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A characterization of strict local minimizers of order one for nonsmooth static minmax problems. (English) Zbl 1121.90123

The authors consider the following problem:



S:={x n |g i (x)0,i=1,,p},
f(x):=sup yY φ(x,y),

φ: n × m Y is a nonempty subset of m , and g i : n .

A point x 0 S is said to be a strict local minimizer of order 1 if there exist ϵ>0 and β>0 such that

f(x)f(x 0 )+βx-x 0 forallxS,x-x 0 β·

Under weak assumptions on φ and the g i , the authors derive a necessary optimality condition for a local minimizer. Moreover, under a certain constraint qualification, a necessary and sufficient condition for a strict local minimizer of order 1 is also established. The optimality conditions are multiplier rules involving Clarke’s generalized gradient.

90C46Optimality conditions, duality
49J35Minimax problems (existence)