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chapter1 - DRAFT Formulation and Analysis of Linear...

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DRAFT Formulation and Analysis of Linear Programs Benjamin Van Roy and Kahn Mason c Benjamin Van Roy and Kahn Mason September 26, 2005 1
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Contents 1 Introduction 7 1.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Linear Programs . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Two-Player Zero-Sum Games . . . . . . . . . . . . . . . . . . 11 1.5 Network Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Markov Decision Problems . . . . . . . . . . . . . . . . . . . . 13 1.7 Linear Programming Algorithms . . . . . . . . . . . . . . . . . 14 2 Linear Algebra 15 2.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Transposition . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 Multiplication . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.5 Linear Systems of Equations . . . . . . . . . . . . . . . 20 2.1.6 Partitioning of Matrices . . . . . . . . . . . . . . . . . 20 2.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Bases and Dimension . . . . . . . . . . . . . . . . . . . 23 2.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Orthogonal Subspaces . . . . . . . . . . . . . . . . . . 27 2.2.4 Vector Spaces Associated with Matrices . . . . . . . . . 29 2.3 Linear Systems of Equations . . . . . . . . . . . . . . . . . . . 31 2.3.1 Solution Sets . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . 34 2.4 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.1 Structured Products and Market Completeness . . . . 38 2.4.2 Pricing and Arbitrage . . . . . . . . . . . . . . . . . . 40 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3
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4 3 Linear Programs 47 3.1 Graphical Examples . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.1 Producing Cars and Trucks . . . . . . . . . . . . . . . 48 3.1.2 Feeding an Army . . . . . . . . . . . . . . . . . . . . . 50 3.1.3 Some Observations . . . . . . . . . . . . . . . . . . . . 50 3.2 Feasible Regions and Basic Feasible Solutions . . . . . . . . . 52 3.2.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Vertices and Basic Solutions . . . . . . . . . . . . . . . 53 3.2.3 Bounded Polyhedra . . . . . . . . . . . . . . . . . . . . 55 3.3 Optimality of Basic Feasible Solutions . . . . . . . . . . . . . 56 3.3.1 Bounded Feasible Regions . . . . . . . . . . . . . . . . 57 3.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . 57 3.3.3 Searching through Basic Feasible Solutions . . . . . . . 58 3.4 Greater-Than and Equality Constraints . . . . . . . . . . . . . 59 3.5 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5.1 Single-Stage Production . . . . . . . . . . . . . . . . . 61 3.5.2 Multi-Stage Production . . . . . . . . . . . . . . . . . 64 3.5.3 Market Stratification and Price Discrimination . . . . . 66 3.6 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6.1 Structured Products in an Incomplete Market . . . . . 68 3.6.2 Finding Arbitrage Opportunities . . . . . . . . . . . . 71 3.7 Pattern Classification . . . . . . . . . . . . . . . . . . . . . . . 72 3.7.1 Linear Separation of Data . . . . . . . . . . . . . . . . 74 3.7.2 Minimizing Violations . . . . . . . . . . . . . . . . . . 75 3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 Duality 85 4.1 A Graphical Example . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . 87 4.1.2 Shadow Prices and Valuation of the Firm . . . . . . . . 89 4.1.3 The Dual Linear Program . . . . . . . . . . . . . . . . 90 4.2 Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.1 Weak Duality . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.2 Strong Duality . . . . . . . . . . . . . . . . . . . . . . 93 4.2.3 First Order Necessary Conditions . . . . . . . . . . . . 93 4.2.4 Complementary Slackness . . . . . . . . . . . . . . . . 96 4.3 Duals of General Linear Programs . . . . . . . . . . . . . . . . 97 4.4 Two-Player Zero-Sum Games . . . . . . . . . . . . . . . . . . 100 4.5 Allocation of a Labor Force . . . . . . . . . . . . . . . . . . . 104 4.5.1 Labor Minimization . . . . . . . . . . . . . . . . . . . . 105
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c Benjamin Van Roy and Kahn Mason 5 4.5.2 Productivity Implies Flexibility . . . . . . . . . . . . . 106 4.5.3 Proof of the Substitution Theorem . . . . . . . . . . . 107 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Network Flows 115 5.1 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Min-Cost-Flow Problems . . . . . . . . . . . . . . . . . . . . . 116 5.2.1 Shortest Path Problems . . . . . . . . . . . . . . . . . 118 5.2.2 Transportation Problems . . . . . . . . . . . . . . . . . 118 5.2.3 Assignment Problems . . . . . . . . . . . . . . . . . . . 119 5.3 Max-Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3.1 Min-Cut Problems . . . . . . . . . . . . . . . . . . . . 121 5.3.2 Matching Problems . . . . . . . . . . . . . . . . . . . . 123 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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Chapter 1 Introduction Optimization is the process of selecting the best among a set of alternatives. An optimization problem is characterized by a set of feasible solutions and an objective function, which assigns a measure of utility to each feasible solution. A simple approach to optimization involves listing the feasible solutions, applying the objective function to each, and choosing one that attains the optimum, which could be the maximum or minimum, depending on what we are looking for. However, this approach does not work for most interesting problems. The reason is that there are usually too many feasible solutions, as we illustrate with the following example.
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