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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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