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 fo...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Giorgi, G. (Giorgio)
Otros Autores: Guerraggio, Angelo, 1948-, Thierfelder, J. (J�org)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Amsterdam ; Boston : Elsevier, 2004.
Temas:
Mathematical optimization.
Nonlinear programming.
Optimisation math�ematique.
Programmation non lin�eaire.
Programa�c�ao matem�atica.
Otimiza�c�ao matem�atica.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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