Generic well-posedness in some classes of optimization problems. (English) Zbl 0644.49023

Let X be a complete metric space and P be the set of all couples (A,f), where A is a nonempty closed subset of X and \(f: X\to R\) is a continuous and real-valued function bounded from below. Every pair (A,f)\(\in P\) gives rise to a constrained optimization problem: find \(x_ 0\in A\) such that \(f(x_ 0)=\inf \{f(x):\) \(x\in A\}\). In this way P can be considered as a set of minimization problems. Let P be endowed with a natural complete metric generated by the Hausdorff metric on the sets A and a uniform metric on the functions f. A problem (A,f)\(\in P\) is said to be well-posed in the sense of Hadamard if it has a unique solution which depends continuously on the data A and f. We do not require X (or A) to be compact, so a particular problem from P may have not even a solution. But since P is a complete metric space the next question makes sense: does the set \(H=\{(A,f)\in P:\) (A,f) is Hadamard well-posed\(\}\) contain a dense and \(G_{\delta}\)-subset of P. In this case we say that most (in the Baire category sense) of the problems in P are Hadamard well-posed. In this paper a positive answer is given to this question. Also, a geometric characterization of Hadamard well-posedness, as well as relations between the Hadamard and other types of well-posedness are given. A class of convex constrained optimization problems is investigated through the above generic point of view.
Reviewer: J.P.Revalski


49K40 Sensitivity, stability, well-posedness
54E52 Baire category, Baire spaces
90C48 Programming in abstract spaces
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