Foundations of Optimization

ISBN-13: 9783540076803 / Angielski / Miękka / 1976 / 193 str.

M. S. Bazaraa;C. M. Shetty

Opis

Current1y there is a vast amount of literature on nonlinear programming in finite dimensions. The pub1ications deal with convex analysis and severa1 aspects of optimization. On the conditions of optima1ity they deal mainly with generali- tions of known results to more general problems and also with less restrictive assumptions. There are also more general results dealing with duality. There are yet other important publications dealing with algorithmic deve10pment and their applications. This book is intended for researchers in nonlinear programming, and deals mainly with convex analysis, optimality conditions and duality in nonlinear programming. It consolidates the classic results in this area and some of the recent results. The book has been divided into two parts. The first part gives a very comp- hensive background material. Assuming a background of matrix algebra and a senior level course in Analysis, the first part on convex analysis is self-contained, and develops some important results needed for subsequent chapters. The second part deals with optimality conditions and duality. The results are developed using extensively the properties of cones discussed in the first part. This has faci- tated derivations of optimality conditions for equality and inequality constrained problems. Further, minimum-principle type conditions are derived under less restrictive assumptions. We also discuss constraint qualifications and treat some of the more general duality theory in nonlinear programming.

Spis treści

I: Convex Analysis.- 1: Linear Subspaces and Affine Manifolds.- 1.1 Linear Subspaces and Orthogonal Complements.- 1.2 Linear Independence and Dimensionality.- 1.3 Projection Theorem.- 1.4 Affine Manifolds.- 2: Convex Sets.- 2.1 Convex Cones, Convex Sets and Convex Hills.- 2.2 Carathéodory Type Theorems.- 2.3 Relative Interior and Related Properties of Convex Sets.- 2.4 Support and Separation Theorems.- 3: Convex Cones.- 3.1 Cones, Convex Cones and Polar Cones.- 3.2 Polyhedral Cones.- 3.3 Cones Generated by Sets.- 3.4 Cone of Tangents.- 3.5 Cone of Attainable Directions, Cone of Feasible Directions and Cone of Interior Directions.- 4: Convex Functions.- 4.1 Definitions and Preliminary Results.- 4.2 Continuity and Directional Differentiability of Convex Functions.- 4.3 Differentiable Convex Functions.- 4.4 Some Examples of Convex Functions.- 4.5 Generalization of Convex Functions.- II: Optimality Conditions and Duality.- 5: Stationary Point Optimality Conditions with Differentiability.- 5.1 Inequality Constrained Problems.- 5.2 Inequality and Equality Constrained Problems.- 5.3 Optimality Criteria of the Minimum Principle Type.- 6: Constraint Qualifications.- 6.1 Inequality Constrained Problems.- 6.2 Equality and Inequality Constrained Problems.- 6.3 Necessary and Sufficient Qualification.- 7: Convex Programming without Differentiability.- 7.1 Saddle Point Optimality Criteria.- 7.2 Stationary Point Optimality Conditions.- 8: Lagrangian Duality.- 8.1 Definitions and Preliminary Results.- 8.2 The Strong Duality Theorem.- 9: Conjugate Duality.- 9.1 Closure of a Function.- 9.2 Conjugate Functions.- 9.3 Main Duality Theorem.- 9.4 Nonlinear Programming via Conjugate Functions.- Selected References.