The document discusses a generalized absolute value optimization problem (GAVP) that generalizes the absolute value programming problem (AVP). It introduces the GAVP primal and dual problems, which allow for both linear and nonlinear terms. It provides examples of norm-like functions that satisfy the assumptions of the GAVP, including applications to conic linear programming and group Lasso problems. It also considers a special case of the GAVP with only nonlinear terms in the objective and constraints. In all cases, it establishes weak duality between the primal and dual problems.
The document summarizes algorithms for solving min-cost linear problems, including min-cost flow problems and min-cost potential problems. It describes how these problems can be formulated and solved using descent methods, where the search direction is chosen as a negative cycle or cut with minimum cost. Iteratively, an optimal solution is found by moving in the direction of negative cycles or cuts and updating residual graphs and data structures. Duality between the min-cost flow and potential problems is also discussed.
Bayesian inversion of deterministic dynamic causal modelskhbrodersen
1. The document discusses various methods for Bayesian inference and model comparison in dynamic causal models, including variational Laplace approximation, sampling methods, and computing model evidence.
2. Variational Laplace approximation involves factorizing the posterior distribution and iteratively optimizing a lower bound on the model evidence called the negative free energy.
3. Sampling methods like Markov chain Monte Carlo generate stochastic approximations to the posterior by constructing a Markov chain with the target distribution as its equilibrium distribution.
This document defines and provides examples of basic concepts in propositional logic, including:
1. Propositions are declarative sentences that are either true or false. Common propositional variables include p, q, r.
2. Logical connectives combine propositions and include negation (¬p), conjunction (p∧q), disjunction (p∨q), conditional (p→q), biconditional (p↔q). Truth tables define the truth values of connected propositions.
3. Relations between conditionals include the converse, contrapositive, and inverse. Examples show how to derive these from a given conditional statement.
This document discusses propositional logic and logical equivalences. It begins by defining tautologies, contradictions, and contingencies. It then discusses logical equivalence and uses truth tables to show several examples of logically equivalent propositions. The document also lists common laws of logical equivalence, such as commutative, associative, distributive, and De Morgan's laws. It provides examples of using these laws to show logical equivalence without truth tables. Finally, the document discusses predicates, universal and existential quantification, and provides several examples of determining truth values of quantified statements.
Couplings of Markov chains and the Poisson equation Pierre Jacob
The document discusses couplings of Markov chains and the Poisson equation. It begins with an outline introducing couplings as a technique to study Markov chain convergence rates. An example is provided of a Gibbs sampler motivated by Dempster-Shafer inference, known as the donkey walk. A common random numbers coupling of the donkey walk yields an explicit bound on the Wasserstein distance between the distribution after t steps and the stationary distribution.
The document discusses rules of inference and proofs in propositional logic. It begins by defining valid arguments and argument forms. It then introduces several common rules of inference like modus ponens, modus tollens, and disjunctive syllogism. The document provides examples of using these rules of inference to determine conclusions given certain premises. It also discusses direct proofs, indirect proofs using contraposition, and proof by cases. Worked examples are provided for each type of proof.
A simple method to find a robust output feedback controller by random search ...ISA Interchange
A random search algorithm is proposed to find a robust output feedback controller for uncertain linear systems. The algorithm generates random feedback gain matrices and evaluates the closed-loop pole locations. If the poles lie in a specified stable region, the gain matrix is a solution. The probability of finding a solution and the number of trials needed can be estimated. Simulation results demonstrate the effectiveness of using this random approach. The method provides a simple way to design output feedback controllers for systems where traditional techniques are intractable, such as problems with nonconvex constraints.
The document summarizes algorithms for solving min-cost linear problems, including min-cost flow problems and min-cost potential problems. It describes how these problems can be formulated and solved using descent methods, where the search direction is chosen as a negative cycle or cut with minimum cost. Iteratively, an optimal solution is found by moving in the direction of negative cycles or cuts and updating residual graphs and data structures. Duality between the min-cost flow and potential problems is also discussed.
Bayesian inversion of deterministic dynamic causal modelskhbrodersen
1. The document discusses various methods for Bayesian inference and model comparison in dynamic causal models, including variational Laplace approximation, sampling methods, and computing model evidence.
2. Variational Laplace approximation involves factorizing the posterior distribution and iteratively optimizing a lower bound on the model evidence called the negative free energy.
3. Sampling methods like Markov chain Monte Carlo generate stochastic approximations to the posterior by constructing a Markov chain with the target distribution as its equilibrium distribution.
This document defines and provides examples of basic concepts in propositional logic, including:
1. Propositions are declarative sentences that are either true or false. Common propositional variables include p, q, r.
2. Logical connectives combine propositions and include negation (¬p), conjunction (p∧q), disjunction (p∨q), conditional (p→q), biconditional (p↔q). Truth tables define the truth values of connected propositions.
3. Relations between conditionals include the converse, contrapositive, and inverse. Examples show how to derive these from a given conditional statement.
This document discusses propositional logic and logical equivalences. It begins by defining tautologies, contradictions, and contingencies. It then discusses logical equivalence and uses truth tables to show several examples of logically equivalent propositions. The document also lists common laws of logical equivalence, such as commutative, associative, distributive, and De Morgan's laws. It provides examples of using these laws to show logical equivalence without truth tables. Finally, the document discusses predicates, universal and existential quantification, and provides several examples of determining truth values of quantified statements.
Couplings of Markov chains and the Poisson equation Pierre Jacob
The document discusses couplings of Markov chains and the Poisson equation. It begins with an outline introducing couplings as a technique to study Markov chain convergence rates. An example is provided of a Gibbs sampler motivated by Dempster-Shafer inference, known as the donkey walk. A common random numbers coupling of the donkey walk yields an explicit bound on the Wasserstein distance between the distribution after t steps and the stationary distribution.
The document discusses rules of inference and proofs in propositional logic. It begins by defining valid arguments and argument forms. It then introduces several common rules of inference like modus ponens, modus tollens, and disjunctive syllogism. The document provides examples of using these rules of inference to determine conclusions given certain premises. It also discusses direct proofs, indirect proofs using contraposition, and proof by cases. Worked examples are provided for each type of proof.
A simple method to find a robust output feedback controller by random search ...ISA Interchange
A random search algorithm is proposed to find a robust output feedback controller for uncertain linear systems. The algorithm generates random feedback gain matrices and evaluates the closed-loop pole locations. If the poles lie in a specified stable region, the gain matrix is a solution. The probability of finding a solution and the number of trials needed can be estimated. Simulation results demonstrate the effectiveness of using this random approach. The method provides a simple way to design output feedback controllers for systems where traditional techniques are intractable, such as problems with nonconvex constraints.
Unbiased Markov chain Monte Carlo methods Pierre Jacob
This document describes unbiased Markov chain Monte Carlo methods for approximating integrals with respect to a target probability distribution π. It introduces the idea of coupling two Markov chains such that their states are equal with positive probability, which can be used to construct an unbiased estimator of integrals of the form Eπ[h(X)]. The document outlines conditions under which the proposed estimator is unbiased and has finite variance. It also discusses implementations of coupled Markov chains for common MCMC algorithms like Metropolis-Hastings and Gibbs sampling.
Monte Carlo methods for some not-quite-but-almost Bayesian problemsPierre Jacob
This document outlines an approach to inference when exact Bayesian methods are not applicable. Specifically, it discusses Dempster-Shafer theory, which defines lower and upper probabilities for hypotheses based on feasible parameter sets. It proposes a Gibbs sampler to sample from the distribution of these feasible sets defined by count data. It represents the feasible set as relations between data points, allowing conditional distributions to be derived. This leads to a Gibbs sampling algorithm for approximating inferences under Dempster-Shafer theory for problems where exact Bayesian computation is difficult.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
Application of the Monte-Carlo Method to Nonlinear Stochastic Optimization wi...SSA KPI
This document describes a method for solving nonlinear stochastic optimization problems with linear constraints using Monte Carlo estimators. The key aspects are:
1) An ε-feasible solution approach is used to avoid "jamming" or "zigzagging" when dealing with linear constraints.
2) The optimality of solutions is tested statistically using the asymptotic normality of Monte Carlo estimators.
3) The Monte Carlo sample size is adjusted iteratively based on the gradient estimate to decrease computational trials while maintaining solution accuracy.
4) Under certain conditions, the method is proven to converge almost surely to a stationary point of the optimization problem.
5) As an example, the method is applied to portfolio optimization with
This document discusses computational issues that arise in Bayesian statistics. It provides examples of latent variable models like mixture models that make computation difficult due to the large number of terms that must be calculated. It also discusses time series models like the AR(p) and MA(q) models, noting that they have complex parameter spaces due to stationarity constraints. The document outlines the Metropolis-Hastings algorithm, Gibbs sampler, and other methods like Population Monte Carlo and Approximate Bayesian Computation that can help address these computational challenges.
The document discusses Bayesian neural networks and related topics. It covers Bayesian neural networks, stochastic neural networks, variational autoencoders, and modeling prediction uncertainty in neural networks. Key points include using Bayesian techniques like MCMC and variational inference to place distributions over the weights of neural networks, modeling both model parameters and predictions as distributions, and how this allows capturing uncertainty in the network's predictions.
(DL hacks輪読) Variational Inference with Rényi DivergenceMasahiro Suzuki
This document discusses variational inference with Rényi divergence. It summarizes variational autoencoders (VAEs), which are deep generative models that parametrize a variational approximation with a recognition network. VAEs define a generative model as a hierarchical latent variable model and approximate the intractable true posterior using variational inference. The document explores using Rényi divergence as an alternative to the evidence lower bound objective of VAEs, as it may provide tighter variational bounds.
This document summarizes a talk given by Heiko Strathmann on using partial posterior paths to estimate expectations from large datasets without full posterior simulation. The key ideas are:
1. Construct a path of "partial posteriors" by sequentially adding mini-batches of data and computing expectations over these posteriors.
2. "Debias" the path of expectations to obtain an unbiased estimator of the true posterior expectation using a technique from stochastic optimization literature.
3. This approach allows estimating posterior expectations with sub-linear computational cost in the number of data points, without requiring full posterior simulation or imposing restrictions on the likelihood.
Experiments on synthetic and real-world examples demonstrate competitive performance versus standard M
This document presents a framework for analyzing the convergence of Galerkin approximations for a class of noncoercive operators. It begins by introducing assumptions on the operators and establishing well-posedness of the continuous problem. It then analyzes a "GAP" condition on the finite element discretization that is sufficient for stability and quasi-optimal convergence. Finally, it discusses two applications of the theory: Maxwell's equations with variable coefficients, and a boundary integral formulation for electromagnetic wave propagation.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
(研究会輪読) Weight Uncertainty in Neural NetworksMasahiro Suzuki
Bayes by Backprop is a method for introducing weight uncertainty into neural networks using variational Bayesian learning. It represents each weight as a probability distribution rather than a fixed value. This allows the model to better assess uncertainty. The paper proposes Bayes by Backprop, which uses a simple approximate learning algorithm similar to backpropagation to learn the distributions over weights. Experiments show it achieves good results on classification, regression, and contextual bandit problems, outperforming standard regularization methods by capturing weight uncertainty.
Do we need a logic of quantum computation?Matthew Leifer
1) The document discusses whether quantum computing needs a formal logic in the same way that classical computing is understood through classical logic. It examines previous proposals for "quantum logics" and focuses on Sequential Quantum Logic (SQL).
2) SQL models sequences of quantum measurements and operations through projection operators and sequential conjunction. The document proposes testing SQL propositions through a quantum algorithm that prepares an encoded "history state" and applies renormalization operations.
3) The proposed algorithm could test SQL propositions with exponentially small probability of success. Several open questions are raised about generalizing and improving SQL as a logic for quantum computing.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
Binary Vector Reconstruction via Discreteness-Aware Approximate Message PassingRyo Hayakawa
The document proposes a Discreteness-Aware Approximate Message Passing (DAMP) algorithm for reconstructing discrete-valued vectors from underdetermined linear measurements. DAMP extends existing AMP algorithms to handle discrete variables by incorporating probability distributions of the elements. The algorithm is analyzed using state evolution to derive conditions for perfect reconstruction. A Bayes optimal version of DAMP is also developed by minimizing mean squared error. Simulation results demonstrate improved reconstruction performance compared to conventional methods.
Unbiased Markov chain Monte Carlo methods Pierre Jacob
This document describes unbiased Markov chain Monte Carlo methods for approximating integrals with respect to a target probability distribution π. It introduces the idea of coupling two Markov chains such that their states are equal with positive probability, which can be used to construct an unbiased estimator of integrals of the form Eπ[h(X)]. The document outlines conditions under which the proposed estimator is unbiased and has finite variance. It also discusses implementations of coupled Markov chains for common MCMC algorithms like Metropolis-Hastings and Gibbs sampling.
Monte Carlo methods for some not-quite-but-almost Bayesian problemsPierre Jacob
This document outlines an approach to inference when exact Bayesian methods are not applicable. Specifically, it discusses Dempster-Shafer theory, which defines lower and upper probabilities for hypotheses based on feasible parameter sets. It proposes a Gibbs sampler to sample from the distribution of these feasible sets defined by count data. It represents the feasible set as relations between data points, allowing conditional distributions to be derived. This leads to a Gibbs sampling algorithm for approximating inferences under Dempster-Shafer theory for problems where exact Bayesian computation is difficult.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
Application of the Monte-Carlo Method to Nonlinear Stochastic Optimization wi...SSA KPI
This document describes a method for solving nonlinear stochastic optimization problems with linear constraints using Monte Carlo estimators. The key aspects are:
1) An ε-feasible solution approach is used to avoid "jamming" or "zigzagging" when dealing with linear constraints.
2) The optimality of solutions is tested statistically using the asymptotic normality of Monte Carlo estimators.
3) The Monte Carlo sample size is adjusted iteratively based on the gradient estimate to decrease computational trials while maintaining solution accuracy.
4) Under certain conditions, the method is proven to converge almost surely to a stationary point of the optimization problem.
5) As an example, the method is applied to portfolio optimization with
This document discusses computational issues that arise in Bayesian statistics. It provides examples of latent variable models like mixture models that make computation difficult due to the large number of terms that must be calculated. It also discusses time series models like the AR(p) and MA(q) models, noting that they have complex parameter spaces due to stationarity constraints. The document outlines the Metropolis-Hastings algorithm, Gibbs sampler, and other methods like Population Monte Carlo and Approximate Bayesian Computation that can help address these computational challenges.
The document discusses Bayesian neural networks and related topics. It covers Bayesian neural networks, stochastic neural networks, variational autoencoders, and modeling prediction uncertainty in neural networks. Key points include using Bayesian techniques like MCMC and variational inference to place distributions over the weights of neural networks, modeling both model parameters and predictions as distributions, and how this allows capturing uncertainty in the network's predictions.
(DL hacks輪読) Variational Inference with Rényi DivergenceMasahiro Suzuki
This document discusses variational inference with Rényi divergence. It summarizes variational autoencoders (VAEs), which are deep generative models that parametrize a variational approximation with a recognition network. VAEs define a generative model as a hierarchical latent variable model and approximate the intractable true posterior using variational inference. The document explores using Rényi divergence as an alternative to the evidence lower bound objective of VAEs, as it may provide tighter variational bounds.
This document summarizes a talk given by Heiko Strathmann on using partial posterior paths to estimate expectations from large datasets without full posterior simulation. The key ideas are:
1. Construct a path of "partial posteriors" by sequentially adding mini-batches of data and computing expectations over these posteriors.
2. "Debias" the path of expectations to obtain an unbiased estimator of the true posterior expectation using a technique from stochastic optimization literature.
3. This approach allows estimating posterior expectations with sub-linear computational cost in the number of data points, without requiring full posterior simulation or imposing restrictions on the likelihood.
Experiments on synthetic and real-world examples demonstrate competitive performance versus standard M
This document presents a framework for analyzing the convergence of Galerkin approximations for a class of noncoercive operators. It begins by introducing assumptions on the operators and establishing well-posedness of the continuous problem. It then analyzes a "GAP" condition on the finite element discretization that is sufficient for stability and quasi-optimal convergence. Finally, it discusses two applications of the theory: Maxwell's equations with variable coefficients, and a boundary integral formulation for electromagnetic wave propagation.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
(研究会輪読) Weight Uncertainty in Neural NetworksMasahiro Suzuki
Bayes by Backprop is a method for introducing weight uncertainty into neural networks using variational Bayesian learning. It represents each weight as a probability distribution rather than a fixed value. This allows the model to better assess uncertainty. The paper proposes Bayes by Backprop, which uses a simple approximate learning algorithm similar to backpropagation to learn the distributions over weights. Experiments show it achieves good results on classification, regression, and contextual bandit problems, outperforming standard regularization methods by capturing weight uncertainty.
Do we need a logic of quantum computation?Matthew Leifer
1) The document discusses whether quantum computing needs a formal logic in the same way that classical computing is understood through classical logic. It examines previous proposals for "quantum logics" and focuses on Sequential Quantum Logic (SQL).
2) SQL models sequences of quantum measurements and operations through projection operators and sequential conjunction. The document proposes testing SQL propositions through a quantum algorithm that prepares an encoded "history state" and applies renormalization operations.
3) The proposed algorithm could test SQL propositions with exponentially small probability of success. Several open questions are raised about generalizing and improving SQL as a logic for quantum computing.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
Binary Vector Reconstruction via Discreteness-Aware Approximate Message PassingRyo Hayakawa
The document proposes a Discreteness-Aware Approximate Message Passing (DAMP) algorithm for reconstructing discrete-valued vectors from underdetermined linear measurements. DAMP extends existing AMP algorithms to handle discrete variables by incorporating probability distributions of the elements. The algorithm is analyzed using state evolution to derive conditions for perfect reconstruction. A Bayes optimal version of DAMP is also developed by minimizing mean squared error. Simulation results demonstrate improved reconstruction performance compared to conventional methods.
Distributed solution of stochastic optimal control problem on GPUsPantelis Sopasakis
Stochastic optimal control problems arise in many
applications and are, in principle,
large-scale involving up to millions of decision variables. Their
applicability in control applications is often limited by the
availability of algorithms that can solve them efficiently and within
the sampling time of the controlled system.
In this paper we propose a dual accelerated proximal
gradient algorithm which is amenable to parallelization and
demonstrate that its GPU implementation affords high speed-up
values (with respect to a CPU implementation) and greatly outperforms
well-established commercial optimizers such as Gurobi.
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
A New Lagrangian Relaxation Approach To The Generalized Assignment ProblemKim Daniels
This summary provides the key details about a new Lagrangian relaxation approach to solving the generalized assignment problem (GAP) in 3 sentences:
The approach reformulates GAP into an equivalent problem by introducing auxiliary variables and coupling constraints, then relaxes the coupling constraints. This results in subproblems where the constraint structures of GAP are both active, providing stronger lower bounds than traditional approaches. The method was tested on small problems and shown to generate upper bounds more easily than traditional Lagrangian relaxation approaches for GAP.
This talk will report briey on some findings from the problem of picking the weights for a weighted function space in QMC. Then it will be mostly about importance sampling. We want to estimate the probability _ of a union of J rare events. The method uses n samples, each of which picks one of the rare events at random, samples conditionally on that rare event happening and counts the total number of rare events that happen. It was used by Naiman and Priebe for scan
statistics, Shi, Siegmund and Yakir for genomic scans and Adler, Blanchet and Liu for extrema of Gaussian processes. We call it ALOE, for `at least one event'. The ALOE estimate is unbiased and we find that it has a coefficient of variation no larger than p (J + J�1 � 2)=(4n). The coefficient of variation is also no larger than p (__=_ � 1)=n where __ is the union bound. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the J rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by 5772
constraints in 326 dimensions, with probability below 10�22, are estimated with a coefficient of variation of about 0:0024 with only n = 10;000 sample values. In a genomic context, the rare events become false discoveries. There we are interested in the possibility of a large number of simultaneous events, not just one or more. Some work with Kenneth Tay will be presented on that problem.
Joint with Yury Maximov and Michael Chertkov Los Alamos National Laboratory and Kenneth Tay, Stanford
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Higher-order factorization machines (HOFMs) provide a framework for modeling feature interactions of arbitrary order in recommendation systems and link prediction tasks. The key ideas are:
(1) HOFMs express the prediction function as a weighted sum of ANOVA kernels of varying orders, capturing interactions between features.
(2) Computing the ANOVA kernel and its gradient can be done in linear time using dynamic programming, enabling efficient learning and prediction.
(3) Experiments on link prediction tasks show HOFMs can effectively model higher-order interactions to improve predictions compared to lower-order models like FM.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
A Level Set Method For Multiobjective Combinatorial Optimization Application...Scott Faria
This document proposes a new algorithm for computing all Pareto optimal solutions to multiobjective combinatorial optimization problems based on the level set method. The algorithm generates level sets in order of increasing objective function values for one objective at a time, checking if each solution is contained in the other level sets and dominates previously found solutions. It relies on the ability to find the K best solutions to a single objective combinatorial problem. The method is applied to the multiobjective quadratic assignment problem and computational results are presented.
This document summarizes a presentation on quaternionic rigid meromorphic cocycles. It discusses generalizing the construction of Darmon–Vonk classes from imaginary quadratic fields to quaternion algebras over totally real fields. This avoids using S-arithmetic groups. It describes defining cohomology classes using norm-one units of a maximal order and constructing pairings between these classes and overconvergent cohomology to obtain algebraic values of period integrals. Examples are provided to illustrate computations with this framework over various number fields and quaternion algebras.
Intro to Quant Trading Strategies (Lecture 7 of 10)Adrian Aley
This document provides an overview of constructing small mean reverting portfolios. It discusses using distance and cointegration methods to construct initial portfolios, but notes their shortcomings. It then formulates the problem as maximizing mean reversion to find sparse portfolios. Various algorithms are presented to solve this, including greedy search, least absolute shrinkage and selection operator (LASSO), and semidefinite programming (SDP) approaches. Key steps involve estimating relationships between assets, selecting subsets of assets, and optimizing portfolio weights to maximize mean reversion.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Talk at Seminari de Teoria de Nombres de Barcelona 2017mmasdeu
1) The document summarizes the construction of Darmon points, which are conjectured to behave like Heegner points for real quadratic fields K where traditional Heegner points are not available.
2) It states the main theorem of Bertolini-Darmon (2009), which proves the "rationality" of a sum of Darmon points and relates it to the rank of the elliptic curve E over K.
3) The proof strategy involves constructing a p-adic L-function for E over K, relating it to Darmon points, factorizing the p-adic L-function, and applying previous results about its factors.
An Algorithm For The Combined Distribution And Assignment ProblemAndrew Parish
This document presents an algorithm for solving the combined distribution and assignment problem using generalized Benders' decomposition. The algorithm formulates the problem as a modified distribution problem with a minimax objective function instead of a linear one. It solves this master problem using the Newton-Kantorovich method for nonlinear concave programming problems with linear constraints. The algorithm iterates between solving the assignment problem given a distribution and solving the modified distribution problem subject to optimality constraints from the assignment problem. When the solution converges, it provides the optimal traffic flows for both distribution and assignment.
STOMA FULL SLIDE (probability of IISc bangalore)2010111
This document provides an overview of a course on stochastic models and applications. It includes:
- A list of reference materials for the course.
- Background prerequisites in calculus, matrix theory, and basic probability concepts.
- Details on course grading with midterm exams, assignments, and a final exam comprising the overall grade.
- An introduction to probability theory and examples of applications in engineering, statistics, and algorithms.
- A review of basic probability concepts like sample space, events, axioms, and examples of finite and countably infinite sample spaces.
This summary provides the key details from the document in 3 sentences:
The document presents a new iterative method (M2 method) for determining the exact solution to a parametric linear programming problem where the objective function and constraints contain parameters. The M2 method exploits the concept of a p-solution to a square linear interval parametric system and iteratively reduces the parameter domain while maintaining upper and lower bounds on the optimal objective value. A numerical example is given to illustrate the new iterative approach for solving parametric linear programming problems.
Similar to Duality of a Generalized Absolute Value Optimization Problem (20)
This presentation by OECD, OECD Secretariat, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
Why Psychological Safety Matters for Software Teams - ACE 2024 - Ben Linders.pdfBen Linders
Psychological safety in teams is important; team members must feel safe and able to communicate and collaborate effectively to deliver value. It’s also necessary to build long-lasting teams since things will happen and relationships will be strained.
But, how safe is a team? How can we determine if there are any factors that make the team unsafe or have an impact on the team’s culture?
In this mini-workshop, we’ll play games for psychological safety and team culture utilizing a deck of coaching cards, The Psychological Safety Cards. We will learn how to use gamification to gain a better understanding of what’s going on in teams. Individuals share what they have learned from working in teams, what has impacted the team’s safety and culture, and what has led to positive change.
Different game formats will be played in groups in parallel. Examples are an ice-breaker to get people talking about psychological safety, a constellation where people take positions about aspects of psychological safety in their team or organization, and collaborative card games where people work together to create an environment that fosters psychological safety.
Carrer goals.pptx and their importance in real lifeartemacademy2
Career goals serve as a roadmap for individuals, guiding them toward achieving long-term professional aspirations and personal fulfillment. Establishing clear career goals enables professionals to focus their efforts on developing specific skills, gaining relevant experience, and making strategic decisions that align with their desired career trajectory. By setting both short-term and long-term objectives, individuals can systematically track their progress, make necessary adjustments, and stay motivated. Short-term goals often include acquiring new qualifications, mastering particular competencies, or securing a specific role, while long-term goals might encompass reaching executive positions, becoming industry experts, or launching entrepreneurial ventures.
Moreover, having well-defined career goals fosters a sense of purpose and direction, enhancing job satisfaction and overall productivity. It encourages continuous learning and adaptation, as professionals remain attuned to industry trends and evolving job market demands. Career goals also facilitate better time management and resource allocation, as individuals prioritize tasks and opportunities that advance their professional growth. In addition, articulating career goals can aid in networking and mentorship, as it allows individuals to communicate their aspirations clearly to potential mentors, colleagues, and employers, thereby opening doors to valuable guidance and support. Ultimately, career goals are integral to personal and professional development, driving individuals toward sustained success and fulfillment in their chosen fields.
XP 2024 presentation: A New Look to Leadershipsamililja
Presentation slides from XP2024 conference, Bolzano IT. The slides describe a new view to leadership and combines it with anthro-complexity (aka cynefin).
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This presentation by OECD, OECD Secretariat, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
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This presentation by Professor Alex Robson, Deputy Chair of Australia’s Productivity Commission, was made during the discussion “Competition and Regulation in Professions and Occupations” held at the 77th meeting of the OECD Working Party No. 2 on Competition and Regulation on 10 June 2024. More papers and presentations on the topic can be found at oe.cd/crps.
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Mastering the Concepts Tested in the Databricks Certified Data Engineer Assoc...SkillCertProExams
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This presentation by Nathaniel Lane, Associate Professor in Economics at Oxford University, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
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Collapsing Narratives: Exploring Non-Linearity • a micro report by Rosie WellsRosie Wells
Insight: In a landscape where traditional narrative structures are giving way to fragmented and non-linear forms of storytelling, there lies immense potential for creativity and exploration.
'Collapsing Narratives: Exploring Non-Linearity' is a micro report from Rosie Wells.
Rosie Wells is an Arts & Cultural Strategist uniquely positioned at the intersection of grassroots and mainstream storytelling.
Their work is focused on developing meaningful and lasting connections that can drive social change.
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This presentation by OECD, OECD Secretariat, was made during the discussion “Competition and Regulation in Professions and Occupations” held at the 77th meeting of the OECD Working Party No. 2 on Competition and Regulation on 10 June 2024. More papers and presentations on the topic can be found at oe.cd/crps.
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Suzanne Lagerweij - Influence Without Power - Why Empathy is Your Best Friend...Suzanne Lagerweij
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This session will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
Abstract:
Let’s talk about powerful conversations! We all know how to lead a constructive conversation, right? Then why is it so difficult to have those conversations with people at work, especially those in powerful positions that show resistance to change?
Learning to control and direct conversations takes understanding and practice.
We can combine our innate empathy with our analytical skills to gain a deeper understanding of complex situations at work. Join this session to learn how to prepare for difficult conversations and how to improve our agile conversations in order to be more influential without power. We will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
In the session you will experience how preparing and reflecting on your conversation can help you be more influential at work. You will learn how to communicate more effectively with the people needed to achieve positive change. You will leave with a self-revised version of a difficult conversation and a practical model to use when you get back to work.
Come learn more on how to become a real influencer!
Suzanne Lagerweij - Influence Without Power - Why Empathy is Your Best Friend...
Duality of a Generalized Absolute Value Optimization Problem
1. Duality of a Generalized Absolute Value
Optimization Problem
Shota Yamanaka* Nobuo Yamashita
Graduate School of Informatics, Kyoto University
2016/8/9
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 1 / 33
2. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 2 / 33
3. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 3 / 33
4. Absolute Value Programming (Mangasarian, 2007)
The absolute value programming (AVP) problem is written as
min ˜cT x ` ˜dT |x|
s.t. ˜Ax ` ˜B|x| “ ˜b,
˜Hx ` ˜K|x| ě ˜p,
(P0)
where ˜c, ˜d P Rn,˜b P Rk, ˜p P Rℓ, ˜A, ˜B P Rkˆn, ˜H, ˜K P Rℓˆn, and
|x| :“ p|x1|, . . . , |xn|qT
.
It is generally nonconvex and nondifferentiable.
It includes the integer optimization problem:
x P t0, 1u ô |x ´ 1| “ 0.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 4 / 33
5. Absolute Value Programming (Mangasarian, 2007)
The absolute value programming (AVP) problem is written as
min ˜cT x ` ˜dT |x|
s.t. ˜Ax ` ˜B|x| “ ˜b,
˜Hx ` ˜K|x| ě ˜p,
(P0)
where ˜c, ˜d P Rn,˜b P Rk, ˜p P Rℓ, ˜A, ˜B P Rkˆn, ˜H, ˜K P Rℓˆn, and
|x| :“ p|x1|, . . . , |xn|qT
.
The equation ˜Ax ` ˜B|x| “ ˜b, called the absolute value equation, is
equivalent to the linear complementarity problem.
The AVP includes the mathematical problem with linear
complementarity constraints.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 5 / 33
6. The dual problem of the AVP
The dual problem of pP0q is given by
max ˜bT u ` ˜pT v
s.t. | ˜AT u ` ˜H T v ´ ˜c| ` ˜B T u ` ˜K T v ď ˜d,
v ě 0.
(D0)
The above problem is always a linear programming (LP):
|x| ď 1 ô ´1 ď x ď 1.
The weak duality theorem holds between pP0q and pD0q.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 6 / 33
7. The weak duality theorem of the AVP
Theorem 1
If x P Rn and pu, vq P Rk ˆ Rℓ are feasible solutions of pP0q and pD0q,
respectively, then the following inequality holds:
˜cT
x ` ˜dT
|x| ě ˜bT
u ` ˜pT
v.
The lower bound of pP0q is obtained by solving pD0q.
pP0q is nonconvex and nondifferentiable, pD0q is always an LP.
pP0q can handle only linear and the absolute value terms.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 7 / 33
8. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 8 / 33
9. A Generalized AVP (GAVP)
A Generalized absolute value optimization problem (GAVP) is written as
min cT x ` dT Ψpxq
s.t. Ax ` BΨpxq “ b,
Hx ` KΨpxq ě p,
(P)
where c P Rn, d P Rm, b P Rk, p P Rℓ, A P Rkˆn, B P Rkˆm,
H P Rℓˆn, K P Rℓˆm, and Ψ: Rn Ñ Rm is nonlinear.
It is generally nonconvex and nondifferentiable.
The function Ψ is the generalization of the absolute value function.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 9 / 33
10. The GAVP dual problem and the assumption of Ψ
The dual problem of pPq is given by
max bT u ` pT v
s.t. Ψ˚pAT u ` HT v ´ cq ` BT u ` KT v ď d,
v ě 0.
(D)
Assumption 1
The function Ψ: Rn Ñ Rm, Ψ˚ : Rn Ñ Rm satisfies the following
conditions:
ΨpxqT Ψ˚pyq ě xT y @x P Rn, @y P Rn,
Ψpxq ě 0 @x P Rn,
Ψ˚pxq ě 0 @x P Rn.
We call this function Ψ a norm-like function.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 10 / 33
11. The weak duality theorem of the GAVP problem
Proposition 2
Suppose that Ψ is a norm-like function. Then, the following inequality
holds:
cT
x ` dT
Ψpxq ě bT
u ` pT
v
for all x P Rn and pu, vq P Rk ˆ Rℓ feasible points of (P) and (D),
respectively.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 11 / 33
12. The property of the function Ψ˚
If Ψpxq “ |x|, then Ψpxq “ Ψ˚pxq.
Let Ψ: Rn Ñ R, Ψ˚ : Rn Ñ R.
If Ψpxq “ }x}p, then Ψ˚pxq “ }x}q, where 1
p ` 1
q “ 1.
The function Ψ can be nonconvex.
We introduce some examples of norm-like function Ψ
that is equivalent to Ψ˚.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 12 / 33
13. Assumption for norm-like functions
Assumption 2
The function Ψ: Rn Ñ Rm is decomposed as follows:
Ψpxq “
»
—
–
ψ1pxI1 q
...
ψmpxIm q
fi
ffi
fl , ψi : Rni
Ñ R, i “ 1, . . . , m,
where Ii Ď t1, . . . , mu is a set of indices satisfying
Ii X Ij “ H, i ‰ j,
mÿ
i“1
|Ii| “ n,
and xIi P Rni is a disjoint subvector of x.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 13 / 33
14. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 14 / 33
15. The generalized AVP
We consider the primal and dual problems that have linear and
nonlinear terms.
min cT x ` dT Ψpxq
s.t. Ax ` BΨpxq “ b,
Hx ` KΨpxq ě p,
(P)
max bT u ` pT v
s.t. ΨpAT u ` HT v ´ cq ` BT u ` KT v ď d,
v ě 0.
(D)
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 15 / 33
16. Example of norm-like function
Ψpxq “
»
—
–
ψ1pxI1 q
...
ψmpxIm q
fi
ffi
fl , ψi : Rni
Ñ R, i “ 1, . . . , m
Proposition 3
Suppose that the function Ψ is described above and Gi P Sni
`` satisfies
minjtλjpGiqu ě 1, where λjpGiq is the j-th eigenvalue of Gi.
Then, the function Ψ given above with
ψipxIi q “
b
xIi
T GixIi , i “ 1, . . . , m,
is a norm-like function.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 16 / 33
17. Example: Conic linear programming
Let x “ px1, x2q P R ˆ Rn´1. We consider the linear second-order cone
programming (SOCP) problem as
min cT x
s.t. Ax “ b,
x1 ´ }x2}2 ě 0.
The above problem is rewritten in GAVP form as
min cT x ` 0T Ψpxq
s.t. Ax ` 0 Ψpxq “ b,
p1, 0, . . . , 0qx ´ p0, 1qΨpxq ě 0,
where Ψ: Rn Ñ R2, Ψpxq “ p|x1|, }x2}2qT .
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 17 / 33
18. Example: Conic linear programming
The dual of the previous problem is
max bT u
s.t. |pAT uq1 ` v ´ c1| ď 0,
}pAT uq2 ´ c2}2 ď v,
v ě 0.
The first constraint of the above problem is v “ c1 ´ pAT uq1. Then,
max bT u
s.t. }pAT uq2 ´ c2}2 ď c1 ´ pAT uq1,
which is the standard form of the SOCP dual problem.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 18 / 33
19. Example of norm-like function
Ψpxq “
»
—
–
ψ1pxI1 q
...
ψmpxIm q
fi
ffi
fl , ψi : Rni
Ñ R, i “ 1, . . . , m
Proposition 4
Suppose that Ψ is described above. For any θi : Rni Ñ R satisfying
θipxIi q ě }xIi }2
2, i “ 1, . . . , m,
and a positive constant αi ě 1
2 , the function Ψ with
ψipxIi q “ θipxIi q ` αi, i “ 1, . . . , m,
is a norm-like function.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 19 / 33
20. Example: Group Lasso type problem
We consider a primal problem written by
min }b ´ Ax}2
2 ` λ
mÿ
i“1
}xIi }2
where λ ě 0. If we denote z :“ b ´ Ax, then the above problem can be
rewritten as:
min }z}2
2 ` λ
mÿ
i“1
}xIi }2
s.t. rA Iksˆx “ b,
where ˆx “ px1, . . . , xn, z1, . . . , zkq P Rn`k and Ik P Rkˆk is the identity.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 20 / 33
21. Example: Group Lasso type problem
The previous problem is described in GAVP form as:
min pλ, . . . , λ, 1qΨpˆxq ´ 1
2
s.t. rA Iksˆx “ b,
where
Ψpˆxq :“
ˆ
}xI1 }2, . . . , }xIm }2, }z}2
2 `
1
2
˙T
.
Then, the dual problem is written by
max bT u ´ 1
2
s.t. ΨprA IksT uq ď pλ, . . . , λ, 1qT .
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 21 / 33
22. Example : Group Lasso type problem
The dual problem is rewritten as
max bT u ´ 1
2
s.t. }pAT qIi u}2 ď λ, i “ 1, . . . , m,
}u}2
2 ď 1
2 .
where pAT qIi is a submatrix of AT with pAT qj, j P Ii as its rows.
The primal problem is written by
min }b ´ Ax}2
2 ` λ
mÿ
i“1
}xIi }2
where λ ě 0.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 22 / 33
23. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 23 / 33
24. GAVP problem with only nonlinear terms
We consider the GAVP primal problem (P) that has no linear terms in its
objective and constraint functions.
min dT Ψpxq
s.t. BΨpxq “ b,
KΨpxq ě p.
(P1)
And, the dual problem of pP1q is given by
max bT u ` pT v
s.t. BT u ` KT v ď d,
v ě 0.
(D1)
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 24 / 33
25. Assumption for the function Ψ
Assumption 3
The function Ψ: Rn Ñ Rm satisfies the following conditions:
ΨpxqT Ψpyq ě xT y @x P F, @y P F,
Ψpxq ě 0 @x P F,
where F is a closed set. We also call this function Ψ a norm-like function.
If we set F :“ tx | x ě 0u, then Ψpxq :“ px1, x2, x2
1, x2
2, x1x2qT is a
norm-like function.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 25 / 33
26. Example: The quadratic optimization problem
We consider the nonconvex quadratic optimization problem:
min x2
1 ´ x2
2
s.t. x2
1 ` x2
2 ď 4
x2
1 ` px2 ´ 1q2 ě 1
x ě 0.
The optimal value is ´4 at x˚ “ p0, 2q.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 26 / 33
27. Example: The quadratic optimization problem
We obtain the GAVP dual problem as
max ´4v1
s.t. v3 ď 0
´2v2 ` v4 ď 0
´v1 ` v2 ď 1
´v1 ` v2 ď ´1
v ě 0,
which is just a linear programming.
The optimal value is ´4 at v˚ “ p1, 0, 0, 0q.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 27 / 33
28. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 28 / 33
29. The dual optimal value
Theorem 5
Suppose that the GAVP dual problem (D) has an optimal solution. Then,
we have
f˚
DL
ě f˚
D
where f˚
DL
and f˚
D are the optimal values of the Lagrangean dual problem
(DL) and the GAVP dual problem (D), respectively.
The GAVP dual gives the lower bound of the Lagrangean dual.
However, there exists a closed form for GAVP dual problem.
We consider conditions under which the GAVP and Lagrangean dual
problems are equivalent.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 29 / 33
30. Assumption for the equivalence
Ψpxq “
»
—
–
ψ1pxI1 q
...
ψmpxIm q
fi
ffi
fl , ψi : Rni
Ñ R, i “ 1, . . . , m
Assumption 4
For any x P Rn, the function ψi : Rni Ñ R satisfies the following
conditions:
1 ψipαxIi q ď αψipxIi q, α ą 0, i “ 1, . . . , m,
2 }xIi }2
2 ě ψipxIi q2, i “ 1, . . . , m,
3 xIi ‰ 0 ñ ψipxIi q ą 0, i “ 1, . . . , m.
For example, the absolute value and the ℓ2 norm functions satisfy the
above conditions.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 30 / 33
31. Sufficient conditions for the equivalence
Theorem 6
Suppose that the Lagrangean dual problem (DL) has a feasible solution,
which is p¯u, ¯vq P Rk ˆ Rℓ, and there exist x˚ P Rn satisfying
pd ´ BT
¯u ´ KT
¯vqT
Ψpx˚
q ´ pAT
¯u ` HT
¯v ´ cqT
x˚
“ 0.
Then, the GAVP dual problem (D) is equivalent to the Lagrangean dual
problem pDLq.
For example, the absolute value function and ℓ2 norm function satisfy
the above equation at x˚ “ 0.
If Ψ and Ψ˚ are a norm function and its dual norm function
respectively, then the GAVP dual problem is equivalent to the
Lagrangean dual problem.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 31 / 33
32. 1 Absolute Value Programming Problem (AVP)
2 A Generalized Absolute Value Optimization Problem (GAVP)
3 Examples with linear and nonlinear terms
4 Examples with only nonlinear terms
5 The Lagrangean duality and GAVP duality
6 Conclusion
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 32 / 33
33. Conclusion
We proposed the generalized absolute value optimization problem
(GAVP), and proved the weak duality theorem for GAVP.
We presented some examples of GAVP problems.
The relation between GAVP duality and the Lagrangean duality were
discussed.
We showed sufficient conditions under which the Lagrangean dual
and GAVP dual problems are equivalent.
As future works, we will investigate norm-like functions Ψ (Ψ˚), and
the relation between GAVP and the Lagrangean dualities.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 33 / 33
34. Example : The quadratic function
Let Mi P Sn
``, and λminpMiq ě 1, where λminpMiq is the minimum
eigenvalue of Mi. Then, the quadratic function given by
ψipxIi q “ xT
Ii
MixIi `
1
2
satisfies the conditions of the previous proposition.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 1 / 4
35. Example: The quadratic function
Let m “ 1, and denote xI1 , Mi as x, M respectively. The primal and dual
problems can be written as
min cT x ` dpxT Mx ` 1
2 q
s.t. AT x ` BpxT Mx ` 1
2 q “ b,
HT x ` KpxT Mx ` 1
2 q ě p,
max bu ` pv
s.t. pAu ` Hv ´ cqT MpAu ` Hv ´ cq ` 1
2 ` Bu ` Kv ď d,
v ě 0.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 2 / 4
36. Example: The sine function
Suppose that Ii “ tiu. Then, the nonconvex function given by
ψipxIi q “ x2
Ii
` sinpxIi q `
3
2
satisfies the conditions of Proposition 4. Notice that we set
θipxIi q “ x2
Ii
` sinpxIi q ` 1,
αi “
1
2
,
respectively.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 3 / 4
37. Example: The sine function
For n “ m “ 1, we denote xIi “ x P R, and we have the primal and dual
problems as
min cx ` dpx2 ` sinpxq ` 3
2 q
s.t. Ax ` Bpx2 ` sinpxq ` 3
2 q “ b,
Hx ` Kpx2 ` sinpxq ` 3
2 q ě p,
max bu ` pv
s.t. pAu ` Hv ´ cq2 ` sinpAu ` Hv ´ cq ` 3
2 ` Bu ` Kv ď d,
v ě 0.
S. Yamanaka*, N. Yamashita Generalized AVP 2016/8/9 4 / 4