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# Notes - Math 171A LINEAR PROGRAMMING Class Notes c 1998...

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Math 171A LINEAR PROGRAMMING Class Notes c circlecopyrt 1998. Philip E. Gill, Walter Murray and Margaret H. Wright Department of Mathematics University of California, San Diego, La Jolla, CA 92093-0112. January 2007

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Contents 1 Background 7 1.1. Definitions and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Matrices with Special Structure . . . . . . . . . . . . . . . . . . . . 15 1.2. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 Linear Dependence and Independence . . . . . . . . . . . . . . . . . 18 1.2.2 Range and Null Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.3 Singular and Nonsingular Matrices . . . . . . . . . . . . . . . . . . . 23 1.3. Solving Rectangular Linear Systems . . . . . . . . . . . . . . . . . . . . . . 25 1.3.1 Full Row Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 Full Column Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.3 Characterization of a Solution . . . . . . . . . . . . . . . . . . . . . 29 2 Linear Programming 31 2.1. Formulating a Linear Program . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.1 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.2 Formulation as a Linear Program . . . . . . . . . . . . . . . . . . . 33 2.2. Properties of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 The Normal Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Level Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.3 One-Dimensional Variation . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.4 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Equality Constraints 41 3.1. Properties of Linear Equality Constraints . . . . . . . . . . . . . . . . . . . 41 3.1.1 Feasible Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Feasible Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2. Optimality for Equality Constraints . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 Feasible Descent Directions . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Derivation of Optimality Conditions . . . . . . . . . . . . . . . . . . 45 3

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4 CONTENTS January 4, 2007 4 Inequality Constraints 49 4.1. Properties of Linear Inequality Constraints . . . . . . . . . . . . . . . . . . 49 4.1.1 Feasible Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.2 Active and Inactive Constraints . . . . . . . . . . . . . . . . . . . . 52 4.1.3 Feasible Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1.4 The Step to the Nearest Constraint . . . . . . . . . . . . . . . . . . 57 4.1.5 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2. Optimality Conditions for Inequality Constraints . . . . . . . . . . . . . . . 63 4.2.1 Feasible Descent Directions . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.2 Farkas’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.3 Summary of Optimality Conditions . . . . . . . . . . . . . . . . . . 71 4.2.4 Vertex Minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3. The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 Motivation for a Simplex Step . . . . . . . . . . . . . . . . . . . . . 78 4.3.2 Definition of the Simplex Method . . . . . . . . . . . . . . . . . . . 82 4.3.3 An Example of the Simplex Method . . . . . . . . . . . . . . . . . . 84 4.3.4 Termination of the Simplex Method . . . . . . . . . . . . . . . . . . 86 4.4. Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4.1 Degeneracy and the Simplex Method . . . . . . . . . . . . . . . . . 87 4.4.2 Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.3 Anti-Cycling Techniques . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5. Complexity of the Simplex Method . . . . . . . . . . . . . . . . . . . . . . . 93 4.5.1 Measurements of Performance . . . . . . . . . . . . . . . . . . . . . 93 4.5.2 Behavior of the Simplex Method . . . . . . . . . . . . . . . . . . . . 94 4.6. Non-Simplex Active-Set Methods . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6.1 Motivation for an Active-Set Strategy . . . . . . . . . . . . . . . . . 96 4.6.2 Finding a Null-Space Descent Direction . . . . . . . . . . . . . . . . 97 4.6.3 Steepest-Descent Null-Space Directions . . . . . . . . . . . . . . . . 98 4.6.4 Definition of a Non-Simplex Method . . . . . . . . . . . . . . . . . . 99 4.6.5 A Non-Simplex Example . . . . . . . . . . . . . . . . . . . . . . . . 101 4.7. Finding a Feasible Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.7.1 Formulation of a Phase-1 Linear Program . . . . . . . . . . . . . . . 103 4.7.2 Adding a Single Artificial Variable to Inequality Form . . . . . . . . 104 4.7.3 Minimizing the Sum of Infeasibilities . . . . . . . . . . . . . . . . . . 108 4.8. Phase-1 Linear Programming Using the Simplex Method . . . . . . . . . . 111 4.8.1 Temporary Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Standard Form 115 5.1. Linear Programs with Mixed Constraints . . . . . . . . . . . . . . . . . . . 115 5.1.1 Form of Mixed Constraints . . . . . . . . . . . . . . . . . . . . . . . 115 5.1.2 Optimality Conditions for Mixed Constraints . . . . . . . . . . . . . 116 5.1.3 Definition of Standard Form . . . . . . . . . . . . . . . . . . . . . . 117 5.2. Simplex Method for Standard Form . . . . . . . . . . . . . . . . . . . . . . 119 5.2.1 Basic and Nonbasic Variables . . . . . . . . . . . . . . . . . . . . . . 119 5.2.2 Motivation for the Standard-Form Simplex Method . . . . . . . . . 122 5.2.3 Summary of the Standard-Form Simplex Method . . . . . . . . . . . 126 5.2.4 An Example of the Standard-Form Simplex Method . . . . . . . . . 128
CONTENTS 5 5.2.5 Standard-Form Simplex Tableaus . . . . . . . . . . . . . . . . . . . 130 5.2.6 Standard-Form Phase-1 Problems . . . . . . . . . . . . . . . . . . . 136 5.2.7 Phase 1 for Standard Form . . . . . . . . . . . . . . . . . . . . . . . 141 6 Duality 143 6.1. Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2. Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.3. All-Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.4. Relations between the Primal and Dual . . . . . . . . . . . . . . . . . . . . 146 6.4.1

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