On solving some classes of multiple minimax problems (Q750317)

The following multiple minimax problem with inequality constraints is considered: \[ (1)\quad \min_{x}\max_{1\leq k\leq r}\min_{y^{(k)}}G(x,k,y^{(k)}),\quad F(x,k,y^{(k)})\leq 0,\quad k=1,...,r. \] The functions \(G(x,k,y^{(k)}\), \(F(x,k,y^{(k)})\) are continuous with respect to x and \(y^{(k)}\) for fixed k, \((x,y^{(k)})\in U\), U is a compact set, \(F(x_ k,k,y^{(k)})\in E^{n_ k}.\) Also considered is the following problem of nonlinear programming: \[ (2)\quad \min_{\nu,x,y^{(k)}}\nu,\quad G(x,k,y^{(k)})\leq \nu,\quad F(x,k,y^{(k)})\leq 0,\quad x=1,...,r. \] Here \(\{y^ k\}\) is a collection of \(y^{(k)}\), \(k=1,...,r\). It is shown that, if \(\nu^*\), \(\{\) \(\bar y^{(k)}\}\), \(x^*\) are the solution of problem (2), then \(x^*\) is the optimal vector for the external minimum of problem (1), and \(\nu^*\) is the optimal value of the objective function of problem (1).