Convex Optimization

This book provides easy access to the basic principles and methods for solving constrained and unconstrained convex optimization problems. Included are sections that cover: basic methods for solving constrained and unconstrained optimization problems with differentiable objective functions; convex sets and their properties; convex functions and their properties and generalizations; and basic principles of sub-differential calculus and convex programming problems. Convex Optimization provides detailed proofs for most of the results presented in the book and also includes many figures and exercises for a better understanding of the material. Exercises are given at the end of each chapter, with solutions and hints to selected exercises given at the end of the book. Undergraduate and graduate students, researchers in different disciplines, as well as practitioners will all benefit from this accessible approach to convex optimization methods.

Notations ix

Introduction xi

Chapter 1. Optimization Problems with Differentiable Objective Functions 1

Chapter 2. Convex Sets 29

Chapter 3. Convex Functions 67

Chapter 4. Generalizations of Convex Functions 111

Chapter 5. Sub-gradient and Sub-differential of Finite Convex Function 137

Chapter 6. Constrained Optimization Problems 163

Solutions, Answers and Hints 207

References 235

Index 237

Mikhail Moklyachuk is Full Professor at the Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine.