Convex Optimization

Introductory Course

Mikhail Moklyachuk

• Publisher: Wiley-ISTE
• Year: 2021
• Print ISBN: 9781786306838
• eBook ISBN: 9781119804086
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

1
Optimization Problems with Differentiable Objective Functions

1.1. Basic concepts

The word “maximum” means the largest, and the word “minimum” means the smallest. These two concepts are combined with the term “extremum”, which means the extreme. Also pertinent is the term “optimal” (from Latin optimus), which means the best. The problems of determining the largest and smallest quantities are called extremum problems. Such problems arise in different areas of activity and therefore different terms are used for the descriptions of the problems. To use the theory of extremum problems, it is necessary to describe problems in the language of mathematics. This process is called the formalization of the problem.

The formalized problem consists of the following elements:

• – objective function
  
  ;
• – domain X of the definition of the objective functional f;
• – constraint: C ⊂ X.

Here,

is an extended real line, that is, the set of all real numbers, supplemented by the values +∞ and –∞, C is a subset of the domain of definition of the objective functional f. So to formalize an optimization problem means to clearly define and describe eleme...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Notations
• Introduction
• 1 Optimization Problems with Differentiable Objective Functions
• 2 Convex Sets
• 3 Convex Functions
• 4 Generalizations of Convex Functions
• 5 Sub-gradient and Sub-differential of Finite Convex Function
• 6 Constrained Optimization Problems
• Solutions, Answers and Hints
• References
• Index
• End User License Agreement