ABSTRACT
CONTENTS 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Convex Functions: Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Examples of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Continuity and Differentiability of Convex Functions . . . . . . . . . . . . . . . . . . . . 12
1.3 What Is Convex Optimization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Linear Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.2 Quadratic and Conic Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.2.1 More Details on the Second-Order Cone . . . . . . . . . . . . . . . . . . . . . . 38 1.4 Optimality and Duality in Convex Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.5 Miscellaneous Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ABSTRACT Convex functions are central to the study of optimization. The main property that makes it pivotal to the study of optimization is that every local minimizer of a convex function over a convex set is a global minimizer. The very large number of problems that arise in engineering, business, economics, and finance can be modeled as a convex optimization problem. In this chapter we first focus on the various examples and important properties of convex functions, which includes the study of subdifferentials and their computation. We also focus on the various important models of convex optimization problems and then on their optimality and duality properties. We end the chapter by a study of monotone operators, variational inequalities, and of the proximal point method for convex optimization problems.
