Vector minimax with dependent variables (Q1281245)

The problem of searching of a minimax of the vector function with connected limitations \[ \text{Min}_{w \in W} \text{Max}_{z\in Z(w)} \Phi ( z) \] where \(\Phi (z)=\{\varphi_{1} (z), \varphi_{2} (z), \ldots , \varphi_{n} (z)\}\), \(\Phi(z)\geq 0\), is considered. The functions \(\varphi_{i} ( z)\) are continuous on compact sets \( Z(w)\in{\mathbb R^{n}}\), \(\forall w\in W.\) The vector \(\varPhi (z)\) is called the vector criterion of the problem. Some definitions of a solution concept are given. For the concept of a guaranteed solution the modes of a parametrization of an optimal set are offered. As a result a set corresponding to the guaranteed value of a solution of the problem is obtained.