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In this paper different classes of unconstrained and constrained minimization problems with max-min objective and/or constraint functions are studied. First we consider simple problems with the max-min objective function, where the explicit description of all local minimizers is given. Then we discuss the application of the cutting angle and discrete gradient methods as well as a special penalization method for solving constrained problems. We report the results of preliminary numerical experiments.doi:10.1007/0-387-24255-4_1 fatcat:gthzdjlgjzaptbbzpvzxavbdvu