Synopses & Reviews
"Optimization on Metric and Normed Spaces" is devoted to the recent progress in optimization on Banach spaces and complete metric spaces. Optimization problems are usually considered on metric spaces satisfying certain compactness assumptions which guarantee the existence of solutions and convergence of algorithms. This book considers spaces that do not satisfy such compactness assumptions. In order to overcome these difficulties, the book uses the Baire category approach and considers approximate solutions. Therefore, it presents a number of new results concerning penalty methods in constrained optimization, existence of solutions in parametric optimization, well-posedness of vector minimization problems, and many other results obtained in the last ten years. The book is intended for mathematicians interested in optimization and applied functional analysis.
Covering recent work on Banach and complete metric spaces, this book uses the Baire approach and considers approximate solutions. It presents new results including penalty methods in constrained optimization and extant solutions in parametric optimization.
Table of Contents
Preface. - Introduction. -1. Exact penalty in constrained optimization. - 1.1. A sufficient condition for exact penalty in constrained optimization. - 1.2. Existence of exact penalty for inequality-constrained problems. - 1.3. Existence of exact penalty for optimization problems with mixed constraints in Banach spaces. - 1.4. Existence of exact penalty for constrained optimization problems in Hilbert spaces with smooth constraint and objective functions. - 1.5. Existence of exact penalty for constrained optimization problems in metric spaces. - 2. Variational principles and generic well-posedness of optimization problems. - 2.1 Generic vriational principles. - 2.2. Porosity and well-posedness of optimization problems. - 2.3. Existence of solutions for a class of minimization problems with a generic objective function. - 2.4. A generic existence result for a class of optimization problems. - 2.5. Well-posedness and porosity in convex optimization. - 2.6. A porosity result in convex optimization in reflexive Banach spaces. - 3. Parametric optimization. - 3.1. Generic existence in parametric optimization. - 3.2. Variational principles and their concretizations. - 3.3. Two generic existence results. - 3.4. A generic existence result for the problem (P). - 3.5. Existence of solutions in parametric optimization and porosity. - 3.6. Variational principles and porosity. - 3.7. Concretization of variational principles. - 3.8. Existence result for the problem (P2). - 3.9. Existence result for the problem (P1). - 3.10. Generic well-posedness in parametric optimization with constraints. - 4. Optimization with increasing objective functions. - 4.1. Generic existence of solutions of minimization problems with increasing objective functions. - 4.2. A variational principle. - 4.3. Spaces of increasing coercive functions. - 4.4. The proof of the first generic existence theorem. - 4.5. Spaces of increasing noncoercive functions. - 4.6. The proof of the second generic existence theorem. - 4.7. Spaces of increasing quasiconvex functions. - 4.8. The proof of the third generic existence result. - 4.9. Spaces of increasing convex functions. - 4.10. The proof of the fourth generic existence result. - 4.11. The generic existence result for the minimization problem (P2). - 4.12. The proof of the generic existence result for the problem (P1). - 4.13. Existence of solutions of minimization problems with an increasing objective functions and porosity. - 4.14. Well-posedness of minimization problems with increasing objective functions. - 4.15. Porosity and variational principles. - 4.16. Porosity results. - 5. Generic well-posedness of minimization problems with constraints. - 5.1. Preliminaries. - 5.2. Problems with continuous objective and constraint functions. - 5.3. Problems with smooth objective and constraint functions. - 5.4. Extensions. - 6. Vector optimization. - 6.1. Preliminaries. - 6.2. A density result. - 6.3. A generic result. - 6.4. A well-posedness result. - 7. Minimal solutions for infinite horizon problems in metric spaces. - 7.1. Preliminaries and main result. - 7.2. Auxiliary results. - 7.3. Proof of the main result. - 7.4. Properties of good sequences. - 7.5. Infinite horizon convex optimization problems in a Banach space. - References. - Index.