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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 $\left(X,J\right)$, to minimize the proper extended real-valued function $J:X\to \left(-\infty ,\infty \right)$ 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 $\left(X,I\left(·,p\right)\right)$ 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., $I\left(x,{p}^{*}\right)=J\left(x\right)$ $\forall x\in X$. Defining the value function $V\left(p\right)=inf\left\{I\left(x,p\right)\mid x\in X\right\}$ the author gives the following definition of well-posedness.

$\left(X,J\right)$ is well-posed with respect to the embedding iff $V\left(p\right)>-\infty$, $\forall p\in L$, and there exists a unique ${x}^{*}=argmin\left(X,J\right)$ and for every sequence ${p}_{n}\to {p}^{*}$ and every sequence ${x}_{n}\in X$ such that $I\left({x}_{n},{p}_{n}\right)-V\left({p}_{n}\right)\to 0$ 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 Optimal control problems in abstract spaces (existence) 49K99 Optimality conditions 90C99 Mathematical programming
##### Keywords:
global optimization problem; well-posedness criteria