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Mathematics of Optimization : Smooth and Nonsmooth Case - J. Thierfelder

Mathematics of Optimization

Smooth and Nonsmooth Case

Hardcover

Published: 24th March 2004
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$415.90
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The book is intended for people (graduates, researchers, but also undergraduates with a good mathematical background) involved in the study of (static) optimization problems (in finite-dimensional spaces). It contains a lot of material, from basic tools of convex analysis to optimality conditions for smooth optimization problems, for non smooth optimization problems and for vector optimization problems.
The development of the subjects are self-contained and the bibliographical references are usually treated in different books (only a few books on optimization theory deal also with vector problems), so the book can be a starting point for further readings in a more specialized literature.
Assuming only a good (even if not advanced) knowledge of mathematical analysis and linear algebra, this book presents various aspects of the mathematical theory in optimization problems. The treatment is performed in finite-dimensional spaces and with no regard to algorithmic questions. After two chapters concerning, respectively, introductory subjects and basic tools and concepts of convex analysis, the book treats extensively mathematical programming problems in the smmoth case, in the nonsmooth case and finally vector optimization problems.
- Self-contained
- Clear style and results are either proved or stated precisely with adequate references
- The authors have several years experience in this field
- Several subjects (some of them non usual in books of this kind) in one single book, including nonsmooth optimization and vector optimization problems
- Useful long references list at the end of each chapter

"To the reader who seeks a comprehensive, rigorous text on optimization in a finite dimensional space, with detailed, clear explanations and examples, the book could be very acttractive." Zvi Artstein (Rehovot), in: Mathematical Reviews, 2005 "The book contains several excellent tables and figures which summarize interrelations between different concepts, like different notions of convexity, or the implications between the numerous constraint quailifications." Mirjam Dur (Darmstadt University of Technology),in: Mathematical Methods of Operational Research, p.2, Vol. 61, 2005)

Contents
Preface
Introduction
Optimization Problems
Basic Mathematical Preliminaries and Notations References to Chapter I
Convex Sets, Convex And Generalized Convex Functions
Convex Sets and Their Main Properties
Separation Theorems
Some Particular Convex Sets. Convex Cone
Theorems of the Alternative for Linear Systems
Convex Functions
Directional Derivatives and Subgradients of Convex Functions
Conjugate Functions
Extrema of Convex Functions
Systems of Convex Functions and Nonlinear Theorems of the Alternative
Generalized Convex Functions
Relationships Between the Various Classes of Generalized Convex Functions. Properties in Optimization Problems
Generalized Monotonicity and Generalized Convexity
Comparison Between Convex and Generalized Convex Functions
Generalized Convexity at a Point
Convexity, Pseudoconvexity and Quasiconvexity of Composite Functions
Convexity, Pseudoconvexity and Quasiconvexity of Quadratic Functions
Other Types of Generalized Convex Functions References to Chapter II
Smooth Optimization Problems Saddle Point Conditions
Introduction
Unconstrained Extremum Problems and Extremum Problems with a Set Constraint
Equality Constrained Extremum Problems
Local Cone Approximations of Sets
Necessary Optimality Conditions for Problem (P) where the Optimal Point is Interior to X
Necessary Optimality Conditions for Problems (P e); and The Case of a Set Constraint
Again on Constraint Qualifications
Necessary Optimality Conditions for (P 1)
Sufficient First-Order Optimality Conditions for (P) and (P 1)
Second-Order Optimality Conditions
Linearization Properties of a Nonlinear Programming Problem
Some Specific Cases
Extensions to Topological Spaces
Optimality Criteria of the Saddle Point Type References to Chapter III
Nonsmooth Optimization Problems
Preliminary Remarks
Differentiability
Directional Derivatives and Subdifferentials for Convex Functions
Generalized Directional Derivatives
Generalized Gradient Mappings
Abstract Cone Approximations of Sets and Relating Differentiability Notions
Special K-Directional Derivative
Generalized Optimality Conditions. References to Chapter IV
Duality
Preliminary Remarks
Duality in Linear Optimization
Duality in Convex Optimization (Wolfe Duality)
Lagrange Duality
Perturbed Optimization Problems. References to Chapter V
Vector Optimization
Vector Optimization Problems
Conical Preference Orders
Optimality (or Efficiency) Notions
Proper Efficiency
Theorems of Existence
Optimality Conditions
Scalarization
The Nondifferentiable Case. References to Chapter VI
Subject
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780444505507
ISBN-10: 0444505504
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 614
Published: 24th March 2004
Publisher: Elsevier Science & Technology
Country of Publication: GB
Dimensions (cm): 22.9 x 15.2  x 3.18
Weight (kg): 1.09