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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.

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