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Well-posedness criteria in optimization with application to the calculus of variations. (English) Zbl 0841.49005

In this paper, for the global optimization problem (X,J), to minimize the proper extended real-valued function J:X(-,) over the given subset X of a normed linear space equipped with the strong convergence, well-posedness criteria are derived. The given problem is embedded into a smoothly parametrized family (X,I(·,p)) of minimization problems, where p is a parameter belonging to a given Banach space L, and p * is the parameter value to which the given unperturbed problem corresponds, i.e., I(x,p * )=J(x) xX. Defining the value function V(p)=inf{I(x,p)xX} the author gives the following definition of well-posedness.

(X,J) is well-posed with respect to the embedding iff V(p)>-, pL, and there exists a unique x * =argmin(X,J) and for every sequence p n p * and every sequence x n X such that I(x n ,p n )-V(p n )0 as n we have x n x * in X.

This definition is stronger than the Tikhonov well-posedness. In the following, the defined well-posedness is related under suitable conditions to the differentiability properties of V at p * . These abstract results are applied to one-dimensional problems of the calculus of variations.


MSC:
49J27Optimal control problems in abstract spaces (existence)
49K99Optimality conditions
90C99Mathematical programming