chapter1 - DRAFT Formulation and Analysis of Linear...

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Unformatted text preview: DRAFT Formulation and Analysis of Linear Programs Benjamin Van Roy and Kahn Mason c Benjamin Van Roy and Kahn Mason September 26, 2005 1 2 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 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....
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