Abstract
We show that duality gaps can be closed under broad hypotheses in minimax problems, provided certain changes are made in the maximum part which increase its value. The primary device is to add a linear perturbation to the saddle function, and send it to zero in the limit. Suprema replace maxima, and infima replace minima. In addition to the usual convexity-concavity type of assumptions on the saddle function and the sets, a form of semireflectivity is required for one of the two spaces of the saddle function. A sharpening of the results is possible when one of the spaces is finite-dimensional. A variant of the proof of the previous results leads to a generalization of a result of Sion, from which the theorem of Kneser and Fan follows.
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Communicated by M. Avriel
This author's work was supported in part by Grant No. DAAG29-77-0024, Army Research Office, Research Triangle Park, North Carolina.
This author's work was supported in part by Grant No. ENG-7900284 of the National Science Foundation.
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Blair, C.E., Duffin, R.J. & Jeroslow, R.G. A limiting infisup theorem. J Optim Theory Appl 37, 163–175 (1982). https://doi.org/10.1007/BF00934766
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DOI: https://doi.org/10.1007/BF00934766