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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 $$J: X\to (- \infty, \infty)$$ 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., $$I(x, p^*)= J(x)$$ $$\forall x\in X$$. Defining the value function $$V(p)= \inf\{I(x, p)\mid x\in X\}$$ the author gives the following definition of well-posedness.
$$(X, J)$$ is well-posed with respect to the embedding iff $$V(p)> -\infty$$, $$\forall p\in L$$, and there exists a unique $$x^*= \arg\min(X, J)$$ and for every sequence $$p_n\to p^*$$ and every sequence $$x_n\in X$$ such that $$I(x_n, p_n)- V(p_n)\to 0$$ as $$n\to \infty$$ we have $$x_n\to x^*$$ 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:
 49J27 Existence theories for problems in abstract spaces 49K99 Optimality conditions 90C99 Mathematical programming
##### Keywords:
global optimization problem; well-posedness criteria
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##### References:
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