This document presents an outer approximation solution algorithm for solving reliable shortest path problems on transportation networks. The algorithm formulates the problem as a mixed integer conic quadratic program to minimize the mean plus standard deviation of path costs. It then uses an outer approximation approach to decompose the problem and solve it efficiently through alternating steps of solving a master problem and subproblem. Computational results on several test networks show the algorithm converges quickly and outperforms directly solving the large-scale mixed integer conic quadratic program. The approach can also be applied to other reliability metrics and joint inventory location problems.
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
Spline interpolation is a problem of "Numerical Methods".
This slide covers the basics of spline interpolation mostly the linear spline and cubic spline interpolation.
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
Fuzzy clustering algorithm can not obtain good clustering effect when the sample characteristic is not obvious and need to determine the number of clusters firstly. For thi0s reason, this paper proposes an adaptive fuzzy kernel clustering algorithm. The algorithm firstly use the adaptive function of clustering number to calculate the optimal clustering number, then the samples of input space is mapped to highdimensional feature space using gaussian kernel and clustering in the feature space. The Matlab simulation results confirmed that the algorithm's performance has greatly improvement than classical clustering algorithm and has faster convergence speed and more accurate clustering results.
Fuzzy clustering algorithm can not obtain good clustering effect when the sample characteristic is not
obvious and need to determine the number of clusters firstly. For thi0s reason, this paper proposes an
adaptive fuzzy kernel clustering algorithm. The algorithm firstly use the adaptive function of clustering
number to calculate the optimal clustering number, then the samples of input space is mapped to highdimensional
feature space using gaussian kernel and clustering in the feature space. The Matlab simulation
results confirmed that the algorithm's performance has greatly improvement than classical clustering algorithm and has faster convergence speed and more accurate clustering results
Gaps between the theory and practice of large-scale matrix-based network comp...David Gleich
I discuss some runtimes for the personalized PageRank vector and how it relates to open questions in how we should tackle these network based measures via matrix computations.
Cycle’s topological optimizations and the iterative decoding problem on gener...Usatyuk Vasiliy
We consider several problem related to graph model related to error-correcting codes. From base problem of cycle broken, trapping set elliminating and bypass to fundamental problem of graph model. Thanks to the hard work of Michail Chertkov, Michail Stepanov and Andrea Montanari which inspirit me...
Slides presented at Applied Mathematics Day, Steklov Mathematical Institute of the Russian Academy of Sciences September 22, 2017 http://www.mathnet.ru/conf1249
Sampling-Based Planning Algorithms for Multi-Objective MissionsMd Mahbubur Rahman
multiobjective path planning has Increasing demand in military missions, rescue operations, construction job-sites.
There is Lack of robotic path planning algorithm that compromises multiple
objectives. Commonly no solution that optimizes all the objective functions. Here we modify RRT, RRT* sampling based algorithm.
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques S...SAJJAD KHUDHUR ABBAS
Episode 50 : Simulation Problem Solution Approaches Convergence Techniques Simulation Strategies
3.2.3.3. Quasi-Newton (QN) Methods
These methods represent a very important class of techniques because of their extensive use in practical alqorithms. They attempt to use an approximation to the Jacobian and then update this at each step thus reducing the overall computational work.
The QN method uses an approximation Hk to the true Jacobian i and computes the step via a Newton-like iteration. That is,
SAJJAD KHUDHUR ABBAS
Ceo , Founder & Head of SHacademy
Chemical Engineering , Al-Muthanna University, Iraq
Oil & Gas Safety and Health Professional – OSHACADEMY
Trainer of Trainers (TOT) - Canadian Center of Human
Development
The MSc defense ceremony was held on 6-7-2017 in Mansoura University, Faculty of Engineering. This presentation is shared to help MSc students in Faculty of Engineering prepare their thesis presentation and ease their tension before their presentation time
Exact network reconstruction from consensus signals and one eigen valueIJCNCJournal
The basic inverse problem in spectral graph theory consists in determining the graph given its eigenvalue
spectrum. In this paper, we are interested in a network of technological agents whose graph is unknown,
communicating by means of a consensus protocol. Recently, the use of artificial noise added to consensus
signals has been proposed to reconstruct the unknown graph, although errors are possible. On the other
hand, some methodologies have been devised to estimate the eigenvalue spectrum, but noise could interfere
with the elaborations. We combine these two techniques in order to simplify calculations and avoid
topological reconstruction errors, using only one eigenvalue. Moreover, we use an high frequency noise to
reconstruct the network, thus it is easy to filter the control signals after the graph identification. Numerical
simulations of several topologies show an exact and robust reconstruction of the graphs.
2. Why Reliability is important in Shortest Path Problems
Reliable Shortest Path Formulation
Outer Approximation Solution Algorithm
Computational Study
Conclusions and Other Applications
3. s t
35
45
STD DEV
MEAN TRAVEL COST
, 5
, 20
ORIGIN DESTINATION
Reliable and risk averse routing suggest taking
the path with least mean+std cost
TC=55
TC=50
5. Directed Graph G=(V,E)
Origin Node r, Destination Node s
aij is the link between node i and node j
cov(aij, alk) is the covariance matrix
Given
Assumptions
Link Travel Cost Distribution is Available
One path with finite variance
8. 𝝈𝒊𝒋
𝟐
𝒙𝒊𝒋
𝒓𝒔 𝟐
+ 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒊𝒋
𝒓𝒔
𝒙𝒍𝒌
𝒓𝒔
𝒂 𝒍𝒌 ∈ 𝑬
𝒍,𝒌 ≠ 𝒊,𝒋
𝒂 𝒊𝒋 ∈ 𝑬𝒂 𝒊𝒋 ∈ 𝑬
≤ 𝒕 𝒓𝒔
Conic Formulation inefficient for Large Networks
Consider a network with 1000 links
1 million quadratic integer terms
9. CQP computationally tractable due to their special Structure
Can be solved by polynomial time interior point algorithms
Solvers like CPLEX and MOSEK offering this capability
X2 + Y2 <= Z2
.
Minimize
Subject to
ii
T
iiii
dxcbxa
i
T
i
xf
10. Outer Approximation (OA)-Linearization and Decomposition
Method
Delivers optimal solution for convex MINLP (Bonami et al.
2008)
ZyRx
yxg
yxf
yx
,
0),(toSubject
Minimize ),(
,
f and g are convex function
Decompose integer and nonlinear part
Solve alternating sequence of Master Problem
(MILP) & Subproblem (NLP)
17. Input: Convergence tolerance ε, Maximum Number of
Iteration H, Upperbound(UB)=+inf, Lowerbound (LB) =-
inf, h=1
Initialization: Solve One-all Shortest path problem
1: If (UB-LB) <=ε or then go to step 6
2: Solve the SP (Closed Form Equations)
3: If ( SP<UB), Update the UP and the best point
4: Add the OA inequalities and solve the MP
(No need to solve MP to optimality, Fletcher and Leyffer; 1994)
5: Update the LB, h=h+1, go to step 1
6:Report the Solutions and Algorithm Stops
18. Networks Number of Links(A) Number of Nodes (N)
SiouxFalls 76 24
Anaheim 914 416
Barcelona 2522 1020
Chicago Sketch 2250 933
Experiments Set up
OA-is coded in GAMS-C++/CPLEX
MILP MP is solved via CPLEX solver
MP is solved within 1% optimality gap
Covariance matrix is randomly generated
20. 0
20
40
60
80
100
120
140
160
180
0.5 0.6 0.8 1 1.5 2
RunningTime(Sec)
Master Problem Relative Optimality Gap(%)
Anaheim Sioux Falls Chicago Sketch Barcelona
Solution time is stable w.r.t. MP optimality gap
21. 0
50
100
150
200
250
0 0.2 0.4 0.6 0.8 1
RunningTime(Sec)
Weight Associated with Mean Tavel Cost
Sioux Falls Anaheim Chicago Sketch Barcelona
Increase in weight on STD term increases the
solution time for larger networks
23. General Reliability Measures
[Shahabi, Unnikrishnan, and Boyles, 2014, 2015] show
application of OA to other forms of convex reliability metrics
Joint Inventory Location Problems
[Shahabi, Unnikrishnan, and Boyles, 2014] show that OA is
efficient in solving multi-echelon joint inventory location
problems with correlations in retailer demand
24. •A general efficient methodology for binary convex
mixed integer programs was presented
•Given the convexity assumption the method
guarantees the global optimality
•The sub-problem is analytically solved
•Minimizes the impact of large correlation matrices
and lead to savings in computational time
Editor's Notes
There are different ways to characterize reliability and risk taking behavior…One such metric is mean-std dev..this ppt is mainly going to focus on mean+std dev shortest path..
CQP computationally tractable due to their special Structure
Can be solved by polynomial time interior point algorithms
Solvers like CPLEX and MOSEK offering this capability