Duality and regularization for inf-sup problems
Let X and Y be nonvoid sets and let L be a real-valued function defined on \(X\times Y\). With \(\alpha:=\inf_{X}\sup_{Y}L\), \(\beta:=\sup_{Y}\inf_{X}L\) the inequality \(\alpha\geq \beta\) is always valid, and one is interested in the duality relation \(\alpha =\beta\). The classical von Neumann minimax theorem, as well as its infinite dimensional extension by \textit{K. Fan} [Proc. Nat. Acad. Sci. USA 39, 42- 47 (1953; Zbl 0050.065)] and more recent refinements due to \textit{B. Fuchssteiner} and \textit{H. König} [in: General inequalities 2, Proc. 2nd int. Conf., Oberwolfach 1978, ISNM 47, 255-266 (1980; Zbl 0452.46002)], and \textit{S. Simons} [in: Game theory and mathematical economics, Proc. Semin., Bonn/Hagen 1980, 379-388 (1981; Zbl 0475.49018)] which guarantee the duality, require among other conditions the compactness of at least one of the two sets X, Y. Since this compactness assumption is too strong for many applications in optimization and related fields, we consider regularized values \(\alpha^*\leq \alpha\) and \(\beta^*\geq \beta\) such that, under suitable assumptions, the regularized duality \(\alpha^*=\beta^*\) holds. This approach is analyzed in this paper within three different regularization schemes and in each case sufficient conditions for regularized duality are provided.
Fenchel-Rockafellar perturbation approach
scientific article; zbMATH DE number 4126163
scientific article