Mathematics of Optimization: Smooth and Nonsmooth Case

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

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