Intro
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