Linear Optimization and Extensions: Theory and Algorithms

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1 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

2 Contents PREFACE 1 INTRODUCTION 1.1 History of Linear Programming The Linear Programming Problem Standard-Form Linear Program, Embedded Assumptions, Converting to Standard Form, Examples of Linear Programming Problems Mastering Linear Programming 9 References for Further Reading 10 Exercises 11 2 GEOMETRY OF LINEAR PROGRAMMING 2.1 Basic Terminologies of Linear Programming Hyperplanes, Halfspaces, and Polyhedral Sets Affine Sets, Convex Sets, and Cones 17

3 VIII Contents 2.4 Extreme Points and Basic Feasible Solutions Nondegeneracy and Adjacency Resolution Theorem for Convex Polyhedrons Fundamental Theorem of Linear Programming Concluding Remarks: Motivations of Different Approaches 25 References for Further Reading 26 Exercises 26 3 THE REVISED SIMPLEX METHOD Elements of an Iterative Scheme Basics of the Simplex Method Algebra of the Simplex Method Stopping the Simplex Method Checking for Optimality, Iterations ofthe Simplex Method Moving for Improvement, Starting the Simplex Method Two-Phase Method, Big-M Method, Degeneracy and Cycling Preventing Cycling Lexicographic Rule, Bland 's Rule, The Revised Simplex Method Concluding Remarks 50 References for Further Reading 50 Exercises 51 4 DUALITY THEORY AND SENSITIVITY ANALYSIS Dual Linear Program Duality Theory 57

4 Contents 4.3 Complementary Slackness and Optimality Conditions An Economic Interpretation of the Dual Problem Dual Variables and Shadow Prices, Interpretation of the Dual Problem, The Dual Simplex Method Basic Idea ofthe Dual Simplex Method, Sherman-Morrison-Woodbury Formula, Computer Implementation of the Dual Simplex Method, Find an Initial Dual Basic Feasible Solution, The Primal Dual Method Step-by-Step Procedure for the Primal-Dual Simplex Method, 75 A.l Sensitivity Analysis Change in the Cost Vector, Change in the Right-Hand-Side Vector, Change in the Constraint Matrix, Concluding Remarks 86 References for Further Reading 87 Exercises 87 5 COMPLEXITY ANALYSIS AND THE ELLIPSOID METHOD 5.1 Concepts of Computational Complexity Complexity of the Simplex Method Basic Ideas of the Ellipsoid Method Ellipsoid Method for Linear Programming Performance of the Ellipsoid Method for LP Modifications of the Basic Algorithm Deep Cuts, Surrogate Cuts, Parallel Cuts, Replacing Ellipsoid by Simplex, Concluding Remarks 108 References for Further Reading 108 Exercises 109

5 X Contents 6 KARMARKAR'S PROJECTIVE SCALING ALGORITHM Basic Ideas of Karmarkar's Algorithm Karmarkar's Standard Form The Simplex Structure, Projective Transformation on the Simplex, Karmarkar's Projective Scaling Algorithm Polynomial-Time Solvability Converting to Karmarkar's Standard Form Handling Problems with Unknown Optimal Objective Values Unconstrained Convex Dual Approach e-optimal Solution, Extension, Concluding Remarks 141 References for Further Reading 141 Exercises AFFINE SCALING ALGORITHMS Primal Affine Scaling Algorithm Basic Ideas of Primal Affine Scaling, Implementing the Primal Affine Scaling Algorithm, Computational Complexity, Dual Affine Scaling Algorithm Basic Ideas of Dual Affine Scaling, Dual Affine Scaling Algorithm, Implementing the Dual Affine Scaling Algorithm, Improving Computational Complexity, The Primal-Dual Algorithm Basic Ideas of the Primal-Dual Algorithm, Direction and Step-Length of Movement, Primal-Dual Algorithm, Polynomial-Time Termination, Starting the Primal-Dual Algorithm, Practical Implementation, 189

6 Contents xi Accelerating via Power-Series Method, Concluding Remarks 194 References for Further Reading 195 Exercises INSIGHTS INTO THE INTERIOR-POINT METHODS Moving Along Different Algebraic Paths Primat Affine Scaling with Logarithmic Barrier Function, Dual Affine Scaling with Logarithmic Barrier Function, The Primal-Dual Algorithm, Missing Information Dual Information in the Primal Approach, Primal Information in the Dual Approach, Extensions of Algebraic Paths Geometrie Interpretation of the Moving Directions Primal Affine Scaling with Logarithmic Barrier Function, Dual Affine Scaling with Logarithmic Barrier Function, The Primal-Dual Algorithm, General Theory General Primal Affine Scaling, General Dual Affine Scaling, Concluding Remarks 220 References for Further Reading 221 Exercises AFFINE SCALING FOR CONVEX QUADRATIC PROGRAMMING Convex Quadratic Program with Linear Constraints Primal Quadratic Program, Dual Quadratic Program, Affine Scaling for Quadratic Programs Primal Affine Scaling for Quadratic Programming, 227

7 xii Contents Improving Primal Affine Scalingfor Quadratic Programming, Primal-Dual Algorithm for Quadratic Programming Basic Concepts, A Step-By-Step Implementation Procedure, Convergence Properties ofthe Primal-Dual Algorithm, Convex Programming with Linear Constraints Basic Concepts, A Step-by-Step Implementation Procedure, Concluding Remarks 249 References for Further Reading 249 Exercises IMPLEMENTATION OF INTERIOR-POINT ALGORITHMS The Computational Bottleneck The Cholesky Factorization Method Computing Cholesky Factor, Block Cholesky Factorization, Sparse Cholesky Factorization, Symbolic Cholesky Factorization, Solving Triangulär Systems, The Conjugate Gradient Method The LQ Factorization Method Concluding Remarks 275 References for Further Reading 276 Exercises 277 BIBLIOGRAPHY 280 INDEX 295

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