Convexity and optimization in Banach spaces

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.