LECTURES ON CONVEX SETS

LECTURES ON CONVEX SETS - LECTURES ON CONVEX SETS Niels...

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Unformatted text preview: LECTURES ON CONVEX SETS Niels Lauritzen There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. D. Zeilberger NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS DENMARK EMAIL: [email protected] URL: http://home.imf.au.dk/niels/ March 2009 March 2010 Contents Preface v Notation vii 1 Introduction 1 1.1 Linear inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Two variables . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Basics 11 2.1 Convex subsets of R n . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Extremal points . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 The characteristic cone for a convex set . . . . . . . . . . . . . 18 2.5 Convex cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Affine independence . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Carath´eodory’s theorem . . . . . . . . . . . . . . . . . . . . . . 22 2.8 The polar cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Separation 33 3.1 Separation of a point from a closed convex set . . . . . . . . . 35 3.2 Supporting hyperplanes . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Separation of disjoint convex sets . . . . . . . . . . . . . . . . . 39 3.4 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 Farkas’ lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Polyhedra 47 4.1 The double description method . . . . . . . . . . . . . . . . . . 48 iii iv Contents 4.1.1 The key lemma . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Extremal and adjacent rays . . . . . . . . . . . . . . . . . . . . 52 4.3 Farkas: from generators to half spaces . . . . . . . . . . . . . . 55 4.4 Polyhedra: general linear inequalities . . . . . . . . . . . . . . 57 4.5 The decomposition theorem for polyhedra . . . . . . . . . . . 58 4.6 Extremal points in polyhedra . . . . . . . . . . . . . . . . . . . 59 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A Linear (in)dependence 65 A.1 Linear dependence and linear equations . . . . . . . . . . . . . 65 A.2 The rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 67 A.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 B Analysis 71 B.1 Measuring distances . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B.2.1 Supremum and infimum . . . . . . . . . . . . . . . . . . 76...
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This note was uploaded on 02/23/2011 for the course ECON 10 taught by Professor Goods during the Spring '11 term at Camosun College.

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LECTURES ON CONVEX SETS - LECTURES ON CONVEX SETS Niels...

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