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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) |