Optimality conditions and duality for a \(P\)-connected minimax programming problem (Q5939125)

The authors investigate the following problem: minimize \(\max \{f_1(x), \dots, f_n(x)\}\) subject to \(x\in X\) and \(g_1(x)\leq 0,\dots, g_m(x)\leq 0\), where \(X\) is a nonempty subset of \(\mathbb{R}^p\) and \(f_1,\dots,f_n\), \(g_1, \dots, g_m\) are real-valued functions defined on \(X\). For the solutions of this problem they derive a necessary optimality condition in terms of the directional derivatives of the functions involved with respect to the same arc. Furthermore, under generalized arcwise connectedness assumptions a sufficient optimality condition and duality results of Mond-Weir type are obtained. Finally, it is claimed that these results can easily be generalized for a certain fractional minimax programming problem.