Well-posed Optimization ProblemsThis book presents in a unified way the mathematical theory of well-posedness in optimization. The basic concepts of well-posedness and the links among them are studied, in particular Hadamard and Tykhonov well-posedness. Abstract optimization problems as well as applications to optimal control, calculus of variations and mathematical programming are considered. Both the pure and applied side of these topics are presented. The main subject is often introduced by heuristics, particular cases and examples. Complete proofs are provided. The expected knowledge of the reader does not extend beyond textbook (real and functional) analysis, some topology and differential equations and basic optimization. References are provided for more advanced topics. The book is addressed to mathematicians interested in optimization and related topics, and also to engineers, control theorists, economists and applied scientists who can find here a mathematical justification of practical procedures they encounter. -- Publisher description. |
Contents
TYKHONOV WELLPOSEDNESS | 1 |
Section 1 | 36 |
GENERIC WELLPOSEDNESS | 81 |
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A(yo arg min X arg min(X assume Banach space best approximation problems bounded characterization Co(X compact consider Conv corollary defined denote diam dist Ekeland equation Example exists fixed fn(un fo(x Fréchet differentiable given Hausdorff hence Hilbert space I(xn I(xo I(yn implies In(n In(un In(x inf I(X lim inf lim sup linear Lipschitz continuity lower semicontinuous Lucchetti Math metric space minimizing sequence Moreover Mosco convergence multifunction nonempty norm obtain optimal control optimal control problems optimal solution optimal value optimization problems perturbations Polyak posed posedness Proof of lemma proof of theorem Proposition prove reflexive rotund sequential solution Hadamard strong convergence strongly Tykhonov well-posed sub lev subset sufficiently large Theorem Let topology trajectory Tykhonov well-posedness uniform convergence uniformly unique upper semicontinuous variational inequality weak yielding