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.


49J27 Existence theories for problems in abstract spaces
49K99 Optimality conditions
90C99 Mathematical programming
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