An example showing that A-lower semi-continuity is essential for minimax continuity theorems

2018/02/07 by Feinberg, Eugene A., Kasyanov, Pavlo O., Zgurovsky, Michael Z.
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1802.02302

Abstract

Recently Feinberg et al. [arXiv:1609.03990] established results on continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Such minimax problems appear in games with perfect information, when the second player knows the move of the first one, in turn-based games, and in robust optimization. Some of the results in [arXiv:1609.03990] are proved under the assumption that the multifunction, defining the domains of the second variable, is A-lower semi-continuous. The A-lower semi-continuity property is stronger than lower semi-continuity, but in several important cases these properties coincide. This note provides an example demonstrating that in general the A-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.

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