Optimality Conditions in Convex Optimization: A Finite-Dimensional View, 1st Edition (Paperback) book cover

Optimality Conditions in Convex Optimization

A Finite-Dimensional View, 1st Edition

By Anulekha Dhara, Joydeep Dutta

CRC Press

444 pages | 17 B/W Illus.

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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

Table of Contents

What Is Convex Optimization?


Basic concepts

Smooth Convex Optimization

Tools for Convex Optimization


Convex Sets

Convex Functions

Subdifferential Calculus

Conjugate Functions


Epigraphical Properties of Conjugate Functions

Basic Optimality Conditions using the Normal Cone


Slater Constraint Qualification

Abadie Constraint Qualification

Convex Problems with Abstract Constraints

Max-Function Approach

Cone-Constrained Convex Programming

Saddle Points, Optimality, and Duality


Basic Saddle Point Theorem

Affine Inequalities and Equalities and Saddle Point Condition

Lagrangian Duality

Fenchel Duality

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


Smooth Case

Nonsmooth Case

Weak Sharp Minima in Convex Optimization


Weak Sharp Minima and Optimality

Approximate Optimality Conditions


ε-Subdifferential Approach

Max-Function Approach

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


Sup-Function Approach

Reduction Approach

Lagrangian Regular Point

Farkas–Minkowski Linearization

Noncompact Scenario: An Alternate Approach

Convexity in Nonconvex Optimization


Maximization of a Convex Function

Minimization of d.c. Functions



About the Authors

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.

Subject Categories

BISAC Subject Codes/Headings:
BUSINESS & ECONOMICS / Operations Research
TECHNOLOGY & ENGINEERING / Operations Research