Optimality Conditions in Convex Optimization

This book explores an extremely important issue in convex optimization: optimality conditions. The text focuses on finite dimensions to allow for much deeper results and a better understanding of the structures involved in a convex optimization problem. The authors include examples, details of major results, and proofs of the main results.

Optimality Conditions in Convex Optimization explores an important and central issue in the field of convex optimization: optimality conditions. It brings together the most important and recent results in this area that have been scattered in the literature-notably in the area of convex analysis-essential in developing many of the important results in this book, and not usually found in conventional texts. Unlike other books on convex optimization, which usually discuss algorithms along with some basic theory, the sole focus of this book is on fundamental and advanced convex optimization theory.

Although many results presented in the book can also be proved in infinite dimensions, the authors focus on finite dimensions to allow for much deeper results and a better understanding of the structures involved in a convex optimization problem. They address semi-infinite optimization problems; approximate solution concepts of convex optimization problems; and some classes of non-convex problems which can be studied using the tools of convex analysis. They include examples wherever needed, provide details of major results, and discuss proofs of the main results.

"It discusses a number of major approaches to the subject, bringing together many results from the past thirty-five years into one handy volume. Researchers in variational analysis should find this book to be a useful reference; for those new to convex optimization, it provides a very accessible entry point to the field. I have begun recommending it to graduate students who would like to learn about convex subdifferential calculus. a valuable book, a most welcome addition to the optimization theory literature."Doug Ward, Mathematical Reviews, January 2013

Autorentext

Anulekha Dhara earned her Ph.d. in IIT Delhi and subsequently moved to IIT Kanpur for her post-doctoral studies. Currently, she is a post-doctoral fellow in Mathematics at the University of Avignon, France. Her main area of interest is optimization theory.

Joydeep Dutta is an Associate Professor of Mathematics at the Indian Institute of Technology, (IIT) Kanpur. His main area of interest is optimization theory and applications.

Inhalt

What Is Convex Optimization?. Tools for Convex Optimization. Basic Optimality Conditions using the Normal Cone. Saddle Points, Optimality and Duality. Enhanced Fritz John Optimality Conditions. Optimality without Constraint Qualification. Sequential Optimality Conditions and Generalized Constraint Qualification. Representation of the Feasible Set and KKT Conditions. Weak Sharp Minima in Convex Optimization. Approximate Optimality Conditions. Convex Semi-infinite Optimization. Convexity in Non-Convex Optimization. Bibliography. Index.