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.
Table of Contents
What Is Convex Optimization?
Smooth Convex Optimization
Tools for Convex Optimization
Epigraphical Properties of Conjugate Functions
Basic Optimality Conditions using the Normal Cone
Slater Constraint Qualification
Abadie Constraint Qualification
Convex Problems with Abstract Constraints
Cone-Constrained Convex Programming
Saddle Points, Optimality, and Duality
Basic Saddle Point Theorem
Affine Inequalities and Equalities and Saddle Point Condition
Equivalence between Lagrangian and Fenchel Duality: Magnanti’s Approach
Enhanced Fritz John Optimality Conditions
Enhanced Fritz John Conditions Using the Subdifferential
Enhanced Fritz John Conditions under Restrictions
Enhanced Fritz John Conditions in the Absence of Optimal Solution
Enhanced Dual Fritz John Optimality Conditions
Optimality without Constraint Qualification
Geometric Optimality Condition: Smooth Case
Geometric Optimality Condition: Nonsmooth Case
Separable Sublinear Case
Sequential Optimality Conditions and Generalized Constraint Qualification
Sequential Optimality: Thibault’s Approach
Fenchel Conjugates and Constraint Qualification
Applications to Bilevel Programming Problems
Representation of the Feasible Set and KKT Conditions
Weak Sharp Minima in Convex Optimization
Weak Sharp Minima and Optimality
Approximate Optimality Conditions
ε-Saddle Point Approach
Exact Penalization Approach
Ekeland’s Variational Principle Approach
Modified ε-KKT Conditions
Duality-Based Approach to ε-Optimality
Convex Semi-Infinite Optimization
Lagrangian Regular Point
Noncompact Scenario: An Alternate Approach
Convexity in Nonconvex Optimization
Maximization of a Convex Function
Minimization of d.c. Functions
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.
"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