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Infinite-dimensional optimization and convexity. (English) Zbl 0565.49003

Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. VIII, 166 p. (1983).
This course is mainly concerned with existence theory. Given an optimization problem, consisting in minimizing a functional over some feasible set, usually defined by constraints, we want to know whether an optimal solution, i.e., a minimizer, can be found. Positive answers to this question rely on growth conditions for the criterion or boundedness of the feasible set in the finite-dimensional case. In the infinite- dimensional case, problems in the calculus of variations for instance, something more is needed, namely convexity. To what extent and precisely why convexity is needed is our main concern. Chapter I is basically an introduction to the existence problem and tries to give, using finite- dimensional geometry and dynamics, a feeling for why and where convexity should appear. Besides rigorous theorems and proofs, it contains some heuristics. Chapter II deals with the so-called direct approach, which consists in showing that minimizing sequences converge. The general theory of existence for optimization problems in Banach spaces is sketched including recent results in the non-convex case, and applications are given to the calculus of variations with a view to the Ramsey problem in welfare economics. Finally, Chapter III deals with duality theory, including recent results on the non-convex case and contains a brief survey of convex analysis. We end - fittingly - with relaxation theory.

MSC:

49J27 Existence theories for problems in abstract spaces
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
90C25 Convex programming
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
46A55 Convex sets in topological linear spaces; Choquet theory
49J45 Methods involving semicontinuity and convergence; relaxation
49N15 Duality theory (optimization)