Elementary Convexity with Optimization

Description

This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines. Preface References Contents 1 Continuity and Existence of Optima 1.1 Some Basic Real Analysis 1.2 Bolzano-Weierstrass Theorem 1.3 Existence of Optima 1.4 More on Continuous Functions 1.5 Exercises References 2 Differentiability and Local Optimality 2.1 Introduction 2.2 Notions of Differentiability 2.3 Conditions for Local Optimality 2.4 Danskin's Theorem 2.5 Parametric Monotonicity of Optimizers 2.6 Ekeland Variational Principle 2.7 Mountain Pass Theorem 2.8 Exercises References 3 Convex Sets 3.1 Introduction 3.2 The Minimum Distance Problem 3.3 Separation Theorems 3.4 Extreme Points 3.5 The Shapley-Folkman Theorem 3.6 Helly's Theorem 3.7 Brouwer Fixed Point Theorem 3.8 Proof of Theorem 3.1 3.9 Exercises References 4 Convex Functions 4.1 Basic Properties 4.2 Continuity 4.3 Differentiability 4.4 An Approximation Theorem 4.5 Convex Extensions 4.6 Further Properties of Gradients of Convex Functions 4.7 Exercises References 5 Convex Optimization 5.1 Introduction 5.2 Legendre Transform and Fenchel Duality 5.3 The Lagrange Multiplier Rule 5.4 The Arrow-Barankin-Blackwell Theorem 5.5 Linear Programming 5.6 Applications to Game Theory 5.6.1 Min–Max Theorem 5.6.2 Existence of Nash Equilibria 5.7 Exercises References 6 Optimization Algorithms: An Overview 6.1 Preliminaries 6.2 Line Search 6.3 Algorithms for Unconstrained Optimization 6.4 Algorithms for Constrained Optimization 6.5 Special Topics 6.6 Other Directions 6.7 Exercises References 7 Epilogue 7.1 What Lies Beyond 7.2 Bibliographical Note References Index