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| MATHEMATICS OF OPTIMIZATION: SMOOTH AND NONSMOOTH CASE
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By
Giorgio Giorgi, University of Pavia, Pavia, Italy.
A. Guerraggio, Insubria University of varese, varese, Italy.
J. Thierfelder, Technical University of Ilmenau, Ilmenau, Germany
Description
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.
Audience
Postgraduate, researchers, university professors, graduates and engineers.
Contents
Contents
Preface.
CHAPTER I.INTRODUCTION.
1.1 Optimization Problems.
1.2 Basic Mathematical Preliminaries and Notations.
References
to Chapter I.
CHAPTER II.CONVEX SETS, CONVEX AND GENERALIZED CONVEX FUNCTIONS.
2.1 Convex Sets and Their Main Properties.
2.2 Separation
Theorems.
2.3 Some Particular Convex Sets. Convex Cone.
2.4 Theorems of the Alternative for Linear Systems.
2.5 Convex Functions.
2.6
Directional Derivatives and Subgradients of Convex Functions.
2.7 Conjugate Functions.
2.8 Extrema of Convex Functions.
2.9 Systems of
Convex Functions and Nonlinear Theorems of the Alternative.
2.10 Generalized Convex Functions.
2.11 Relationships Between the Various
Classes of Generalized Convex Functions. Properties in Optimization Problems.
2.12 Generalized Monotonicity and Generalized Convexity.
2.13 Comparison Between Convex and Generalized Convex Functions.
2.14 Generalized Convexity at a Point.
2.15 Convexity, Pseudoconvexity
and Quasiconvexity of Composite Functions.
2.16 Convexity, Pseudoconvexity and Quasiconvexity of Quadratic Functions.
2.17 Other Types
of Generalized Convex Functions References to Chapter II.
CHAPTER III.SMOOTH OPTIMIZATION PROBLEMS
SADDLE POINT CONDITIONS.
3.1 Introduction.
3.2 Unconstrained Extremum Problems and Extremum
Problems with a Set Constraint.
3.3 Equality Constrained Extremum Problems.
3.4 Local
Cone Approximations of Sets.
3.5 Necessary Optimality Conditions for Problem (P) where the Optimal Point is Interior to X.
3.6 Necessary
Optimality Conditions for Problems (P e); and The Case of a Set Constraint.
3.7 Again on Constraint Qualifications.
3.8 Necessary Optimality
Conditions for (P 1).
3.9 Sufficient First-Order Optimality Conditions for (P) and (P 1).
3.10 Second-Order Optimality Conditions.
3.11
Linearization Properties of a Nonlinear Programming Problem.
3.12 Some Specific Cases.
3.13 Extensions to Topological Spaces.
3.14 Optimality
Criteria of the Saddle Point Type References to Chapter III
CHAPTER IV. NONSMOOTH OPTIMIZATION PROBLEMS.
4.1 Preliminary Remarks.
4.2
Differentiability.
4.3 Directional Derivatives and Subdifferentials for Convex Functions.
4.4 Generalized Directional Derivatives.
4.5
Generalized Gradient Mappings.
4.6 Abstract Cone Approximations of Sets and Relating Differentiability Notions.
4.7 Special K-Directional
Derivative.
4.8 Generalized Optimality Conditions.
References to Chapter IV
CHAPTER V. DUALITY.
5.1 Preliminary Remarks.
5.2 Duality
in Linear Optimization.
5.3 Duality in Convex Optimization (Wolfe Duality).
5.4 Lagrange Duality.
5.5 Perturbed Optimization Problems.
References to Chapter V
CHAPTER VI. VECTOR OPTIMIZATION.
6.1 Vector Optimization Problems.
6.2 Conical Preference Orders.
6.3 Optimality
(or Efficiency) Notions.
6.4 Proper Efficiency.
6.5 Theorems of Existence.
6.6 Optimality Conditions.
6.7 Scalarization.
6.8 The Nondifferentiable
Case.
References to Chapter VI.
SUBJECT INDEX
| Bibliographic details |
Hardbound, 614 pages, publication date: MAR-2004
ISBN-13: 978-0-444-50550-7
ISBN-10: 0-444-50550-4
Imprint: ELSEVIER
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| Price and Ordering |
Price:
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USD 190
EUR 135.95
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Last update: 7 Sep 2009
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