A simple approach to optimality conditions in minmax programming

Abstract

Considering the minmax programming problem, lower and upper subdifferential optimality conditions, in the sense of Mordukhovich, are derived. The approach here, mainly based on the nonsmooth dual objects of Mordukhovich, is completely different from that of most of the previous works where generalizations of the alternative theorem of Farkas have been applied. The results obtained are close to those known in the literature. However, one of the main achievements of this article is that we could also derive necessary optimality conditions for the minmax program of the usual Karush–Kuhn–Tucker type, which seems to be new in this field of study.

Keywords:

• minmax programming
• optimal value function
• optimality conditions
• inner semicompact/semicontinuous set-valued mapping
• convexity
• basic subdifferential and normal cone

AMS Subject Classifications::

• 49K35
• 90C47
• 90C30

Acknowledgements

The author is indebted to the referee for his/her remarks and suggestions that led to important adjustments in this article. The author is also grateful to Boris S. Mordukhovich for inspiring discussion on the upper subdifferential optimality conditions and to Stephan Dempe for insights on bilevel programming. Financial support by the Deutscher Akademischer Austausch Dienst is also acknowledged.