Duality and regularization for inf-sup problems
type: Item
instance of: scholarly article
title: Duality and regularization for inf-sup problems
zbMATH Open document ID: 0687.49014
DOI: 10.1007/BF03322614
author: Werner Oettli
author: Joachim Gwinner
published in: Results in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 4126163
zbMATH Keywords: Fenchel-Rockafellar perturbation approach
zbMATH Keywords: homogeneous
zbMATH Keywords: optimization
zbMATH Keywords: von Neumann minimax theorem
zbMATH Keywords: regularized duality
MaRDI profile type: Publication
cites work: Q4077092
cites work: Minimax Theorems
cites work: Exact penalty functions in nonlinear programming
cites work: Q3956760
cites work: Q4050397
full work available at URL: https://doi.org/10.1007/bf03322614
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description: scientific article; zbMATH DE number 4126163