CO227-Chpt1and2

CO227-Chpt1and2 - Introduction to Optimization Course Notes...

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Introduction to Optimization Course Notes for CO 227 Spring 2011 c ± Department of Combinatorics and Optimization University of Waterloo June 24, 2011
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2 c ± Department of Combinatorics and Optimization, University of Waterloo Fall 2011
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Contents 1I n t r o d u c t i o n 7 1.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 A production planning example . . . . . . . . . . . . . . . . . . . . .8 1.1.2 Multiperiod models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 3 1.2.1 Maximum Weight Matching . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 0 1.3.1 Pricing a DVD Player . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Overview of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.5 Further reading and notes ............................ 2 2 2S o l v i n g l i n e a r p r o g r a m s 25 2.1 Possible outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Infeasible linear programs . . . . . . . . . . . . . . . . . . . . . ... 2 6 2.1.2 Linear programs with optimal solutions . . . . . . . . . . . ...... 2 9 2.1.3 Unbounded linear programs . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Standard equality form . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 3 4 2.3 A Simplex iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 8 2.4 Bases and canonical forms . . . . . . . . . . . . . . . . . . . . . . . . . ... 4 0 2.4.1 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3
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CONTENTS 4 2.4.2 Canonical forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 The Simplex algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 7 2.5.1 An example with an optimal solution . . . . . . . . . . . . . . . ... 4 7 2.5.2 An unbounded example . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.3 Formalizing the procedure . . . . . . . . . . . . . . . . . . . . . . . .5 1 2.6 Finding feasible solutions . . . . . . . . . . . . . . . . . . . . . . . ...... 5 4 2.6.1 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.6.2 The Two Phase method - an example . . . . . . . . . . . . . . . . . . 59 2.6.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.7 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.8 Further reading and notes ............................ 6 5 3I n t r o d u c t i o n t o d u a l i t y 67 3.1 A ±rst example: Maximum weight matching . . . . . . . . . . . . . ...... 6 8 3.2 Bounding the optimal value of a linear program . . . . . . . . ......... 69 3.3 The matching example revisited . . . . . . . . . . . . . . . . . . . . ..... 74 3.4 A second example: Network ²ow . . . . . . . . . . . . . . . . . . . . . . . .7 9 3.5 A third example: Scheduling . . . . . . . . . . . . . . . . . . . . . . . .... 8 4 3.6 Duals of general linear programs . . . . . . . . . . . . . . . . . . . ...... 8 7 3.7 The duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2 3.7.1 Integer programs and the duality gap . . . . . . . . . . . . . . ..... 92 3.7.2 Linear programs and the duality theorem . . . . . . . . . . . ..... 93 3.8 Complementary Slackness . . . . . . . . . . . . . . . . . . . . . . . . . ... 9 5 3.9 Complementary Slackness and combinatorial examples . ............ 100 3.9.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.9.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.10 Further reading and notes 1 0 4 c ± Department of Combinatorics and Optimization, University of Waterloo Fall 2011
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CONTENTS 5 4S o l v i n g I n t e g e r P r o g r a m s 107 4.1 Maximum weight matching algorithm . . . . . . . . . . . . . . . . . ..... 108 4.1.1 Hall’s condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.2 An optimality condition . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.1.3 The Hungarian matching algorithm . . . . . . . . . . . . . . . . ... 1 1 1 4.2 Cutting planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2.1 General scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.2 Valid inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2.3 Cutting plane and simplex . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3 Branch & Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3.1 A discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4 Further reading and notes ............................ 1 2 7 5G e o m e t r y o f o p t i m i z a t i o n 129 5.1 Feasible solutions to linear programs and polyhedra . . ............. 1 2 9 5.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3 Extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4 Geometric interpretation of Simplex algorithm . . . . . . ............ 137 5.5 Cutting planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.6 Further reading and notes 1 4 1 c ± Department of Combinatorics and Optimization, University of Waterloo Fall 2011
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CONTENTS 6 c ± Department of Combinatorics and Optimization, University of Waterloo Fall 2011
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Chapter 1 Introduction Optimization problems are abundant in every day life, and al lo fusfacesuchp rob lemsf re - quently, although we may not always be aware of the fact! Obvious examples are, for instance, the use of your GPS to Fnd a shortest route from your home to yourworkplaceinthemorning, or the scheduling of trains on the rail connections between Waterloo and Toronto. There are however many more, less obvious examples. How, for example, does the region of Waterloo determine the structure of its electricity network? How are the schedules for buses determined?
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CO227-Chpt1and2 - Introduction to Optimization Course Notes...

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