Dhara / Dutta Optimality Conditions in Convex Optimization

A Finite-Dimensional View
Erscheinungsjahr 2011
ISBN: 978-1-4398-6823-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 444 Seiten

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.

Zielgruppe

Graduate students and researchers in optimization theory and algorithms; engineers, physicists, and economists; professionals in finance, business, and industry.

Autoren/Hrsg.

Dhara, Anulekha

Dutta, Joydeep

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Wirtschaftswissenschaften Betriebswirtschaft Unternehmensforschung
• Mathematik | Informatik Mathematik Operations Research
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

What Is Convex Optimization?
Introduction
Basic concepts
Smooth Convex Optimization

Tools for Convex Optimization
Introduction
Convex Sets

Convex Functions
Subdifferential Calculus
Conjugate Functions
e-Subdifferential
Epigraphical Properties of Conjugate Functions

Basic Optimality Conditions using the Normal Cone
Introduction
Slater Constraint Qualification
Abadie Constraint Qualification
Convex Problems with Abstract Constraints
Max-Function Approach
Cone-Constrained Convex Programming

Saddle Points, Optimality, and Duality
Introduction
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
Introduction
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
Introduction
Geometric Optimality Condition: Smooth Case
Geometric Optimality Condition: Nonsmooth Case
Separable Sublinear Case

Sequential Optimality Conditions and Generalized Constraint Qualification
Introduction
Sequential Optimality: Thibault’s Approach
Fenchel Conjugates and Constraint Qualification
Applications to Bilevel Programming Problems

Representation of the Feasible Set and KKT Conditions
Introduction
Smooth Case
Nonsmooth Case

Weak Sharp Minima in Convex Optimization
Introduction
Weak Sharp Minima and Optimality

Approximate Optimality Conditions
Introduction
e-Subdifferential Approach
Max-Function Approach
e-Saddle Point Approach
Exact Penalization Approach
Ekeland’s Variational Principle Approach
Modified e-KKT Conditions
Duality-Based Approach to e-Optimality

Convex Semi-Infinite Optimization
Introduction
Sup-Function Approach
Reduction Approach
Lagrangian Regular Point
Farkas–Minkowski Linearization
Noncompact Scenario: An Alternate Approach

Convexity in Nonconvex Optimization
Introduction
Maximization of a Convex Function
Minimization of d.c. Functions

Bibliography
Index

Über Autor(innen)

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.