The convolution method in the problem of the search for a vector maximin
A formal approach to a problem of finding of the minimax of the vector function \(\Phi(z,w)= \{\varphi_1(z,w), \varphi_2(z,w),\dots, \varphi_Q(z,w)\}, \Phi(z,w)\geq 0\) is presented. Two formal values of the vectorial maximin, i.e. the sets \[ \Phi_\ast(z,w) = \max \bigcup_{w\in W} \min \bigcup_{z\in Z(w) \{\psi \mid\Phi(z,w)}\leq \psi \} \tag{1} \] and \[ \Phi_\ast^\prime(z,w) = \min \bigcap_{w\in W}\bigcap_{z\in Z(w)} \{\psi\mid\Phi(z,w)\leq\psi\} \tag{2} \] are compared. Here \(\max\) and \(\min\) denote sets of maximal and minimal elements with respect to an order relation of the type ``\(>\)'' or ``\(\geq\)''. It is found that for relations of the type ``\(>\)'' is available the equality \(\Phi_\ast^\prime=\Phi_\ast.\)
scientific article; zbMATH DE number 1569654