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Smooth transformation of the generalized minimax problem. (English) Zbl 0890.90165
Summary: We consider the generalized minimax problem, that is, the problem of minimizing a function $$\varphi(x)= F(g_1,(x),\dots, g_m(x))$$, where $$F$$ is a smooth function and each $$g_i$$ is the maximum of a finite number of smooth functions. We prove that, under suitable assumptions, it is possible to construct a continuously differentiable exact barrier function, whose minimizers yield the minimizers of the function $$\varphi$$. In this way, the nonsmooth original problem can be solved by usual minimization techniques for unconstrained differentiable functions.

##### MSC:
 90C30 Nonlinear programming 49J52 Nonsmooth analysis 49J35 Existence of solutions for minimax problems
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##### References:
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