Towards a practical simplex method for second order cone programming
|
|
- Anthony Patterson
- 5 years ago
- Views:
Transcription
1 Towards a practical simplex method for second order cone programming Kartik Krishnan Department of Computing and Software McMaster University Joint work with Gábor Pataki (UNC), Neha Gupta (IIT Delhi), and Tamás Terlaky (Mac) INFORMS Annual Meeting Denver Oct 26, 2004
2 Overview Second Order Cone Programming (SOCP) Contrast simplex-like approaches and IPMs for conic optimization The geometry of SOCP Simplex algorithm for SOCP Special case: Simplex method for LP Properties of the algorithm Preliminary computational results Conclusions and future work 1
3 Second order cone optimization Primal min c T 1 x 1 + c T 2 x c T r x r A 1 x 1 + A 2 x A r x r = b, (SOCP) x i K i. Dual max b T y A T i y + s i = c i, i = 1,..., r, s i K i. (SOCD) Notation A = (A 1, A 2,..., A r ) R m n with full row rank. K = K 1... K r. Each K i = { x R n i : x 1 of size n i, i = 1,..., r. x 2:ni } is a second order cone 2
4 Optimality conditions (x, y, s ) are optimal iff Ax = b, x K, A T y + s = c, s K, x i s i = 0, i = 1,..., r. (CS) (PF) (DF) For an SOCP cone in R n x s = ( x T s x(1)s(2 : n) + s(1)x(2 : n) ). For any cone K x K, s K, x T s = 0 x s = 0. 3
5 Eigenvalues and Eigenvectors Given x R n, we have ( x = 1 2 ( x 1 x 2:n ) 1; x ) 2:n + 1 x 2:n 2 ( x 1 + x 2:n ) = λ min ( x)v min ( x) + λ max ( x)v max ( x). Index classification: Given x i K i we have i O (zero blocks) if λ max ( x i ) = 0. i R (boundary blocks) if λ min ( x i ) = 0. i I (interior blocks) if λ min ( x i ) > 0. ( 1; ) x 2:n x 2:n 4
6 Contrast simplex and IPMs for CP I. Start Extreme Points Start Central Path Our Algorithm Simplex Method Finish Interior Point Method Finish Feasible Direction Method Interior Point Methods deal with matrices of full rank. In the simplex method, the rank of the extreme points satisfy (1) r <= m (Linear Programming) (2) r(r+1)/2 <= m (Semidefinite Programming) 5
7 Contrast simplex and IPMs for CP II. Why a simplex method for conic programming?. 1. Listed as an important open problem in conic programming. 2. Warm start after branching or the addition of cutting planes using the dual simplex method. 3. It is possible to do every simplex iteration more quickly than an IPM iteration using fast basis LU updates to factorize the basis matrix. 6
8 Terminology I. Given a closed convex cone K R n. Consider x K Lineality space: B x = {d R n : x ± ǫd K, ǫ > 0}. Tangent space: T x = { d R n : dist( x ± ǫd, K) = O(ǫ 2 ), ǫ > 0 }. Residual space: R x = T x \ B x. 7
9 Terminology II. Cone of feasible directions: Tangent cone: dir( x, K) = (K + B x ) = {d R n : x + ǫd K, ǫ > 0}. TC( x, K) = cl(dir( x, K)) = { d R n : dist( x + ǫd, K) = O(ǫ 2 ), ǫ > 0 }. Null space: N = {x R n : Ax = 0}. 8
10 Demo Given x i R n i on the boundary of the SOCP cone K i. B xi = α x i for α R. T xi = lin ( B xi ( 0 {( 0 w ) : w T x 2:ni = 0 Note x i + ǫ / K w i. ( ) ( 0 However x i + ǫ + ǫ w ) ) }). int(k i ). 9
11 Notions of nondegeneracy A feasible x in (SOCP) is c-nondegenerate This is a generic property. T x + N = R n. f-nondegenerate B x + N = R n. Extreme point B x N = {0}. 10
12 Simplex algorithm for SOCP I. Given a feasible c-nondegenerate x = ( x I ; x R ; x O ) in (SOCP). Indices i I and j R are ranked on λ min ( x i ) and x j1 respectively. 1. Select basis: Construct the following basis elements: Decompose x = M B x B + M N x N, where M B R n m and M N R n (n m). M B is chosen with Range(M B ) T x such that A B = AM B is a nonsingular basis matrix of size m. Also, c B = M T B c. Let B,N be the index set of cones K i,i = 1,..., r with columns in M B and M N respectively. (Note: B N φ). 2. Construct dual solution: Solve A T B y = c B for ȳ. Compute s i = c i A T i ȳi, i N. (Note: s i = 0, i / N). 11
13 Simplex algorithm for SOCP II. 3. Pricing: Compute α = max{ λ min ( s i ) : s i / K i, i N}. β = max{ x T i s i : s i K i, i N}. If α = β = 0 STOP; ( x, ȳ, s) is an optimal solution. Else k = { an index s.t. α = λmin ( s k ) if α > β. an index s.t. β = x T k s k if α < β. 4. Find improving direction: Improving component: Compute d N R n where d Ni = min d =1 d TC( x k,k ) st k d k (IC) if i = k 0 for i = 1,..., r and i k. (Note: dir( x k, K k ) is not CLOSED!) 12
14 Simplex algorithm for SOCP III. 4. Find improving direction (continued): Centering component: Construct d C R n satisfying d Ci = (α;0) i R B, a nonzero subset of R xi Range(A B ) (α;0) i = k if k R 0 otherwise with α > 0 chosen so that s T k ( d N + d C ) < 0. Basic component: Compute d B R m by solving A B d B = (Ad N + Ad C ). The improving direction d = (M B d B + d C + d N ). 13
15 Simplex algorithm for SOCP IV. 5. Line search: Compute ᾱ where ᾱ = max{α i : x i + α i d i K i, i = 1,..., r}. If ᾱ = the primal is unbounded. STOP. Else set x = x + ᾱ d and return to step 1. (Note: The new x is assumed to be c-nondegenerate). 14
16 Computing the improving component I. The solution d Nk to (IC) is If k O: d Nk = s k s k if λ max ( s k ) < 0 v min ( s k ) if λ min ( s k ) < 0 and λ max ( s k ) 0 0 if s k K k If k I: d Nk = s k s k if s k 0 0 if s k = 0 15
17 Computing the improving component II. If k R: We have TC( x k, K k ) = {d R n k : x 1 d 1 n k j=2 x jd j 0} In this case d Nk is d Nk = 0 if s k = ( x 1 ; x 2:nk ) min d =1, d TC( x k,k ) st k d otherwise k 16
18 Special case: Simplex method for LP 1. The initial iterate is an nondegenerate extreme point solution. If this iterate is not an extreme point, the method is instead a feasible direction method. 2. Given a non-degenerate extreme point, the simplex method chooses the basis matrix as A B = A(:,support( x)). This is nonsingular with Range(A B ) = B x. 3. For the improving direction d N = e k (k is the index for which s i is the most negative) and d B the solution to A B d B = A k. No centering term d C is needed. The resulting direction d is along an edge of the feasible set. 17
19 Properties of the algorithm Theorem. Let {(x k, y k, s k )} be a sequence generated by the algorithm. Then for all k, x k is primal feasible. At the k-th iteration, one of the following alternative cases arises: 1. If it stops in Step 2 then x k, (y k, s k ) are primal and dual optimal solutions to (SOCP) and (SOCD) respectively. 2. If it stops in Step 5, then (SOCP) is unbounded and (SOCD) is infeasible. 3. Otherwise, if x k is also c-nondegenerate we have c T x k+1 < c T x k. 18
20 Convergence of the algorithm Some practical issues affecting convergence include: One also adds a centering term d Ci to the cones i I which are very nearly in R. This is to prevent the algorithm from getting jammed at a suboptimal point. There is also the issue of zigzagging with the algorithm. Conjecture. If {x k } contains a nondegenerate subsequence, then the algorithm with the anti-jamming safeguard either terminates in a finite number of steps, or all accumulation points of {x k } are optimal solutions to (SOCP). 19
21 Comments on the simplex algorithm The simplex algorithm for SOCP is a feasible direction method that generates search directions in the tangent cone. This direction is suitably centered so as to generate a feasible direction. The simplex iterates are not always extreme points and the search directions may traverse the interior of the feasible region. The method resembles the convex simplex method of Zangwill for minimizing a convex function over a polyhedron. The algorithm maintains primal feasibility in every iteration while dual feasibility and complementary slackness are attained at optimality. The step length calculation in Step 5 has a closed analytic expression. 20
22 Preliminary computational results Prob m n lp r Opt Obj(0) Obj(iter) Iter hs [12] (*) 10 slp (3) slp (3) slp [50 10] (*) 183 slp (3) (*) 896 slp (3) (*) 153 slp (5) slp (5) slp [10] (*) 157 slp [10] (*) 128 slp [ ] slp [ ]
23 Conclusions and future work 1. A primal simplex approach for conic optimization of which the primal simplex method for LP is a special case. 2. The simplex approach exploits the well known facial structure of SOCP problems developed in Pataki (2000). 3. We have the framework for solving conic optimization problems over LP, SOCP and SDP cones. 4. We also have a dual simplex variant which mimics the dual simplex method for LP. 5. Currently investigating fast basis inverse (LU) updates to speed up the algorithm. 6. Future use in warm start after branching or the addition of cutting planes. 22
24 References F. Alizadeh and D. Goldfarb. Second-order cone programming, Mathematical Programming, 95(2003), pp F. Alizadeh, J.P.A. Haeberly, and M.L. Overton. Complementarity and Nondegeneracy in SDP, Mathematical Programming, 77(1997), pp M.S. Bazaraa, H.D. Sherali and C.M. Shetty. Nonlinear Programming: Theory and Algorithms, 2nd edition, John Wiley K. Krishnan, G. Pataki, N. Gupta and T. Terlaky. A feasible direction method for SOCP, forthcoming. 23
25 I. Maros. Computational techniques of the simplex method. Kluwer International Series, G. Pataki. Cone-LP s and Semidefinite Programs: Geometry and a Simplex-type Method, Proceedings of the 5th IPCO conference, Springer Verlag G. Pataki. The Geometry of Semidefinite Programming, Handbook of SDP edited by H. Wolkowicz et al., Kluwer Academic Publishers, 2000, pp
Linear programming and duality theory
Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables
More informationDavid G. Luenberger Yinyu Ye. Linear and Nonlinear. Programming. Fourth Edition. ö Springer
David G. Luenberger Yinyu Ye Linear and Nonlinear Programming Fourth Edition ö Springer Contents 1 Introduction 1 1.1 Optimization 1 1.2 Types of Problems 2 1.3 Size of Problems 5 1.4 Iterative Algorithms
More informationResearch Interests Optimization:
Mitchell: Research interests 1 Research Interests Optimization: looking for the best solution from among a number of candidates. Prototypical optimization problem: min f(x) subject to g(x) 0 x X IR n Here,
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More informationIntroduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs
Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling
More informationA primal-dual Dikin affine scaling method for symmetric conic optimization
A primal-dual Dikin affine scaling method for symmetric conic optimization Ali Mohammad-Nezhad Tamás Terlaky Department of Industrial and Systems Engineering Lehigh University July 15, 2015 A primal-dual
More informationOutline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014
5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationMath 5593 Linear Programming Lecture Notes
Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................
More informationA PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS
Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 123-132 DOI:10.2298/YUJOR0901123S A PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS Nikolaos SAMARAS Angelo SIFELARAS
More informationSome Advanced Topics in Linear Programming
Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,
More informationLinear Optimization. Andongwisye John. November 17, Linkoping University. Andongwisye John (Linkoping University) November 17, / 25
Linear Optimization Andongwisye John Linkoping University November 17, 2016 Andongwisye John (Linkoping University) November 17, 2016 1 / 25 Overview 1 Egdes, One-Dimensional Faces, Adjacency of Extreme
More informationLinear Programming. Course review MS-E2140. v. 1.1
Linear Programming MS-E2140 Course review v. 1.1 Course structure Modeling techniques Linear programming theory and the Simplex method Duality theory Dual Simplex algorithm and sensitivity analysis Integer
More informationAdvanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material
More informationContents. I Basics 1. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.
page v Preface xiii I Basics 1 1 Optimization Models 3 1.1 Introduction... 3 1.2 Optimization: An Informal Introduction... 4 1.3 Linear Equations... 7 1.4 Linear Optimization... 10 Exercises... 12 1.5
More informationLecture Notes 2: The Simplex Algorithm
Algorithmic Methods 25/10/2010 Lecture Notes 2: The Simplex Algorithm Professor: Yossi Azar Scribe:Kiril Solovey 1 Introduction In this lecture we will present the Simplex algorithm, finish some unresolved
More informationRead: H&L chapters 1-6
Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Fall 2006 (Oct 16): Midterm Review http://www-scf.usc.edu/~ise330
More informationLinear Optimization and Extensions: Theory and Algorithms
AT&T Linear Optimization and Extensions: Theory and Algorithms Shu-Cherng Fang North Carolina State University Sarai Puthenpura AT&T Bell Labs Prentice Hall, Englewood Cliffs, New Jersey 07632 Contents
More informationSection Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017
Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures
More informationOpen problems in convex geometry
Open problems in convex geometry 10 March 2017, Monash University Seminar talk Vera Roshchina, RMIT University Based on joint work with Tian Sang (RMIT University), Levent Tunçel (University of Waterloo)
More informationThe Ascendance of the Dual Simplex Method: A Geometric View
The Ascendance of the Dual Simplex Method: A Geometric View Robert Fourer 4er@ampl.com AMPL Optimization Inc. www.ampl.com +1 773-336-AMPL U.S.-Mexico Workshop on Optimization and Its Applications Huatulco
More informationDual-fitting analysis of Greedy for Set Cover
Dual-fitting analysis of Greedy for Set Cover We showed earlier that the greedy algorithm for set cover gives a H n approximation We will show that greedy produces a solution of cost at most H n OPT LP
More informationCS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm Instructor: Shaddin Dughmi Algorithms for Convex Optimization We will look at 2 algorithms in detail: Simplex and Ellipsoid.
More informationTHEORY OF LINEAR AND INTEGER PROGRAMMING
THEORY OF LINEAR AND INTEGER PROGRAMMING ALEXANDER SCHRIJVER Centrum voor Wiskunde en Informatica, Amsterdam A Wiley-Inter science Publication JOHN WILEY & SONS^ Chichester New York Weinheim Brisbane Singapore
More informationLinear Programming in Small Dimensions
Linear Programming in Small Dimensions Lekcija 7 sergio.cabello@fmf.uni-lj.si FMF Univerza v Ljubljani Edited from slides by Antoine Vigneron Outline linear programming, motivation and definition one dimensional
More informationINTRODUCTION TO LINEAR AND NONLINEAR PROGRAMMING
INTRODUCTION TO LINEAR AND NONLINEAR PROGRAMMING DAVID G. LUENBERGER Stanford University TT ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California London Don Mills, Ontario CONTENTS
More informationLecture 2. Topology of Sets in R n. August 27, 2008
Lecture 2 Topology of Sets in R n August 27, 2008 Outline Vectors, Matrices, Norms, Convergence Open and Closed Sets Special Sets: Subspace, Affine Set, Cone, Convex Set Special Convex Sets: Hyperplane,
More informationOpen problems in convex optimisation
Open problems in convex optimisation 26 30 June 2017 AMSI Optimise Vera Roshchina RMIT University and Federation University Australia Perceptron algorithm and its complexity Find an x R n such that a T
More informationDetecting Infeasibility in Infeasible-Interior-Point. Methods for Optimization
FOCM 02 Infeasible Interior Point Methods 1 Detecting Infeasibility in Infeasible-Interior-Point Methods for Optimization Slide 1 Michael J. Todd, School of Operations Research and Industrial Engineering,
More informationProperties of a Cutting Plane Method for Semidefinite Programming 1
Properties of a Cutting Plane Method for Semidefinite Programming 1 Kartik Krishnan Sivaramakrishnan Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 kksivara@ncsu.edu http://www4.ncsu.edu/
More informationGraphs that have the feasible bases of a given linear
Algorithmic Operations Research Vol.1 (2006) 46 51 Simplex Adjacency Graphs in Linear Optimization Gerard Sierksma and Gert A. Tijssen University of Groningen, Faculty of Economics, P.O. Box 800, 9700
More informationProgramming, numerics and optimization
Programming, numerics and optimization Lecture C-4: Constrained optimization Łukasz Jankowski ljank@ippt.pan.pl Institute of Fundamental Technological Research Room 4.32, Phone +22.8261281 ext. 428 June
More informationConic Duality. yyye
Conic Linear Optimization and Appl. MS&E314 Lecture Note #02 1 Conic Duality Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/
More informationLesson 17. Geometry and Algebra of Corner Points
SA305 Linear Programming Spring 2016 Asst. Prof. Nelson Uhan 0 Warm up Lesson 17. Geometry and Algebra of Corner Points Example 1. Consider the system of equations 3 + 7x 3 = 17 + 5 = 1 2 + 11x 3 = 24
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino e-mail: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More informationCivil Engineering Systems Analysis Lecture XIV. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics
Civil Engineering Systems Analysis Lecture XIV Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Dual 2 Linear Programming Dual Problem 3
More informationExtensions of Semidefinite Coordinate Direction Algorithm. for Detecting Necessary Constraints to Unbounded Regions
Extensions of Semidefinite Coordinate Direction Algorithm for Detecting Necessary Constraints to Unbounded Regions Susan Perrone Department of Mathematics and Statistics Northern Arizona University, Flagstaff,
More information4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
4 LINEAR PROGRAMMING (LP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 Mathematical programming (optimization) problem: min f (x) s.t. x X R n set of feasible solutions with linear objective function
More informationConvex Sets. CSCI5254: Convex Optimization & Its Applications. subspaces, affine sets, and convex sets. operations that preserve convexity
CSCI5254: Convex Optimization & Its Applications Convex Sets subspaces, affine sets, and convex sets operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More information4.1 Graphical solution of a linear program and standard form
4.1 Graphical solution of a linear program and standard form Consider the problem min c T x Ax b x where x = ( x1 x ) ( 16, c = 5 ), b = 4 5 9, A = 1 7 1 5 1. Solve the problem graphically and determine
More informationStability of closedness of convex cones under linear mappings
Stability of closedness of convex cones under linear mappings Jonathan M. Borwein and Warren B. Moors 1 Abstract. In this paper we reconsider the question of when the continuous linear image of a closed
More informationLARGE SCALE LINEAR AND INTEGER OPTIMIZATION: A UNIFIED APPROACH
LARGE SCALE LINEAR AND INTEGER OPTIMIZATION: A UNIFIED APPROACH Richard Kipp Martin Graduate School of Business University of Chicago % Kluwer Academic Publishers Boston/Dordrecht/London CONTENTS Preface
More informationPart 4. Decomposition Algorithms Dantzig-Wolf Decomposition Algorithm
In the name of God Part 4. 4.1. Dantzig-Wolf Decomposition Algorithm Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Introduction Real world linear programs having thousands of rows and columns.
More informationPrograms. Introduction
16 Interior Point I: Linear Programs Lab Objective: For decades after its invention, the Simplex algorithm was the only competitive method for linear programming. The past 30 years, however, have seen
More informationCS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension
CS 372: Computational Geometry Lecture 10 Linear Programming in Fixed Dimension Antoine Vigneron King Abdullah University of Science and Technology November 7, 2012 Antoine Vigneron (KAUST) CS 372 Lecture
More informationMathematical and Algorithmic Foundations Linear Programming and Matchings
Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis
More informationAMS : Combinatorial Optimization Homework Problems - Week V
AMS 553.766: Combinatorial Optimization Homework Problems - Week V For the following problems, A R m n will be m n matrices, and b R m. An affine subspace is the set of solutions to a a system of linear
More informationMarginal and Sensitivity Analyses
8.1 Marginal and Sensitivity Analyses Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor, Winter 1997. Consider LP in standard form: min z = cx, subject to Ax = b, x 0 where A m n and rank m. Theorem:
More informationConic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding
Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding B. O Donoghue E. Chu N. Parikh S. Boyd Convex Optimization and Beyond, Edinburgh, 11/6/2104 1 Outline Cone programming Homogeneous
More informationLinear Programming. Larry Blume. Cornell University & The Santa Fe Institute & IHS
Linear Programming Larry Blume Cornell University & The Santa Fe Institute & IHS Linear Programs The general linear program is a constrained optimization problem where objectives and constraints are all
More informationBilinear Programming
Bilinear Programming Artyom G. Nahapetyan Center for Applied Optimization Industrial and Systems Engineering Department University of Florida Gainesville, Florida 32611-6595 Email address: artyom@ufl.edu
More informationConvex Optimization M2
Convex Optimization M2 Lecture 1 A. d Aspremont. Convex Optimization M2. 1/49 Today Convex optimization: introduction Course organization and other gory details... Convex sets, basic definitions. A. d
More information2. Convex sets. x 1. x 2. affine set: contains the line through any two distinct points in the set
2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More informationAone-phaseinteriorpointmethodfornonconvexoptimization
Aone-phaseinteriorpointmethodfornonconvexoptimization Oliver Hinder, Yinyu Ye Department of Management Science and Engineering Stanford University April 28, 2018 Sections 1 Motivation 2 Why a one-phase
More informationMath Models of OR: The Simplex Algorithm: Practical Considerations
Math Models of OR: The Simplex Algorithm: Practical Considerations John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Simplex Algorithm: Practical Considerations
More informationA Generic Separation Algorithm and Its Application to the Vehicle Routing Problem
A Generic Separation Algorithm and Its Application to the Vehicle Routing Problem Presented by: Ted Ralphs Joint work with: Leo Kopman Les Trotter Bill Pulleyblank 1 Outline of Talk Introduction Description
More informationDiscrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity
Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity Marc Uetz University of Twente m.uetz@utwente.nl Lecture 5: sheet 1 / 26 Marc Uetz Discrete Optimization Outline 1 Min-Cost Flows
More informationDEGENERACY AND THE FUNDAMENTAL THEOREM
DEGENERACY AND THE FUNDAMENTAL THEOREM The Standard Simplex Method in Matrix Notation: we start with the standard form of the linear program in matrix notation: (SLP) m n we assume (SLP) is feasible, and
More informationConvex Optimization. Convex Sets. ENSAE: Optimisation 1/24
Convex Optimization Convex Sets ENSAE: Optimisation 1/24 Today affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 6
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 6 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 19, 2012 Andre Tkacenko
More informationLecture 5: Duality Theory
Lecture 5: Duality Theory Rajat Mittal IIT Kanpur The objective of this lecture note will be to learn duality theory of linear programming. We are planning to answer following questions. What are hyperplane
More informationWhat is linear programming (LP)? NATCOR Convex Optimization Linear Programming 1. Solving LP problems: The standard simplex method
NATCOR Convex Optimization Linear Programming 1 Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk 14 June 2016 What is linear programming (LP)? The most important model used in
More informationLECTURE 18 LECTURE OUTLINE
LECTURE 18 LECTURE OUTLINE Generalized polyhedral approximation methods Combined cutting plane and simplicial decomposition methods Lecture based on the paper D. P. Bertsekas and H. Yu, A Unifying Polyhedral
More informationA PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM. 1.
ACTA MATHEMATICA VIETNAMICA Volume 21, Number 1, 1996, pp. 59 67 59 A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM NGUYEN DINH DAN AND
More informationLinear Programming. Linear programming provides methods for allocating limited resources among competing activities in an optimal way.
University of Southern California Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research - Deterministic Models Fall
More informationLinear Programming Problems
Linear Programming Problems Two common formulations of linear programming (LP) problems are: min Subject to: 1,,, 1,2,,;, max Subject to: 1,,, 1,2,,;, Linear Programming Problems The standard LP problem
More informationSurrogate Gradient Algorithm for Lagrangian Relaxation 1,2
Surrogate Gradient Algorithm for Lagrangian Relaxation 1,2 X. Zhao 3, P. B. Luh 4, and J. Wang 5 Communicated by W.B. Gong and D. D. Yao 1 This paper is dedicated to Professor Yu-Chi Ho for his 65th birthday.
More informationNATCOR Convex Optimization Linear Programming 1
NATCOR Convex Optimization Linear Programming 1 Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk 5 June 2018 What is linear programming (LP)? The most important model used in
More information2. Convex sets. affine and convex sets. some important examples. operations that preserve convexity. generalized inequalities
2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More informationLinear and Integer Programming (ADM II) Script. Rolf Möhring WS 2010/11
Linear and Integer Programming (ADM II) Script Rolf Möhring WS 200/ Contents -. Algorithmic Discrete Mathematics (ADM)... 3... 4.3 Winter term 200/... 5 2. Optimization problems 2. Examples... 7 2.2 Neighborhoods
More informationNew Penalty Approaches for Bilevel Optimization Problems arising in Transportation Network Design
New Penalty Approaches for Bilevel Optimization Problems arising in Transportation Network Design Jörg Fliege, Konstantinos Kaparis, & Huifu Xu Melbourne, 2013 Contents 1 Problem Statement 2 A Slight Detour:
More informationMixed Integer Second Order Cone Optimization (MISOCO): Cuts, Warm Start, and Rounding
Mixed Integer Second Order Cone Optimization (MISOCO): Conic Tamás Terlaky, Mohammad Shahabsafa, Julio C. Góez, Sertalp Cay, Imre Pólik OPDGTP, Tel Aviv, Israel April 2018 1/35 Outline 1 Disjunctive Conic
More informationConstrained optimization
Constrained optimization Problem in standard form minimize f(x) subject to a i (x) = 0, for i =1, 2, p c j (x) 0 for j =1, 2,,q f : R n R a i : R n R c j : R n R f(x )=, if problem is infeasible f(x )=,
More informationLinear Programming: Mathematics, Theory and Algorithms
Linear Programming: Mathematics, Theory and Algorithms Applied Optimization Volume 2 The titles published in this series are listed at the end of this volume. Linear Programming: Mathematics, Theory and
More informationVARIANTS OF THE SIMPLEX METHOD
C H A P T E R 6 VARIANTS OF THE SIMPLEX METHOD By a variant of the Simplex Method (in this chapter) we mean an algorithm consisting of a sequence of pivot steps in the primal system using alternative rules
More informationCS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 36
CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 36 CS 473: Algorithms, Spring 2018 LP Duality Lecture 20 April 3, 2018 Some of the
More informationA Simplex-Cosine Method for Solving Hard Linear Problems
A Simplex-Cosine Method for Solving Hard Linear Problems FEDERICO TRIGOS 1, JUAN FRAUSTO-SOLIS 2 and RAFAEL RIVERA-LOPEZ 3 1 Division of Engineering and Sciences ITESM, Campus Toluca Av. Eduardo Monroy
More informationLinear Programming. Readings: Read text section 11.6, and sections 1 and 2 of Tom Ferguson s notes (see course homepage).
Linear Programming Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory: Feasible Set, Vertices, Existence of Solutions. Equivalent formulations. Outline
More informationPrimal and Dual Methods for Optimisation over the Non-dominated Set of a Multi-objective Programme and Computing the Nadir Point
Primal and Dual Methods for Optimisation over the Non-dominated Set of a Multi-objective Programme and Computing the Nadir Point Ethan Liu Supervisor: Professor Matthias Ehrgott Lancaster University Outline
More informationTribhuvan University Institute Of Science and Technology Tribhuvan University Institute of Science and Technology
Tribhuvan University Institute Of Science and Technology Tribhuvan University Institute of Science and Technology Course Title: Linear Programming Full Marks: 50 Course No. : Math 403 Pass Mark: 17.5 Level
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More informationOptimization Methods. Final Examination. 1. There are 5 problems each w i t h 20 p o i n ts for a maximum of 100 points.
5.93 Optimization Methods Final Examination Instructions:. There are 5 problems each w i t h 2 p o i n ts for a maximum of points. 2. You are allowed to use class notes, your homeworks, solutions to homework
More informationAdvanced Algorithms Linear Programming
Reading: Advanced Algorithms Linear Programming CLRS, Chapter29 (2 nd ed. onward). Linear Algebra and Its Applications, by Gilbert Strang, chapter 8 Linear Programming, by Vasek Chvatal Introduction to
More informationNonlinear Programming
Nonlinear Programming SECOND EDITION Dimitri P. Bertsekas Massachusetts Institute of Technology WWW site for book Information and Orders http://world.std.com/~athenasc/index.html Athena Scientific, Belmont,
More informationGeorge B. Dantzig Mukund N. Thapa. Linear Programming. 1: Introduction. With 87 Illustrations. Springer
George B. Dantzig Mukund N. Thapa Linear Programming 1: Introduction With 87 Illustrations Springer Contents FOREWORD PREFACE DEFINITION OF SYMBOLS xxi xxxiii xxxvii 1 THE LINEAR PROGRAMMING PROBLEM 1
More information60 2 Convex sets. {x a T x b} {x ã T x b}
60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity
More informationR n a T i x = b i} is a Hyperplane.
Geometry of LPs Consider the following LP : min {c T x a T i x b i The feasible region is i =1,...,m}. X := {x R n a T i x b i i =1,...,m} = m i=1 {x Rn a T i x b i} }{{} X i The set X i is a Half-space.
More informationInterpretation of Dual Model for Piecewise Linear. Programming Problem Robert Hlavatý
Interpretation of Dual Model for Piecewise Linear 1 Introduction Programming Problem Robert Hlavatý Abstract. Piecewise linear programming models are suitable tools for solving situations of non-linear
More informationSimplex Algorithm in 1 Slide
Administrivia 1 Canonical form: Simplex Algorithm in 1 Slide If we do pivot in A r,s >0, where c s
More informationThe Affine Scaling Method
MA33 Linear Programming W. J. Martin October 9, 8 The Affine Scaling Method Overview Given a linear programming problem in equality form with full rank constraint matrix and a strictly positive feasible
More informationSelected Topics in Column Generation
Selected Topics in Column Generation February 1, 2007 Choosing a solver for the Master Solve in the dual space(kelly s method) by applying a cutting plane algorithm In the bundle method(lemarechal), a
More informationArtificial Intelligence
Artificial Intelligence Combinatorial Optimization G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/34 Outline 1 Motivation 2 Geometric resolution
More informationConvex Optimization - Chapter 1-2. Xiangru Lian August 28, 2015
Convex Optimization - Chapter 1-2 Xiangru Lian August 28, 2015 1 Mathematical optimization minimize f 0 (x) s.t. f j (x) 0, j=1,,m, (1) x S x. (x 1,,x n ). optimization variable. f 0. R n R. objective
More informationCollege of Computer & Information Science Fall 2007 Northeastern University 14 September 2007
College of Computer & Information Science Fall 2007 Northeastern University 14 September 2007 CS G399: Algorithmic Power Tools I Scribe: Eric Robinson Lecture Outline: Linear Programming: Vertex Definitions
More informationConvex Optimization. 2. Convex Sets. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University. SJTU Ying Cui 1 / 33
Convex Optimization 2. Convex Sets Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2018 SJTU Ying Cui 1 / 33 Outline Affine and convex sets Some important examples Operations
More informationAMATH 383 Lecture Notes Linear Programming
AMATH 8 Lecture Notes Linear Programming Jakob Kotas (jkotas@uw.edu) University of Washington February 4, 014 Based on lecture notes for IND E 51 by Zelda Zabinsky, available from http://courses.washington.edu/inde51/notesindex.htm.
More informationDETERMINISTIC OPERATIONS RESEARCH
DETERMINISTIC OPERATIONS RESEARCH Models and Methods in Optimization Linear DAVID J. RADER, JR. Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN WILEY A JOHN WILEY & SONS,
More informationInteger Programming Theory
Integer Programming Theory Laura Galli October 24, 2016 In the following we assume all functions are linear, hence we often drop the term linear. In discrete optimization, we seek to find a solution x
More informationImproved Gomory Cuts for Primal Cutting Plane Algorithms
Improved Gomory Cuts for Primal Cutting Plane Algorithms S. Dey J-P. Richard Industrial Engineering Purdue University INFORMS, 2005 Outline 1 Motivation The Basic Idea Set up the Lifting Problem How to
More informationApplied Integer Programming
Applied Integer Programming D.S. Chen; R.G. Batson; Y. Dang Fahimeh 8.2 8.7 April 21, 2015 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces,
More information