Convexity and Optimization in Banach Spaces

ISBN-13: 9789401029209 / Angielski / Miękka / 2011 / 316 str.

V. Barbu; Th Precupanu

Opis

It was our intention to make the book seIf-contained and accessible to a large number of readerso To achieve this in Chapter I we have summarized with or without proofs some basic results in functional analysis and non-linear operator equations in Banach spaces. The list of references is not intended to be complete. It refers only to papers which were used or are directly connected with the subjects treated in this book. V. BARBU Jassy, July 1974 TH. PRECUPANU x Pre tace to the English edition This English edition differs from the Romanian version in that several ehanges have been made. Several seetions in Chapters III and IV have been entirely rewritten and several errors and inaccuracies in the first edition were correeted. The authors wish to express their gratitude to Dr. V. Popescu, from the Jassy University, who kindly assisted them in reading various parts of the manuseript, eorreeting errors and improv- ing the presentation.

Spis treści

1 Fundamentals of Functional Analysis.- 1 Convexity in topological linear spaces.- 1.1 Classes of topological linear spaces.- 1.2 Convex sets.- 1.3 Separation of convex sets.- 2 Duality in linear normed spaces.- 2.1 Dual systems of the linear spaces.- 2.2 Weak topologies on linear normed spaces.- 2.3 Reflexive Banach spaces.- 2.4 Duality mapping.- 3 Vector-valued functions and distributions.- 3.1 The Bochner integral.- 3.2 Bounded variation vector functions.- 3.3 Vector distributions on the real axis.- 3.4 Vector distributions and Wk, P spaces.- 3.5 Sobolev spaces.- 4 Maximal monotone operators.- 4.1 Definitions and fundamental results.- 4.2 Evolution equations in Hilbert spaces.- 2 Convex Functions.- 1 General properties of convex functions.- 1.1 Definitions and basic properties.- 1.2 Lower-semicontinuous functions.- 1.3 Lower-semicontinuous convex functions.- 1.4 Conjugate functions.- 2 The subdifferential of a convex function.- 2.1 Definition and fundamental results.- 2.2 Further properties of subdifferential mappings.- 2.3 Regularization of the convex function.- 2.4 Perturbations of cyclically monotone operators.- 2.5 Variational inequalities.- 3 Concave-convex functions.- 3.1 Saddle points and minimax equality.- 3.2 Saddle functions.- 3.3 Minimax theorems.- Bibliographical notes.- 3 Convex Programming.- 1 Optimality conditions.- 1.1 The case of a finite number of constraints.- 1.2 Operatorial convex constraints.- 1.3 Non-linear programming in the case of Fréchet-differentiability.- 1.4 Examples.- 2 Duality in convex programming.- 2.1 Dual problems.- 2.2 Fenchel duality theorem.- 2.3 Examples.- 3 Applications of the duality theory.- 3.1 Linear programming.- 3.2 The best approximation problem.- Bibliographical notes.- 4 Convex Control Problems in Hilbert Spaces.- 1 Necessary and sufficient conditions for optimality.- 1.1 Basic assumptions.- 1.2 Optimality theorem.- 1.3 Proof of Theorem 1.1.- 1.4 Proof of Theorem 1.2.- 1.5 Further remarks on optimality theorems.- 2 The dual optimal control problem.- 2.1 Formulation of the dual problem.- 2.2 The duality theorem.- 2.3 Some examples.- 3 Convex control problems associated with linear evolutionary processes in Hilbert space.- 3.1 Statement of the problem.- 3.2 The optimality theorem.- 3.3 Optimal control of linear hereditary systems.- 4 Synthesis of optimal control.- 4.1 Optimal synthesis function.- 4.2 Hamilton-Jacobi equations.- Bibliographical notes.