Mathematics of Optimization: Smooth and Nonsmooth Case

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 PROBLEMSSADDLE POINT CONDITIONS.3.1 Introduction.3.2 Unconstrained Extremum Problems and ExtremumProblems 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 IIICHAPTER 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 IVCHAPTER 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 VCHAPTER 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