjustment performs best only at high speeds with large packets. The robust technique provides stability but cannot adapt to interference variations. Both methods may be needed for a flexible system.
The document is a project report for a Master's thesis investigating conjugate heat transfer for electronic cooling using the open-source software OpenFOAM. The student, Avinash Gorde, modeled a server system with solid components like RAM and a PCB coupled to an air domain. The project involved setting up the geometry, meshing, boundary conditions, and solving the conjugate heat transfer equations using OpenFOAM's chtMultiRegionSimpleFoam solver. Results showed temperature distributions within the heat sink and across solid components like the socket and RAM. The project provided experience applying OpenFOAM to model complex fluid-solid interaction problems for electronic cooling applications.
This document provides a summary of the ITU-T Teletraffic Engineering Handbook. It discusses telecommunication systems modeling, conventional telephone systems, communication networks including telephone networks, data networks, local area networks and the internet. It also covers mobile communication systems, the international organization of telephony and ITU-T recommendations. The handbook contains technical information on traffic concepts, probability theory, time interval distributions, arrival processes, Erlang's loss model and loss systems with full accessibility. It is intended as a reference for teletraffic engineers and was drafted in 2001 by Villy B. Iversen of the Technical University of Denmark.
This thesis provides a nonstationary statistical analysis of annual maximum temperature records to evaluate global warming. It demonstrates that a nonstationary extreme value Weibull model with a linear trend in the location parameter best explains the data among various parametric and nonparametric models. However, other modeling techniques using splines in a generalized additive model previously showed that the trend in annual maxima is not simultaneously significant over time. The thesis develops theoretical backgrounds on state-of-the-art extreme value analysis methods and presents their careful application in a reusable R package and Shiny application.
This document describes Kerry Steven Hall's dissertation research on using air-coupled ultrasonic tomography to image concrete elements. The research aims to integrate recent developments in air-coupled ultrasonic measurements with advanced tomography technology to apply them to concrete structures. Finite element models are developed and used to simulate measurement configurations and optimize data collection procedures. Non-contact and semi-contact ultrasonic sensors are developed and tested on concrete cylinder and block specimens. Tomographic reconstructions with error calculations are performed to image inclusions and defects within the concrete. Issues related to applying the techniques to full-scale concrete structures are also discussed.
This document outlines the contents of a course on neural networks and deep learning across multiple weeks. It covers topics such as neural network basics including logistic regression and activation functions, deep neural networks, improving networks through techniques like regularization and optimization algorithms, convolutional neural networks including applications to object detection, and face recognition. Specific algorithms and architectures discussed include residual networks, Inception networks, YOLO, R-CNN, and Siamese networks.
This document is a thesis presented to obtain a doctorate degree in electronics and communications from Ecole Nationale Supérieure des Télécommunications. The thesis examines non-systematic codes on graphs for redundant data. Chapter 1 introduces the topic, outlines motivations and contributions, and discusses information theoretical limits, design principles, and simulations of non-systematic LDPC codes. Chapter 2 analyzes density evolution of split-LDPC codes and examines their analytic properties and stability. Chapter 3 presents exit chart analysis and design of irregular split-LDPC codes. Chapter 4 proposes enhancing iterative decoding using EM source-channel estimation.
Similar to Approximate Algorithms for the Network Pricing Problem with Congestion - MS thesis (20)
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
OpenID AuthZEN Interop Read Out - AuthorizationDavid Brossard
During Identiverse 2024 and EIC 2024, members of the OpenID AuthZEN WG got together and demoed their authorization endpoints conforming to the AuthZEN API
Main news related to the CCS TSI 2023 (2023/1695)Jakub Marek
An English 🇬🇧 translation of a presentation to the speech I gave about the main changes brought by CCS TSI 2023 at the biggest Czech conference on Communications and signalling systems on Railways, which was held in Clarion Hotel Olomouc from 7th to 9th November 2023 (konferenceszt.cz). Attended by around 500 participants and 200 on-line followers.
The original Czech 🇨🇿 version of the presentation can be found here: https://www.slideshare.net/slideshow/hlavni-novinky-souvisejici-s-ccs-tsi-2023-2023-1695/269688092 .
The videorecording (in Czech) from the presentation is available here: https://youtu.be/WzjJWm4IyPk?si=SImb06tuXGb30BEH .
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
GraphRAG for Life Science to increase LLM accuracyTomaz Bratanic
GraphRAG for life science domain, where you retriever information from biomedical knowledge graphs using LLMs to increase the accuracy and performance of generated answers
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
Webinar: Designing a schema for a Data WarehouseFederico Razzoli
Are you new to data warehouses (DWH)? Do you need to check whether your data warehouse follows the best practices for a good design? In both cases, this webinar is for you.
A data warehouse is a central relational database that contains all measurements about a business or an organisation. This data comes from a variety of heterogeneous data sources, which includes databases of any type that back the applications used by the company, data files exported by some applications, or APIs provided by internal or external services.
But designing a data warehouse correctly is a hard task, which requires gathering information about the business processes that need to be analysed in the first place. These processes must be translated into so-called star schemas, which means, denormalised databases where each table represents a dimension or facts.
We will discuss these topics:
- How to gather information about a business;
- Understanding dictionaries and how to identify business entities;
- Dimensions and facts;
- Setting a table granularity;
- Types of facts;
- Types of dimensions;
- Snowflakes and how to avoid them;
- Expanding existing dimensions and facts.
Generating privacy-protected synthetic data using Secludy and MilvusZilliz
During this demo, the founders of Secludy will demonstrate how their system utilizes Milvus to store and manipulate embeddings for generating privacy-protected synthetic data. Their approach not only maintains the confidentiality of the original data but also enhances the utility and scalability of LLMs under privacy constraints. Attendees, including machine learning engineers, data scientists, and data managers, will witness first-hand how Secludy's integration with Milvus empowers organizations to harness the power of LLMs securely and efficiently.
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
Introduction of Cybersecurity with OSS at Code Europe 2024Hiroshi SHIBATA
I develop the Ruby programming language, RubyGems, and Bundler, which are package managers for Ruby. Today, I will introduce how to enhance the security of your application using open-source software (OSS) examples from Ruby and RubyGems.
The first topic is CVE (Common Vulnerabilities and Exposures). I have published CVEs many times. But what exactly is a CVE? I'll provide a basic understanding of CVEs and explain how to detect and handle vulnerabilities in OSS.
Next, let's discuss package managers. Package managers play a critical role in the OSS ecosystem. I'll explain how to manage library dependencies in your application.
I'll share insights into how the Ruby and RubyGems core team works to keep our ecosystem safe. By the end of this talk, you'll have a better understanding of how to safeguard your code.
Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
Monitoring and Managing Anomaly Detection on OpenShift.pdfTosin Akinosho
Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Approximate Algorithms for the Network Pricing Problem with Congestion - MS thesis
1. Università degli Studi di Trieste
FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di Laurea Magistrale in Informatica
Curriculum Ingegneria Informatica
Approximate algorithms
for the
Network Pricing Problem with Congestion
Tesi di Laurea in
Ricerca Operativa
Relatore: Laureanda:
Chiar.mo Prof. Desirée Rigonat
Lorenzo Castelli
Sessione Estiva
Anno Accademico 2011 - 2012
2.
3. Ringraziamenti
E così abbiamo finalmente raggiunto questo nuovo traguardo.
Sarei egoista se parlassi al singolare, il merito di questo risultato è di tante, tante persone, che
in mille modi mi hanno aiutata, sostenuta, incoraggiata durante questi tre anni.
Un sentito ringraziamento va prima di tutto al mio relatore, Professor Lorenzo Castelli per
avermi dato l’opportunità di svolgere questo lavoro e per la pazienza, l’aiuto e il sostegno con cui
mi ha seguita durante quest’ultimo anno e per avermi proposto il periodo di studi all’estero, che
mi ha permesso di avvicinarmi al mondo della ricerca.
Ringrazio poi la Professoressa Martine Labbé e tutto il gruppo di ricerca del G.O.M. dell’Université
Libre de Bruxelles, per avermi dato la possibilità di lavorare con loro. È stato un onore e so-
prattutto un piacere.
Niente di tutto questo sarebbe stato possibile se non avessi avuto il sostegno costante, sia
morale che materiale, della mia famiglia: grazie mamma e papà per aver sempre creduto in me,
Pietro per aver condiviso gioie e dolori della vita domestica, Gattila per avermi sempre ricordato
quando era ora di staccare e andare a dormire, Oscar per aver sostituito efficacemente la sveglia
e Napo per avermi costretta a fare ogni tanto una pausa!
Come non sfruttare poi quest’occasione per dire a tutti i miei amici quanto sono stati e
sono importanti per me: è bello sapere che, non importa in quale angolo di mondo ci si trovi, a
casa ci sono delle persone su cui puoi sempre contare. Non mi basterebbe davvero il tempo per
citare ognuno, ma sappiate che vi ho pensati tutti. Qualche eccezione però è doverosa e quindi
ringrazio Marta: sei la migliore amica che si possa avere, grazie per le lunghe chiacchierate e
per condividere con me momenti di follia, di regressione infantile (Pasqua e Natale, tu sai cosa
intendo!) e per essere la persona allegra e positiva che sei! Grazie a te e a tutta la tua famiglia!
Poi devo proprio ringraziare Federico: c’è poco da dire, sei un vero amico, grazie per tutte le volte
che mi hai ascoltata, per le chiacchierate senza fine, per tutti gli hobby che abbiamo condiviso e
per le serate a base di sushi e babezzi!
Ringrazio poi tutti i miei compagni di avventure di EESTEC: Erni, Mariela, Filippo, Carlo,
Nicola, Alessandra, Enrico, Francesca, conoscervi e condividere esperienze con voi mi ha davvero
aperto un mondo, mi ha fatto imparare un sacco di cose e mi ha fatto crescere tantissimo. Siete
grandissimi!
Infine, un grazie speciale a Matteo, che più di tutti mi ha aiutata a credere in me stessa e a
trovare la forza di continuare quando credevo di non farcela. Grazie per i tuoi incoraggiamenti,
per l’entusiasmo, per il tuo humour, lasciatelo dire..un po’ British (ma che io apprezzo enorme-
mente) e per la tua arte culinaria, quest’ultima invece prettamente Italica (per fortuna)! Vorrei
ringraziarti per tante tante altre cose, ma quella a cui tengo di più è ringraziarti semplicemente
di esserci, perchè mi rendi una persona migliore.
Grazie anche alla tua stupenda famiglia: Laura, Danilo, Chiara, John e George, grazie per tutti
i bei momenti passati assieme, e per gli incoraggiamenti che mi avete sempre dato!
Ancora una volta, grazie di cuore a tutti quanti!
I wish you all the best!
4.
5. Acknowledgements
The present thesis is the result of a research carried on from September 2011 to
August 2012; part of this work has been developed at the Graphes et Optimisation
Mathématique (G.O.M.) group of the Université Libre de Bruxelles, under the supervi-
sion of Professor Martine Labbé; further development and experimentation have been
carried on at the Laboratorio di Ricerca Operativa of the Università degli studi di Trieste
with Professor Lorenzo Castelli as supervisor.
9. 9
Introduction
Bilevel programs are nowadays a well known field of research. Studies have been
carried on this particular class of optimization problems since the 1970s. The Network
Pricing Problem, a particular case of which is the main subject of the present work, has
been formulated in the late 1990s (see Labbé et al. (1998)). It belongs, as the name
suggests, to the class of network optimization problems where prices have to be set on
the links of a network in order to maximize the profit of the owner. The bilevel structure
of the NPP implies that these prices will be influenced by the distribution on the network
of one or more users that want to travel on it at the minimum cost.
The NPP usually assumes that arc costs are independent of flows . In road transportation
systems, when arc costs depend on flows, the network is usually referred to as congested.
In the present work, we illustrate two asymptotically converging algorithms to solve the
NPP in the case of congested networks, hereafter referred to as the Congested Network
Pricing Problem (CNPP).
In particular, we propose to identify an equilibrium point for the CNPP using the Frank-
Wolfe algorithm (see Frank and Wolfe (1956)) by reformulating the bilevel CNPP into a
sequence of approximating single level linear problems. One of the algorithms uses these
linear approximations to solve only the second level problem (that is, the problem of the
users) while the other applies the linearization procedure to the whole CNPP.
The Frank-Wolfe linearization scheme was also used by Brotcorne et al. (2001) in the
design of a primal-dual heuristic to solve the NPP. In their work however arc costs are
still supposed to be independent of flows.
In this thesis, Chapter 1 introduces the NPP in its generic, uncongested formulation,
together with a linearization procedure to obtain an equivalent linear single level prob-
lem.
In Chapter 2 we first define a nonlinear arc cost function in order to introduce the
congestion element; then we formulate the lower level of the CNPP as an optimization
problem with nonlinear objective function and linear constraints. Finally we present a
simplification procedure to transform the CNPP in a non linear single level problem.
In Chapter 3 a brief introduction to the Franke-Wolfe algorithm is given; then the two
F-W based algorithms to solve the CNPP are presented, together with a review of the
(very brief) pre-existent literature and some hints for future improvements.
Chapter 4 deals with implementation details, for both of the presented algorithms and
for the software used for generating the networks to be used for computational tests.
Finally, Chapter 5 is dedicated to experimental data and results, from which we desume
our final considerations and discuss future developments.
11. Chapter 1
The Network Pricing Problem
In the present section we will introduce the Network Pricing Problem (NPP) in its
generic formulation, which does not take congestion into account. As its initial form is
that of a bilinear bilevel problem, we will first give a brief introduction to this class of
problems. Then the NPP will be introduced and we will illustrate how it can be refor-
mulated first into a single level bilinear, then into a linear problem with a mixed integer
formulation. In the present chapter we will refer to the formulations and notations in
Labbé et al. (1998), Brotcorne et al. (2000) and Brotcorne et al. (2001). Heilporn (2008)
in her PHD thesis gives a detailed analisys of the geometric structure of the problem and
the particular cases that have so far been proved to be easier to solve.
1.1 Bilevel programming
Bilevel programs belong to a class of Stackelberg sequential games with two players,
where a leader plays first, taking into account the possible reactions of the second player,
called the follower. By denoting x and y respectively the leader’s and follower’s decision
variables vectors, this situation can be described mathematically by:
min F (x, y) (1.1)
x,y
s.t. G(x, y) ≤ 0 (1.2)
y ∈ arg min f (x, y) (1.3)
y
s.t. g(x, y) ≤ 0 (1.4)
Note that the formulation above assumes that if there are multiple optimal solutions
for the lower level problem, the solution that is most profitable for the leader is selected;
this is an optimistic approach, in opposition to a pessimistic approach where the leader
chooses the solution that protects himself against the follower’s worst possible reaction.
Both scenarios have been investigated in literature (see Heilporn (2008) for a detailed
bibliography on this).
11
12. 12 CHAPTER 1. THE NETWORK PRICING PROBLEM
Bilevel programs first appeared in 1973 in an article by Bracken and McGill (1973),
while the complete formulation, as described aboved, was first introduced in an article
by Shimizu and Aiyoshi (1981). An annotated bibliography containing more than one
hundred references on bilevel programming has been compiled by Vicente and Calamai
(1994), while the books by Shimizu et al. (1997) and Luo et al. (1996) are devoted, in
full or in part, to this subject.
Generically non differentiable and non convex, bilevel problems are, by nature, hard.
Even the linear bilevel problem, where the objective functions and the constraints are
linear, was proved to be N P-hard by Jeroslow (1985). Hansen et al. (1992) prove strong
N P-hardness. Vicente et al. (1994) strengthen these results and prove that merely check-
ing strict or local optimality is strongly N P-hard.
1.2 The Network Pricing Problem (NPP)
Let us define a transportation network as a set of nodes (cities) and a set of arcs
(routes) linking some of these nodes together. At the upper and lower level, consider an
authority and a set of network users respectively. We also define a commodity as a set
of network users travelling from the same origin to the same destination. In addition to
a fixed cost associated with every arc, tolls are imposed by the authority on a specified
subset of arcs of the network. Hence the Network Pricing Problem consists of devising toll
levels on the specified subset of toll arcs in order to maximize the authority’s revenues.
Then, reacting to the tolls, each commodity travels on the shortest path from its origin
to its destination, with respect to a cost equal to the sum of tolls and initial costs. More
specifically, the formulation we will refer to in the present work assumes that only a subset
of the links has taxes and that the network is a multicommodity transportation network;
moreover, in order to avoid trivial solutions leading to infinite revenues for the authority,
we will assume that there always exists a toll-free path for each origin/destination pair.
No assumption is made regarding the non-negativity of the toll: as shown by Labbé et al.
(1998) a schema which allows negative tolls (that can be interpreted as incentives) can
lead to a better solution than one with only positive taxes.
An optimal tolling policy is such that tolls are low enough not to deter the users from
using those links (rather than alternative routes with no toll arcs) while still generating
high profits. In this model, it is generally assumed that the users will travel on shortest
(cheapest) origin-destination routes and congestion is not taken into consideration.
1.3 The bilevel formulation of the NPP
Let G = (N , A ∪ B) be a transportation network where N denotes the set of nodes
and A ∪ B is the set of arcs, where A is the subset of toll arcs and B the subset of toll-free
arcs. Each arc of A has a travel cost composed of a fixed part ca and an unknown toll
ta . Each arc of B bears only a fixed travel cost, identified by da .
13. 1.3. THE BILEVEL FORMULATION OF THE NPP 13
Let K denote the set of commodities, where each commodity k is associated with an
origin/destination pair (ok , dk ); the demand vector bk associated with each commodity
k is defined by:
k
η if i = o(k)
bk =
i −η k if i = d(k) ∀i ∈ N , ∀k ∈ K (1.5)
0 otherwise
where η k represents the number of users of commodity k.
Finally xk denotes the number of users of commodity k on arc a ∈ A ∪ B (that is, the
a
amount of flow on arc a for the origin/destination pair k).
The NPP can thus be formulated as a bilevel program with bilinear objective func-
tions and linear constraints, where the flows xk denote the optimal solution of the second
a
level problem parametrized by the upper level toll vector t.
max t a · xk
a (1.6)
t,x
k∈K a∈A
s.t. min ( (ca + ta ) · xk +
a da · xk )
a (1.7)
x
k∈K a∈A a∈B
xk
a − xk
a = bk
i ∀k ∈ K, ∀i ∈ N (1.8)
a=i+ ∈A∪B a=i− ∈A∪B
xk ≥ 0
a ∀k ∈ K, ∀a ∈ A ∪ B (1.9)
1.3.1 The Tmax upper bound
It is usually assumed that there cannot exist a toll setting scheme that generates
profits and creates negative cost cycles in the network and that there exists at least
one path composed solely of untolled arcs for each origin-destination pair (the "toll-free
path" previously mentioned). These conditions avoid the degenerative and unrealistic
cases where looping in a cycle drains users’ traveling costs to zero and where a non-
alternative path scenario would allow the leader to put an infinitely high toll on one or
more links, thus leading to an infinite profit.
In a practical scenario, by the way, such an assumption could not always hold; for
example the toll-free path could be significantly longer than the tolled path, and even if
we consider the time as a factor in the cost function of our model, such a choice could
not be likely for most "real" users (i.e. they would in any case prefer the quicker route).
Moreover, often tariffs are set by companies considering a whole set of policies that are
just partially related to traffic. In such a scenario, it is perfectly reasonable to assume
that an upper bound for the tariffs exists, much more reasonable than assuming that they
can be "arbitrarily high" or "limited by the cost of a toll-free path". Finally, whether or
14. 14 CHAPTER 1. THE NETWORK PRICING PROBLEM
not such constraints could undermine feasibility of specific instances of NPP is a matter
related with the specific istance itself.
The common formulation of the Network Pricing Problem however allows for an up-
per bound T max to be set for the tariffs. Since this bound can be set to infinite, it will
not alter the generality of the model and will help to find a realistic solution in case
the network is not fully compliant with the above requirements regarding toll-free paths.
This implies adding the following constraint to the above model:
ta ≤ T max ∀a ∈ A ∪ B (1.10)
In the present work we will assume that a feasible toll-free path always exixsts; as a
consequence the above constraint will be omitted in our formulations.
1.3.2 Complexity of the NPP
As demonstrated by Labbé et al. (1998) this problem in its general form is strongly
N P-Hard. However a linearization scheme illustrated in the same work leads to a mixed
integer programming formulation that involves a small number of binary variables. This
formulation can be solved using standard algorithms such as branch and bound that will
lead to an exact solution in reasonable time for small instances of the problem, or allows
for efficient heuristic procedures to be developed, as shown by Brotcorne et al. (2000
and 2001). Such algorithms lead to far better results on large network since they are
able to exploit the particular structure of the network in order to quicken the solving
procedure.
1.4 Linearization scheme for the NPP
The aim of the procedure that will briefly be illustrated in the present section (for a
more comprehensive description see Labbé et al. (1998)) is to obtain a single level linear
problem from the bilevel bilinear NPP. The process is carried on in two phases:
1. From bilevel to single level through the theory of Duality;
2. From bilinear to linear.
1.4.1 From bilevel NPP to single level NPP
Under the assumptions that strong duality holds for the second level problem, and
that there exists a toll-free path for each origin/destination pair (or, alternatively, that
there exists an upper bound to tariffs), it is possible to reduce the bilevel bilinear NPP
to a single level bilinear problem.
The process implies the replacement of the second level problem by its dual constraints
15. 1.4. LINEARIZATION SCHEME FOR THE NPP 15
(while mantaining the primal feasibility constraint) and its objective by the complemen-
tary slackness condition, that is, adding another constraint which imposes primal and
dual feasibility for the respective objectives.
Dual constraints: according to Equation 1.9 for each arc a ∈ A ∪ B and for each
commodity k ∈ K a non negative primal variable xk exists. Thus the dual problem will
a
have as many dual constraints associated with each arc and each commodity:
λk − λk ≤ ca + ta
i j ∀a = (i, j) ∈ A, ∀k ∈ K (1.11)
λk − λk ≤ da
i j ∀a = (i, j) ∈ B, ∀k ∈ K (1.12)
Where i and j are respectively the head and tail nodes for arc a ∈ A ∪ B.
Moreover, since the primal constraints of flow conservation are equalities, the correspond-
ing variables λk are free.
i
Primal-dual feasibility: the dual objective of the second level problem is:
max η k λkk −
d η k λkk
o (1.13)
k∈K k∈K
According to the strong duality theorem, if:
1. vector x is a feasible solution for the primal problem,
¯
¯
2. vector λ is a feasible solution for the dual problem,
3. the values of both the primal and the dual objective function coincide,
¯
then the vectors x and λ are optimal solutions for both of the problems.
¯
Thus we can substitute the second level objective with a constraint that imposes equality
between the values of the primal and the dual objective, which implies optimality:
Zprimal = Zdual (1.14)
that is:
(ca + ta ) · xk +
a da · xk = η k · λkk − λkk
a d o ∀k ∈ K (1.15)
a∈A a∈B
The resulting problem, equivalent to the NPP as described previously, is thus the
following:
16. 16 CHAPTER 1. THE NETWORK PRICING PROBLEM
max t a · xk
a (1.16)
t,x
k∈K a∈A
s.t. (ca + ta ) · xk +
a d a · xk =
a
a∈A a∈B
k
=η · λkk
d − λkk
o ∀k ∈ K (1.17)
xk −
a xk = bk
a i ∀k ∈ K, ∀i ∈ N (1.18)
a=i+ ∈A∪B a=i− ∈A∪B
λk − λk ≤ ca + ta
i j ∀a = (i, j) ∈ A, ∀k ∈ K (1.19)
λk
i − λk
j ≤ da ∀a = (i, j) ∈ B, ∀k ∈ K (1.20)
k
xa ≥0 ∀k ∈ K, ∀a ∈ A ∪ B (1.21)
ta ≥ 0 ∀a ∈ A (1.22)
λk
i free ∀i ∈ N , ∀k ∈ K (1.23)
1.4.2 From bilinear NPP to linear NPP
The problem obtained so far still contains bilinear terms in both the objective and
the complementary slackness constraint. What we want to obtain in this section is a
single level linear problem.
First of all it is necessary to introduce a binary variable and a slack variable to
transform the bilinear term ta · xk . The binary variable is used to re-define the integer
a
variable.
Since the second level problem is a flow assignment problem, each flow demand of
the commodities is distributed on the network by following the path of minimum cost. It
follows that the binary variable introduced for each arc a ∈ A∪B and for each commodity
k ∈ K is defined as follows:
k 1 if a ∈ A ∪ B belongs to the minimum cost path for commodity k ∈ K
ra =
0 otherwise
(1.24)
Thus resulting in:
xk = η k · ra
a
k
(1.25)
The following constraint for the new variable needs to be added to the problem:
k
ra ∈ {0; 1} ∀a ∈ A ∪ B, ∀k ∈ K (1.26)
Then, a slack variable is used to re-define the continuous variable. Since each com-
modity is associated with a single path (the one with minimum cost), the leader imposes
the fee on a tariffed arc a ∈ A only if this is used by at least one commodity, for which
that particular arc belongs to the path of minimum cost. Therefore, the slack variable
introduced for each arc a ∈ A and for each commodity k ∈ K is as follows:
17. 1.4. LINEARIZATION SCHEME FOR THE NPP 17
k
ta if ra = 1
pk =
a (1.27)
0 otherwise
The following constraints need to be added:
pk − ta ≤ 0
a ∀a ∈ A, ∀k ∈ K (1.28)
− pk + ta − M · (1 − ra ) ≤ 0
a
k
∀a ∈ A, ∀k ∈ K (1.29)
pk
a −N · k
ra ≤0 ∀a ∈ A, ∀k ∈ K (1.30)
where M e N are arbitrary big − M parameters.
Equation 1.28 forces the value of pk to zero if ta is zero. Equation 1.29 is a Big-M type
a
constraint. If ra is one, the constraint must be satisfied. In fact pk will be equal to ta
k
a
k
according to Equations 1.28 and 1.29. If ra is zero, the constraint is relaxed. Equation
1.30 denotes an upper bound for pk . If ra equals one, pk is less than Na . If ra is zero, pk
a
k
a
k
a
is zero too.
According to the transformation carried out, the formulation of the single level NPP
is as follows:
max η k · pk
a (1.31)
p
k∈K a∈A
k
s.t. ra − ra = ek
k
i ∀i ∈ N , ∀k ∈ K (1.32)
a=i− ∈A∪B a=i+ ∈A∪B
(ca · ra + pk ) +
k
a da · ra = λkk − λkk
k
d o ∀k ∈ K (1.33)
a∈A a∈B
k k
λi − λj ≤ ca + ta ∀a = (i, j) ∈ A, ∀k ∈ K (1.34)
λk − λk ≤ da
i j ∀a = (i, j) ∈ B, ∀k ∈ K (1.35)
k
pa − ta ≤ 0 ∀a ∈ A, ∀k ∈ K (1.36)
− pk + ta − M · (1 − ra )
a
k
≤0 ∀a ∈ A, ∀k ∈ K (1.37)
pk − N · ra ≤ 0
a
k
∀a ∈ A, ∀k ∈ K (1.38)
ta ≥ 0 ∀a ∈ A (1.39)
pk ≥ 0
a ∀a ∈ A, ∀k ∈ K (1.40)
k
ra ∈ {0; 1} ∀a ∈ A ∪ B, ∀k ∈ K (1.41)
λk
i free ∀i ∈ N, ∀k ∈ K (1.42)
where ek is the new demand vector, defined as follows:
i
−1 if i = ok
k
ei = 1 if i = dk ∀i ∈ N, ∀k ∈ K (1.43)
0 otherwise
19. Chapter 2
The Congested Network Pricing
Problem
In this section we will describe a variant to the Network Pricing Problem which takes
congestion levels into account. We will refer to this as the Congested Network Pricing
Prooblem (CNPP). The peculiarity of considering congestion lays in the fact that arc
costs are no longer considered as constants. Instead, they depend upon arc flows: they
get higher as flow levels approach the capacity of the links (which are fixed) and get lower
as flow is moved to other links. Congestion thus implies a mutual dependence between
arc costs and arc flows, and here lies the difficulty of the model. Our particular case
deals with a road transportation network, such as a highway system, so that the first
level problem remains a profit maximization problem and the second level problem will
be a Traffic Assignment Problem, which has been covered by a vast amount of literature
over the past decades. For a complete mathematical analysis of the Traffic Assignment
problem we refer to the omonymous work by Patriksson (1994) and for a more generic
introduction on optimization problems on transportation networks we refer to the work
by Sheffi (1985). No previous work covers the CNPP as presented here, however a sim-
ilar model, applied on a telecommunications network can be found in Julsain (1998).
Instead, issues related to congestion have been thoroughly investigated in the field of
Network Design Problems by Marcotte (1986).
2.1 Congestion
The generic definition of congestion on a network, either a physical transportation
network or a data telecommunication network states that such a condition occurs when
a link or node is carrying so much data that its quality of service deteriorates. This
can lead to various disadvantages for the users of the network, namely delay, increase
in transportation costs and can eventually lead, in worst case scenarios, to the complete
halt of the service. Network congestion is thus tightly related to the concept of network
capacity, that represents the maximum amount of units of flow that a link or a node of
19
20. 20 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
the network is able to sustain before congestion occurs (or before the system comes to a
halt because of it). In order to introduce congestion in the NPP we have to take all this
into consideration. The scenario to which we will refer is that of a road transportation
network (i.e. a highway system) where link costs will depend upon the amount of flow
on that specific link: as the amount of flow on that link increases, the cost will increase
as well in order to avoid congestion; thus the users will be induced into moving part
of the flow to another, now cheaper link; the cost of the previously loaded link will
consequently decrease proportionally with the flow that has been removed from it, and
so on. This deviation of flow and consequent fluctuation of link costs finally converge
to a configuration of balance, that is referred to as equilibrium. An equilibrium can be
referred to either the users (thus referred to as User Equilibrium) or to the network
(System Equilibrium) depending on the final configuration we want to obtain.
All of these concepts will be anayzed in detail in the present chapter in order to apply
what explained above to the NPP.
For the proper definitions of Equilibria on a trasnsportation network we will refer to
(Wardrop, 1952).
2.2 CNPP lower level: traffic assignment problem
In this section we will analyze the second level of the CNPP, that is, how the users of
the network (followers) spread across the arcs according to the supply (per-arc cost) and
the demand (necessity to travel form origin to destination). We will formulate assign-
ment assumptions in order to formulate the follwers’ problem as a Traffic Assignment
problem (see i.e. Sheffi (1985)).
According to transportation system theory, the system we are considering is mono-
modal and continuous (meaning that all users/vehicles belong to the same category),
and we assume that users travel with private individual vehicles. Demand is represented
by an origin-destination (O/D) matrix and the supply is defined by the transportation
network itself (nodes, links and assigned costs).
Interaction between demand and supply is simulated according to an appropriate as-
signment model, which must be coherent with the objectives, the given constraints and
possible simplification hypotheses.
Where not otherwise specified, notation is consistent with the one used in the previous
chapter for the NPP.
2.2.1 Assignment hypotheses
In order to build a model that is consistent with the structure of a traffic assignment
problem, we rely on the following hypotheses:
1. Cost-flow functions: we are considering congestion, so we define the congested
network, where the arc-cost vector depends upon the arc-flow vector (C = C(f )).
21. 2.2. CNPP LOWER LEVEL: TRAFFIC ASSIGNMENT PROBLEM 21
2. Demand-offer interaction: since we are considering a congested network and
we are not taking into account any inter-period dynamics on the evolution of the
system, we will refer to a User Equilibrium assignment (UE); in this model we will
specifically consider the equilibrium configurations where demand, path and arc
flows are equal to their respective costs.
3. Path choice behavior: users choose their path preemptively and don’t modify
it while traveling.
4. Choice model: it is completely deterministic, because the perceived usefulness is
deterministic (not aleatory); all users choose a maximum usefulness path, that is
the one with the minimum cost.
5. Classes of users: although it is possible to divide users in classes according to
specific criteria, we are considering here as an only differentiation criterion the
O/D pair, thus determining a mono-class assignment for the users.
6. Demand characteristics: demand flows are constant in time, since they are not
dependent upon cost fluctuations due to congestion. Thus we are considering a
rigid demand.
Regarding transportation costs, we are considering the following hypotheses:
1. cost functions are separable with respect to the links, that is, the cost of a single
¯
arc is independent from the cost of other arcs (Ca (f ) = Ca (fa ), ∀a ∈ A).
2. non additive path costs will not be taken into consideration; specifically, we will
ignore all costs that cannot be obtained through a sum of single-arc costs over the
path.
2.2.2 Taking congestion into account: D.U.E. Assignment
Considering a congested network implies that there is not only dependence of arc
flows from arc costs (as in assignments to non-congested networks) but there is also
dependence of arc costs from arc flows (through the arc cost functions). This mutual
dependence of flows and arcs determines a final configuration of the system where flows
on paths are consistent with the cost of the paths themselves. This configuration is seen
as the state towards which the system evolves in conditions of recurring congestion, that
is, when congestion occurs in a systematic way for a sufficiently long period of analysis.
Given the above considerations and previously stated hypotheses, literature suggests that
we choose as a traffic assignment model a single-class, mono-modal and rigid demand
Deterministic User Equilibrium (D.U.E.).
A D.U.E. assignment is consistent with Wardrop’s first principle (Wardrop, 1952),
which states that “The journey times in all routes actually used are equal and less than
those which would be experienced by a single vehicle on any unused route” . In other
words, when an equilibrium is reached, no individual is able to reduce their costs by
22. 22 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
choosing an alternative route over the network. Thus, unlike other models such as De-
terministic Uncongested Networks (D.U.N.) or Deterministic Network Loading (D.N.L.)
models, D.U.E. models do not assign all the demand to the path of minimum cost for
each O/D pair, but distribute it on different paths to take into account the effect of
congestion at the equilibrium condition.
2.2.3 Another type of equilibrium: System Optimum
A possible alternative to a D.U.E. assignment is represented by a System Optimum
(S.O.) assignment, which is consistent with Wardrop’s second principle (Wardrop, 1952).
This states that which “At equilibrium the average journey time is minimum”. This
means that users must cooperate to the minimization of the total cost of the network.
This approach can be used, for example, to calibrate the instruments of control
available to the manager of the system because flows and costs at the system optimum are
consistent with the objectives to which the operator aims. However, the resulting choice
behavior is likely to be unrealistic because some users, just to reach the minimum cost of
the system, would not choose individual paths of minimum cost. Therefore, in general,
solutions of an S.O. assignment do not coincide with those of a D.U.E. assignment.
Thus this model, although providing a formulation and a resolution algorithm that
are similar to the D.U.E. and despite being able to provide good solutions when applied
to the road pricing problem, is not consistent to the objective of wanting to recreate a
sequential game between the parties where each seeks to achieve its purpose without any
mutual collaboration.
2.2.4 Cost functions
Cost functions express the cost of a path or arc based on the performance of the
network. Because network congestion is taken into account, the costs vary with the
flows on which they depend. Assuming separable functions and the absence of non-
additive path costs, each arc of the network has a cost that is a function only of the flow
on the arc itself.
According to (Wardrop, 1952), a generic arc cost function, separable and free of
non-additive costs, has the following formulation:
Ca (fa ) = β1 · tra (fa ) + β2 · twa (fa ) + β3 · mca (fa ) (2.1)
where tra is the travel time, twa is the waiting time, mca is the monetary cost, β1 ,
β2 , β3 are coefficients of homogenization.
Travel time on a directed arc of a highway network is obtained through the following
empirical relation:
β
fa
tra (fa ) = tr0a · 1 + α · (2.2)
qa
23. 2.2. CNPP LOWER LEVEL: TRAFFIC ASSIGNMENT PROBLEM 23
where tr0a is the travel time in unconstrained conditions (i.e. optimal traffic and
weather conditions), qa is the capacity of an arc (calculated through appropriate analysis
tools provided by the technical literature(i.e. Transportation Research Board (2000)), α
e β are parameters that have to be calibrated.
The function defined by eq. 2.2 is not linear, continuous, strictly positive and strictly
increasing.
Waiting time can be attributed to the presence of barriers for the collection of tolls,
crossroads with traffic lights or parking areas. Since we are considering the case of a
highway network, there are no intersections and parking areas to generate waiting times,
therefore we neglect the related component of generalized cost.
Monetary cost, borne by the individual user who travels along the transportation
network, can be further decomposed into the sum of the monetary cost of the toll and
the monetary cost of fuel (depending on the level of congestion), that is:
mca (fa ) = mctoll + mcf uel (fa ) (2.3)
Since this would not cause any loss in generality for our model, here we neglect the
component of the cost due to fuel consumption. As a consequence, the monetary cost is
reduced to mca (fa ) = mctoll that from now on will simply be referred to as "arc tariff" ta .
Cost function is thus reduced to:
β
fa
Ca (fa ) = tr0a · 1 + α · + δ · ta (2.4)
qa
where δ is a binary parameter with value 1 if the arc has costs, 0 otherwise.
Note that the charge included in the cost function specified is considered independent
of the distance traveled and of the time evolution of the system (the so-called inter-and
intra-period dynamics).
2.2.5 Second level of the CNPP: flows assignment
Let G = (N , A ∪ B) be a transportation network comprising set N of the nodes and
the union A ∪ B of disjoint sets A and B (A ∩ B = ∅), where A is the set of the arcs
where the leader imposes a tariff and B is the set of the toll-free arcs. Let K be the set of
commodities, that in this case correspond to the O/D pairs. A set of O/D pairs is thus
defined, (ok , dk ) : k ∈ K and each (ok , dk ) pair has a constant demand flow η k , that is
the number of users for the commodity k ∈ K. Let x and t be the flow and tariffs vector
respectively. We denote by xk the flow on arc a ∈ A ∪ B for the commodity (O/D pair)
a
k ∈ K; thus the flow on an arc is given by xa = k∈K xk . We denote by Ca (xa ) the arc
a
cost function specified as follows:
β
xa
Ca (xa ) = tr0a · 1 + α · ∀a ∈ A ∪ B (2.5)
qa
24. 24 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
where:
xa = xk
a ∀a ∈ A ∪ B (2.6)
k∈K
Note that for every arc a ∈ A we will also have to consider the tariff ta ; such tariff is
added to the cost Ca as shown by eq.2.4.
Moreover, let bk be the demand vector associated with each commodity k ∈ K, whose
components (one for each node i ∈ N ) are:
−η k if i = ok
k
bi = ηk if i = dk ∀i ∈ N , ∀k ∈ K (2.7)
0 otherwise
The arc formulation of the deterministic equilibrium assignment problem as an opti-
mization model is as follows:
xa xa
min (Ca (ωa ) + ta )dωa + Ca (ωa )dωa (2.8)
x
a∈A 0 a∈B 0
s.t. xk +
a xk −
a xk −
a xk = bk
a i ∀i ∈ N , ∀k ∈ K (2.9)
a∈i− ∩A a∈i− ∩B a∈i+ ∩A a∈i+ ∩B
k
xa ≥ 0 ∀a ∈ A ∪ B, ∀k ∈ K (2.10)
where i− ∈ N and i+ ∈ N are respectively the entering and exiting arcs for node
i ∈ N.
Equation 2.8 expresses the objective function of the problem, which is nonlinear
because of the cost function adopted. Equation 2.9 imposes a number of linear constraints
which is equal to the number of network nodes in order to express conservation of flows
at the nodes. Finally, equation 2.10 imposes the non-negativity of the arc flow variables
for each commodity.
The problem above is known as minimum cost multi-commodity flow convex prob-
lem ((LeBlanc et al., 1975)) or Traffic Assignment problem (Petersen (1975), Patriksson
(1994)). In literature there are several variations to the basic problem presented here.
Daganzo (1977a,b) proposes the introduction of capacity constraints on arc flow variables
and linearization of the cost function from a non-zero value of the flow below the limit
of capacity. However, it is believed that such changes are unnecessary here, given that
the cost function grows more than proportionally with the increase of the flow/capacity
ratio, thus already conditioning in an effective and appropriate way the distribution of
flows on the network. In addition, changes should be made to the solution algorithm and
to the convergence conditions that would make too complex and uncertain the definition
of the problem (see also Hearn and Ribera, 1981).
A stochastic variant of this problem, based on the concept of Stochastic User Equilib-
rium (S.U.E.) was also investigated thoroughly in literature (see Sheffi (1985) for the
25. 2.3. FIRST LEVEL OF THE CNPP: LEADER PROFITS 25
theoretical bases and Polyak (1990) and Damberg et al. (1996) for resolution algorithms).
2.3 First level of the CNPP: leader profits
In this section the first level of the CNPP is considered, that is, how the leader
maximizes his or her profit by imposing fees on the tariffed arcs and dependently on the
distribution of the followers on the network at the equilibrium.
For the original problem the objective function for the first level of the problem (the
leader’s problem) is the following:
max xk · t a
a (2.11)
t,x
k∈K a∈A∪B
As said before, it is a non-linear objective function, since it contains a bilinear term.
In addition, the following constraint will apply:
ta ≥ 0 ∀a ∈ A (2.12)
Equation 2.12 being imposed by the non negativity of the cost functions stated in
sec.2.2.4.
The resulting bilevel problem is thus the following:
max xk · ta
a (2.13)
t,x
k∈K a∈A∪B
xa xa
min (Ca (ωa ) + ta )dωa + Ca (ωa )dωa (2.14)
x
a∈A 0 a∈B 0
xk
a − xk
a = bk
i ∀i ∈ N , ∀k ∈ K (2.15)
a=i− ∈A∪B a=i+ ∈A∪B
xk ≥ 0
a ∀a ∈ A ∪ B, ∀k ∈ K (2.16)
ta ≥ 0 ∀a ∈ A (2.17)
(2.18)
With cost/flow interdependence as expressed by equation 2.5 and demand vector bk
defined by equation 2.7.
2.4 Reformulating the CNPP
The aim of the present section is to apply a simplification scheme to the CNPP, in
order to obtain a formulation that is easier to handle. The process is carried out in two
phases:
26. 26 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
1. From bilevel to single level through Lagrangian Duality;
2. From bilinear to mixed-integer non-linear.
2.4.1 From bilevel CNPP to single level CNPP
Symmetrically to the procedure that was carried on for the linear NPP, in this phase
the second level problem will be replaced by its KKT conditions. This is a legitimate
operation, assuming that objective and constraints are ∈ C 1 and have the characteristics
described below (see Sheffi (1985) and Bertsekas (1995)). Be such the case, a vector x
that satifies the KKT conditions, is known to be optimal for the considered problem.
Given a generical convex program:
min f (x) (2.19)
s.t. h(x) = 0 (2.20)
g(x) ≥ 0 (2.21)
where f (x) and h(x) are convex and g(x) are linear.
Its Lagrangian is defined as:
L(x, λ, µ) = f (x) + λT h(x) − µT g(x) (2.22)
At a minimum point, x, we have L(x, λ, µ) = 0 which implies that the following with
respect to the partial derivatives:
xL : f (x) + λT h(x) − µT g(x) = 0 (2.23)
λL : h(x) = 0 (2.24)
µL : g(x) ≥ 0 (2.25)
Plus the following complementarity condition must hold:
µT g(x) = 0 (2.26)
In the typical instance of a Traffic Assignment problem, which is our second level prob-
lem, the corresponding constraints are the following:
h(x) = 0 ⇒ xk −
a xk − bk = 0
a i ∀i ∈ N , ∀k ∈ K (2.27)
a=i− ∈A∪B a=i+ ∈A∪B
g(x) ≥ 0 ⇒ xk ≥ 0
a ∀a ∈ A ∪ B, ∀k ∈ K (2.28)
27. 2.4. REFORMULATING THE CNPP 27
Consequently, the partials derivatives are:
xL : Ca (xa ) + ta + λk − λk − µk = 0
i j a ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.29)
λL : xk −
a xk − bk = 0
a i ∀i ∈ N , ∀k ∈ K (2.30)
a=i− ∈A∪B a=i+ ∈A∪B
µL : xk ≥ 0
a ∀a ∈ A ∪ B, ∀k ∈ K (2.31)
(2.32)
The complementarity condition will be: µT xk = 0.
a a
Equations 2.23 and 2.26 are the Karush Kuhn Tucker Conditions (KKT ) for our second
level problem.
By substituting the problem with its KKT conditions, we obtain the following reformu-
lation for the CNPP:
max ta · xk
a (2.33)
t,x
a,k
s.t. xk −
a xk = bk
a i ∀i ∈ N , ∀k ∈ K (2.34)
a=i− ∈A∪B a=i+ ∈A∪B
Ca (xa ) + ta + λ i − λ k − µk =
k
j a 0 ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.35)
µ k · xk = 0
a a ∀a ∈ A ∪ B, ∀k ∈ K (2.36)
xk
a ≥0 ∀a ∈ A ∪ B, ∀k ∈ K (2.37)
µk
a ≥0 ∀a ∈ A ∪ B, ∀k ∈ K (2.38)
λk
i free ∀i ∈ N , ∀k ∈ K (2.39)
where Ca (xa ) is the derivative of the cost function expressed by eq.2.5
2.4.2 From bilinear CNPP to Mixed Integer non-linear CNPP
The problem obtained so far still contains bilinear terms in both the objective and
the complementary constraint. What we want to obtain in this section is a single level
non-linear problem with a mixed-integer formulation.
Simplification of the objective (ta · xk term)
a
In the present section we will substitute the ta · xk term in the objective with terms
a
derived from the equality constraints of the problem in order to obtain an equivalent
formulation that will be non-linear in only one variable.
28. 28 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
From the first KKT condition (eq. 2.35) we have that:
µk = Ca (xa ) + ta + λk − λk
a i j ∀a = (i, j) ∈ A, ∀k ∈ K (2.40)
µk = Ca (xa ) + λk − λk
a i j ∀a = (i, j) ∈ B, ∀k ∈ K (2.41)
Thus the second KKT condition (eq. 2.36) can be formulated as:
(Ca (xk ) + ta + λk − λk ) · xk = 0
a i j a ∀a = (i, j) ∈ A, ∀k ∈ K (2.42)
(Ca (xk ) + λk − λk ) · xk = 0
a i j a ∀a = (i, j) ∈ B, ∀k ∈ K (2.43)
by substituting µk with eq. 2.40 and eq. 2.41. These are equal to:
a
Ca (xk ) · xk + ta · xk + (λk − λk ) · xk = 0
a a a i j a ∀a = (i, j) ∈ A, ∀k ∈ K (2.44)
Ca (xk )
a · xk
a + (λk
i − λk )
j · xk
a =0 ∀a = (i, j) ∈ B, ∀k ∈ K (2.45)
From 2.44 we thus obtain:
ta · xk = −Ca (xk ) · xk + (λk − λk ) · xk
a a a j i a ∀a = (i, j) ∈ A, ∀k ∈ K (2.46)
So the objective (eq.) becomes:
max −(Ca (xk ) · xk + (λk − λk ) · xk )
a a j i a (2.47)
λ,x
a∈A k∈K
Note that the following equality holds:
(λk − λk ) · xk =
j i a λk · (
i xk −
a xk )
a (2.48)
a∈A k∈K k∈K i∈N a=i− ∈A a=i+ ∈A
Which is even more evident if we express it in the equivalent matrix-vector formulation:
k k )i∈N λk · (Ai,a · xk ).
a∈A k∈K i∈N xa · (Ai,a · λi ) = a∈A k∈K i a
The flow conservation constraint (eq. 2.34) states that:
xk −
a xk +
a xk −
a xk = bk
a i ∀i ∈ N , ∀k ∈ K (2.49)
a=i− ∈A a=i+ ∈A a=i− ∈B a=i+ ∈B
Thus we have:
xk −
a xk = bk −
a i xk +
a xk
a ∀i ∈ N , ∀k ∈ K (2.50)
a=i− ∈A a=i+ ∈A a=i− ∈B a=i+ ∈B
Following from eq.2.48 and 2.50:
29. 2.4. REFORMULATING THE CNPP 29
λk · (
i xk −
a xk ) =
a (2.51)
k∈K i∈N a=i− ∈A a=i+ ∈A
λk · bk −
i i λk · (
i xk +
a xk ) ∀i ∈ N , ∀k ∈ K
a (2.52)
k∈K i∈N k∈K i∈N a=i− ∈B a=i+ ∈B
Eq.2.45 states that:
Ca (xk ) · xk = (λk − λk ) · xk
a a j i a ∀a = (i, j) ∈ B, ∀k ∈ K (2.53)
k k k
We can thus substitute the a∈A k∈K (λj −λi )·xa term in the objective, that becomes:
max λk · bk −
i i Ca (xk ) · xk −
a a Ca (xk ) · xk
a a (2.54)
λ,x
k∈K i∈N a∈A b∈B
Linearization of the µk · xk term
a a
The problem obtained so far still contains the nonlinear constraint µk ·xk = 0 deriving
a a
from the complementarity slackness condition. However, this term can be linearized
through a relaxation, in the following way:
k
First of all, it is necessary to introduce a binary variable za , defined as:
k 1 if µk = 0
a
za = (2.55)
0 if µk = 0
a
Since we don’t really need to know the exact value of µk but only have to impose that
a
it will be zero if xk = 0 and vice versa, we can write the KKT conditions as:
a
Ca (xa ) + ta + λk − λk ≤ M · za
i j
k
∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.56)
xk
a ≤ M · (1 − k
za ) ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.57)
Where M is an arbitrary Big-M constant. However, for the problem not to be unbounded
the following must also hold:
Ca (xa ) + ta + λk − λk ≥ 0
i j ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.58)
xk ≥ 0
a ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K (2.59)
Thus the final reformulation for the CNPP will be:
30. 30 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
max ( λk · bk −
i i Ca (xa ) · xk −
a Ca (xa ) · xk )
a (2.60)
λ,x
k∈K i∈N a∈A b∈B
s.t. xk
a − xk
a = bk
i ∀i ∈ N , ∀k ∈ K
a=i− ∈A∪B a=i+ ∈A∪B
(2.61)
Ca (xa ) + ta + λk
i − λk
j ≤M· k
za ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K
(2.62)
Ca (xa ) + ta + λk − λk ≥ 0
i j ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K
(2.63)
xk ≤ M · (1 − za )
a
k
∀a = (i, j) ∈ A ∪ B, ∀k ∈ K
(2.64)
xk ≥ 0
a ∀a = (i, j) ∈ A ∪ B, ∀k ∈ K
(2.65)
k
za ∈ {0; 1} ∀a ∈ A ∪ B, ∀k ∈ K
(2.66)
ta ≥ 0 ∀a ∈ A, ∀k ∈ K
(2.67)
λk
i free ∀i ∈ N , ∀k ∈ K
(2.68)
2.4.3 Complexity of the CNPP
The complexity of the CNPP, as formulated above, depends heavily on the choice for
the cost function. For the particular cost function that was illustrated in section 2.2.4,
this complexity relies on the choice for the positive integer parameters α and β. We can
distinguish three situations that can occurr and that are of some interest:
1. α = 0
β
fa
In this case the congestion-dependent term α · qa equals zero, and the CNPP
instance becomes a standard NPP one.
In fact, while the constraints and leader objective face no change, the follower’s
objective becomes:
31. 2.4. REFORMULATING THE CNPP 31
xa
min (Ca (ωa ) + ta )dωa = (2.69)
x 0
a∈A∪B
xa
= min (tr0a + ta )dωa = (2.70)
x 0
a∈A∪B
= min (tr0a + ta ) · xa (2.71)
x
a∈A∪B
Thus resulting in a formulation that is identical to the one illustrated in the pre-
vious chapter for the uncongested NPP.
The case β = 0 is similar and does not offer any interesting particolarity.
2. α > 0 and β = 1
In this case the CNPP instance described above represents a quadratic problem
(note that the constraints are all linear). Its complexity is superior to the uncon-
gested NPP, but many efficient resolutive algorithms exist for this class of problems.
3. α > 0 and β > 1
In this case the CNPP instance is non linear and we can expect it to be much harder
to solve than the uncongested NPP. Several solving approaches could be possible,
from penalizing the nonlinear constraint in the objective (thus obtaining a convex
polyhedron as feasible region) to heuristic procedures that use local approximations
of the nonlinear term to generate a succession af approximating subproblems, and
so on. Two procedures of this last type will be presented in the next chapter.
32. 32 CHAPTER 2. THE CONGESTED NETWORK PRICING PROBLEM
33. Chapter 3
Solving the CNPP
3.1 The Conditional Gradient method
(Frank-Wolfe Algorithm)
Although the bilevel formulation for the CNPP presented in the previous chapter
has not been studied so far, the User Equilibriun problem which constitutes the second
level of our problem (that is, the followers’ problem) is well known and has been studied
thoroughly in the past decades, with many resolutive algorithms developed so far. Thw
most widely known is probably the Frank-Wolfe algorithm (Frank and Wolfe (1956)),
also known as Conditional Gradient method (Bertsekas (1995)). This algorithm solves a
sequence of linear problems that approximate the original non linear one, and generates
a sequence of admissible flow vectors from the solutions of the approximating instances.
Asymptotically, this sequence will converge to the optimum solution. The fact that
it uses a linear approximation of what is here the second level part makes the F-W
algorithm particularly interesting for the resolution of the bilevel problem.
3.1.1 Applying Frank-Wolfe to the D.U.E. assignment problem
In the present section, we will briefly illustrate how the Frank-Wolfe algorithm can
be applied to a D.U.E. resolution.
The algorithm solves a sequence of linear problems that approximate the original
problem and then generates a sequence of admissible flows arc vectors x(j) from a feasi-
¯
ble solution of the original problem x(0) ∈ Sx , where Sx is the set of admissible flows arcs.
¯
The solution of the linear subproblems identifies, with respect to the current solution
x(j−1) , a direction along which to minimize the objective function in order to determine
¯
the new point x(j) .
¯
Using the Taylor expansion stopped at the first term, the objective function z(¯) can
x
be approximated in a point y ∈ Sx by a linear function zl (¯):
¯ x
z(¯) ∼ zl (¯) = z(¯) +
x = x y z(¯)T (¯ − y )
y x ¯ (3.1)
33
34. 34 CHAPTER 3. SOLVING THE CNPP
Therefore, the optimization problem with nonlinear objective function z(¯) and lin-
x
ear constraints can be approximated by a succession of problems with linear objective
function zl (¯) and linear constraints for every point y ∈ Sx . In fact we have:
x ¯
argmin z(¯) ∼ argmin zl (¯) = argmin z(¯) +
x = x y z(¯)T (¯ − y ) = argmin
y x ¯ z(¯)T x (3.2)
y ¯
Since in one point y the gradient of the objective function
¯ z(¯) equals the arc cost
x
vector calculated in the same point, z(¯) = c(¯), we obtain:
y ¯y
argmin z(¯) ∼ argmin c(¯)T x
x = y ¯ (3.3)
The objective function described above, paired with the non-negativity and flow
conservation constraints of the Traffic Assignment Problem (eq. 2.9 and 2.10) denote an
optimization problem that corresponds to the model of D.U.N / D.N.L, the minimum
cost - multi-commodity flow linear problem. This problem can be solved through the well-
known Dijkstra’s algorithm (henceforth referred as All-or-nothing assignment (AON)).
It follows that through the application of Frank-Wolfe algorithm it is possible to
reduce a problem of D.U.E. assignment (convex) to a sequence of approximating D.U.N.
assignment subproblems (linear), thus ignoring locally the cost-flow inter-dependence
due to congestion.
The j-th subproblem, with j = 1, . . . , m, where m is the number of the subproblems
in the succession, can be formulated as follows:
(j) (j)
min Ca (x(j−1) ) · xa,AON +
(j)
a Ca (x(j−1) ) · xa,AON
(j)
a (3.4)
x
a∈A a∈B
k,(j) k,(j)
s.t. xa,AON − xa,AON = bk
i ∀i ∈ N , ∀k ∈ K
a=i− ∈A∪B a=i+ ∈A∪B
(3.5)
k,(j)
xa,AON ≥ 0 ∀a ∈ A ∪ B, ∀k ∈ K
(3.6)
(j−1)
where xa is the flow on arc a ∈ A ∪ B of the (j − 1)-th problem, specified as
follows:
x(j−1) =
a xk,(j−1)
a (3.7)
k∈K
(j) k,(j)
xa,AON = xa,AON (3.8)
k∈K
(j) (j−1)
Ca (xa ) is the cost of arc a ∈ A ∪ B calculated through the specified cost function
(j−1)
from the arc flow xa :
35. 3.1. THE CONDITIONAL GRADIENT METHOD(FRANK-WOLFE ALGORITHM)35
(j−1) β
(j) (j−1) xa
Ca (xa ) = tr0a · 1 + α · ∀a ∈ A ∪ B (3.9)
qa
(j)
xa,AON is the flow on arc a ∈ A ∪ B, solution of the j-th problem obtained through
the A.O.N. resolution algorithm.
(j−1)
Note that xa arc flows of the (j − 1)-th problem are constant (since they are the
(j) (j−1)
solution of this problem) and consequently Ca (xa ) arc costs are constant for the j-th
(j)
problem. It follows that xa,AON arc flows are the only variables in the j-th problem, thus
resulting in a linear objective function for the j-th problem.
3.1.2 F-W algorithm steps
The steps of the Frank-Wolfe algorithm applied to D.U.E. assignment are the follow-
ing:
Initialization An admissible starting solution is found x(0) ∈ Sx (see §3.4.1) and a
¯
threshold is chosen for the stopping criterion (see §3.1.3).
j-th iteration Given the flow vector x(j−1) , solution of the problem at the (j − 1)-th
¯
iteration:
¯ ¯x
1. The cost vector is determined in function of the flow vector, as C (j) = C(¯(j−1) ),
through equation 2.5.
2. The approximating linear problem denoted by Equations 3.4, 3.5, 3.6 is solved
through the All-or-nothing algorithm, since arc costs (denoted as arc labels)
are constant for the j-th iteration. Thus the flow vector for the non congested
(j)
network is obtained: xAON .
¯
3. The descent step is executed in order to determine the final solution for the
j-th iteration. It consists in the following one-dimensional non-linear research
problem:
(j)
µ(j) ∈ argminµ∈[0,1] ψ(µ) = z x(j−1) + µ · (¯AON − x(j−1) )
¯ x ¯ (3.10)
where µ is a scalar variable. The problem can be solved through the bisection
algorithm or any other one-dimensional minimization procedure.
4. Vector x(j) is found as a solution for the j-th iteration. It is a convex combi-
¯
(j)
nation of the previous j All-or-nothing assignment xAON :
¯
(j)
x(j) = x(j−1) + µ(j) · (¯AON − x(j−1) )
¯ ¯ x ¯ (3.11)
Stopping criterion The stopping test is executed against a threshold (see §3.1.3). If
the test fails, a new iteration is executed with vector x(j) as the starting solution;
¯
otherwise the algorithm is stopped and x ¯ (j) is the deterministic flows vector.
36. 36 CHAPTER 3. SOLVING THE CNPP
3.1.3 Stopping criterion
As mentioned in Chapter 3.1.2, at the end of each iteration a stopping test is run;
when it is verified the sequence of approximating subproblems is stopped and the vectors
¯
x(j) and t(j) of the j-th subproblem are the optimal solutions of the CNPP problem.
¯
In order for x(j) to be a local minumum (in our case a global minimum since the
¯
objective function is convex) it has to be a stationary point for z(x). This means that
the following condition must hold:
(j)
z(¯(j) )T (¯AON − x(j) ) ≥ 0
x x ¯ (3.12)
However, from a practical point of view, such a termination criterion may not be
so valuable. The conditional gradient algorithm in fact is known for slowing down the
convergence rate significantly as it approaches the optimal solutiol. Moreover, in most
applications, an approximated solution that is close enough to the optimum is more
than sufficient, especially if we can obtain it more quickly, through a lower number of
iterations. Given the considerations above, we are more eager to use a threshold-based
stopping criterion, so that the algorithm will stop once a given threshold is reached.
From the convexity of the objective function we can obtain upper and lower bound
inequalities from which we will derive such a criterion.
We know z(x) to be convex only if the following holds:
z(y) ≥ z(x) + z(x)T (y − x) (3.13)
We therefore have that:
(j) (j)
zl (¯AON ) = z(¯(j) ) +
x x z(¯(j) )T (¯AON − x(j) )
x x ¯ (3.14)
(j) (j) T (j) (j)
≤ z(¯
x )+ z(¯
x ) (¯∗
x −x
¯ ) (3.15)
(j)
≤ z(¯∗ )
x (3.16)
where the equality follows from the definition of the optimal solution to the LP
(j)
problem, the first inequality by the fact that xAON solves this LP problem but not
¯
(j)
necessarily x∗ , and the second inequality from the convexity of z. So, we have that at
¯
every iteration j,
(j) (j)
zl (¯AON ) ≤ z(¯∗ ) ≤ z(¯(j) )
x x x (3.17)
where the last inequality holds because x(j) is a feasible solution. In the condi-
¯
tional gradient algorithm the value of z descends after each iteration, thus the se-
(j)
quence of {z(¯(j) )} strictly monotonically decreases towards z(¯∗ ), while the sequence
x x
(j) (j)
of {zl (¯AON )} approaches z(¯∗ ) from below, though not necessarily monotonically. We
x x
(j)
therefore always have an interval, [zl (¯AON ), z(¯(j) )], which contains the optimal objec-
x x
tive value. From this we can derive a threshold-based criterion of the type: let be a
chosen value for an acceptable relative objective error, then the stopping condition for
the j-th iteration is as follows:
37. 3.2. CONVERGENCE CONDITIONS 37
(j)
|z(¯(j) ) − zl (¯AON )|
x x
(j)
< (3.18)
|zl (¯AON )|
x
3.2 Convergence conditions
In order for the Frank-Wolfe algorithm to converge to a vector of optimal solutions
after a certain number of iterations, it is necessary that certain conditions for convergence
are met.
Since in the present work the algorithm is used for a User Equilibrium allocation, we
refer to LeBlanc et al. (1975). In the article it is recommended that the specified cost
function meets the following requirements:
1. continuous and differentiable at least once at all points,
2. non-negative,
3. non-decreasing.
As can be easily verified, the cost function used in the present work (eq. 3.9) meets
all three requirements, in particular, is strictly positive and strictly increasing.
Typically, the Frank-Wolfe algorithm is adopted to find an equilibrium point on
congested networks in which all arcs are priced (so that the tariff component is added in
the function of generalized cost) or where there are no tariffs. In literature (Yang et al.
(2004)) there is an example of application to a mixed network with tariffed and non-
tariffed arcs, including a bilevel programming approach. However no account is taken of
the congestion and still the objective function of the optimization model is nonlinear, as
demand is elastic; moreover, there are also non-additive costs and cost functions are not
separable. The Frank-Wolfe algorithm is adapted successfully to the case examined, as
the only additional condition states that it must be convex programming.
Further necessary conditions for the convergence of Frank-Wolfe algorithm applied
to this case could not be found in literature.
3.3 Accelerating convergence
As it is known from numerous examples in literature, the time required for con-
vergence of the Frank-Wolfe algorithm with D.U.E. allocation can vary considerably
depending on the size and complexity of the network, on the descending step and on the
threshold for the stopping test. As the number of iterations grow, the algorithm tends
to zigzag increasingly.
While the algorithm has the historical advantage of a low memory usage (useful for
computers especially a few decades ago), on the other hand, the computational effort
tends to increase as you get closer to the optimal solution, thus making the resolution
38. 38 CHAPTER 3. SOLVING THE CNPP
in real time more critical . This can be partly overcome by setting a less restrictive
threshold although this implies to sacrifice final accuracy.
Leaving aside the particular structure of the modeled network to more specific cases
than those of this discussion, it is possible to accelerate the convergence of the algorithm
by choosing an appropriate technique for the descending step.
The µ(j) parameter introduced in subsection 3.1.2 was first used in LeBlanc et al.
(1975); for each j-th iteration this parameter is determined through a one-dimensional
linear research, by solving a problem determined by Equation 3.10.
There are some alternatives in literature to the descending step described above, in
order to accelerate the convergence of the original Frank-Wolfe algorithm. In particular,
a variant of the of the parallel tangent method (PARTAN) LeBlanc et al. (1985) can be
used, where two one-dimensional minimizations are made for each iteration. In an ar-
ticle Lupi (1986) introduces another modification to the descending step that, although
providing a convergence rate of the same order of the PARTAN, requires only one one-
dimensional minimization. Variants proposed by Fukushima (1984) and Weintraub et al.
(1985) are also interesting. Please refer to the cited articles cited for a comparison of
different methods based on numerical examples.
The Frank-Wolfe algorithm for D.U.E. assignments has some resemblance with the
Method of Successive Averages (M.S.A.) algorithm for S.U.E. allocations (Stochastic User
Equilibrium) for stochastic equilibrium, except for the cost functions used (that include
random components), the algorithm for solving approximating subproblems approxi-
mants (Dial algorithm for stochastic assignments on non congested networks (S.N.L))
and the descending step. Regarding the latter, a simple µ(j) = 1 is used, where j is the
j
index of the iteration. By adopting this simple solution at the expense of mathematical
sophistication, the zigzag behavior is eliminated but the optimal solution we want the
algorithm to converge to may be missed.
Again, there are alternatives in literature to the 1 descending step for the M.S.A
j
algorithm for S.U.E. allocation; for example Polyak (1990) suggests a µ(j) = j −β de-
scending step, where 0 < β < 1.
39. 3.4. TWO-STEPS ALGORITHM 39
3.4 Two-steps algorithm
The procedure described in the present section is a Gauss-Seidel-decomposition-type
algorithm. It solves the problem described by eq. 2.13 - 2.17 by iteratively solving first
the followers’ problem (through Frank-Wolfe) with respect to arc flows x(j) and then
¯
the leader’s problem with respect to the tariff vector t(j) . It is probably the most ob-
vious and immediate way to approach a resolution for our problem. It is also the only
bilevel/congestion solution approach for which some reference exists in literature. A
similar algorithm was indeed used by Julsain (1998) to find an equilibrium point on a
congested telecommunication network.
The basic steps of the algorithm are the following:
Initialization An admissible starting solution is found x(0) ∈ Sx (see §3.4.1) and a
¯
threshold is chosen for the stopping criterion (see §3.1.3).
j-th iteration Given the flow vector x(j−1) and the tariff vector t(j−1) , solution of the
¯
problem at the (j − 1)-th iteration:
1. The cost vector is determined in function of the flow vector and the tariff
(j)
vector, as Ca , through:
(j−1) β
(j) xa
Ca = tr0a · 1 + α · if a ∈ A ∪ B (3.19)
qa
(j−1) k,(j−1)
where xa = k∈K xa .
2. The linear approximation for the second level problem is solved through the
All-or-nothing algorithm, since arc costs (and tariffs) are constant for the j-th
iteration.
(j) (j)
min (Ca + t(j−1) ) · xa,AON +
(j)
a
(j)
Ca · xa,AON (3.20)
x
a∈A a∈B
k,(j) k,(j)
s.t. xa,AON − xa,AON = bk
i ∀i ∈ N , ∀k ∈ K
a=i− ∈A∪B a=i+ ∈A∪B
(3.21)
k,(j)
xa,AON ≥ 0 ∀a ∈ A ∪ B, ∀k ∈ K
(3.22)
(j)
Thus the flow vector for the non congested network is obtained: xAON .
¯