*(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 $(X,I(\xb7,p\left)\right)$ 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., $I(x,{p}^{*})=J\left(x\right)$ $\forall x\in X$. Defining the value function $V\left(p\right)=inf\left\{I\right(x,p)\mid x\in X\}$ the author gives the following definition of well-posedness.

$(X,J)$ is well-posed with respect to the embedding iff $V\left(p\right)>-\infty $, $\forall p\in L$, and there exists a unique ${x}^{*}=argmin(X,J)$ and for every sequence ${p}_{n}\to {p}^{*}$ and every sequence ${x}_{n}\in X$ such that $I({x}_{n},{p}_{n})-V\left({p}_{n}\right)\to 0$ 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.