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Enterprise content management class lectures_266879736-Lectures-on-Convex-Sets

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LECTURES ON CONVEX SETS Niels Lauritzen N IELS L AURITZEN D EPARTMENT OF M ATHEMATICAL S CIENCES U NIVERSITY OF A ARHUS D ENMARK E MAIL : niels@imf.au.dk URL: http://home.imf.au.dk/niels/ March 2009 Contents Preface v Notation 1 2 3 4 Introduction 1.1 Linear inequalities . 1.1.1 Two variables 1.2 Polyhedra . . . . . . 1.3 Exercises . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics 2.1 Convex subsets of R n . . . . . . . . . . . 2.2 The convex hull . . . . . . . . . . . . . . 2.2.1 Intersections of convex subsets . 2.3 Extremal points . . . . . . . . . . . . . . 2.4 The characteristic cone for a convex set 2.5 Convex cones . . . . . . . . . . . . . . . 2.6 Affine independence . . . . . . . . . . . 2.7 Carathéodory’s theorem . . . . . . . . . 2.8 The dual cone . . . . . . . . . . . . . . . 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separation 3.1 Separation of a point from a closed convex set 3.2 Supporting hyperplanes . . . . . . . . . . . . . 3.3 Separation of disjoint convex sets . . . . . . . . 3.4 An application . . . . . . . . . . . . . . . . . . . 3.5 Farkas’ lemma . . . . . . . . . . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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