Fundamentals of Convex Analysis: Duality, Separation, Representation, and Resolution
Fundamentals of Convex Analysis offers an in-depth look at some of the fundamental themes covered within an area of mathematical analysis called convex analysis. In particular, it explores the topics of duality, separation, representation, and resolution. The work is intended for students of economics, management science, engineering, and mathematics who need exposure to the mathematical foundations of matrix games, optimization, and general equilibrium analysis. It is written at the advanced undergraduate to beginning graduate level and the only formal preparation required is some familiarity with set operations and with linear algebra and matrix theory. Fundamentals of Convex Analysis is self-contained in that a brief review of the essentials of these tool areas is provided in Chapter 1. Chapter exercises are also provided.
Topics covered include: convex sets and their properties; separation and support theorems; theorems of the alternative; convex cones; dual homogeneous systems; basic solutions and complementary slackness; extreme points and directions; resolution and representation of polyhedra; simplicial topology; and fixed point theorems, among others. A strength of this work is how these topics are developed in a fully integrated fashion.
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Fundamentals of Convex Analysis: Duality, Separation, Representation, and ...
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A'y+1ys affine hull affine set affinely independent alternative barycentric subdivision basic feasible solution basis closed convex set closed half-spaces columns conical combination convergent convex combination convex hull convex polyhedron convex polytope COROLLARY derived subsimplex dimension equations equivalent existence theorem exists a vector extreme direction extreme points extreme vectors Farkas Figure finite cone finite number finite set fixed point follows function F given half-line hemicontinuity Hence hyperplane JG implies intersection k-simplex lemma Let f limit point linear combination linear inequalities linear subspace linearly independent linearly independent vectors matrix Moreover nonnegative null vector point of f point-to-set function polyhedral convex cone polyhedral convex set PROOF recession directions scalar semi-positive sequence xk set f set of extreme set of vectors simplex simplicial solution set solution to Ax solution xe Sperner's lemma subspace supporting hyperplane unique variables vertex vertices Weyl's theorem x|Ax x|Axis xe f