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Well-posedness of constrained minimization problems via saddle-points

Authors:	Biagio Ricceri

Institution:	(1) Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Abstract:	In this paper, it is proved a very general well-posedness result for a class of constrained minimization problems of which the following is a particular case: Let X be a Hausdorff topological space and let $J, \Phi{:} X\to {\mathbb{R}}$ be two non-constant functions such that, for each $\lambda\in {\mathbb{R}}$ , the function $J+\lambda\Phi$ has sequentially compact sub-level sets and admits a unique global minimum in X. Then, for each $r\in ]\inf_X\Phi,\sup_X\Phi$ , the restriction of J to $\Phi^{-1}(r)$ has a unique global minimum, say $\hat x_r$ , toward which every minimizing sequence converges. Moreover, the functions $r\to \hat x_r$ and $r\to J(\hat x_r)$ are continuous in $]\inf_X\Phi,\sup_X\Phi$ .

Keywords:	Constrained minimization problem Well-posedness Minimax Saddle-point
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