An inverse logical convolution in the problem of the search for a multiple vector minimax (Q1848650)

The authors consider the problem of searching for the value of consecutive multiple vector minimax: \[ \Phi^* =\underset{\omega^1 \in W^1}{\text{Min}} \underset{u^1\in U^1(\omega^1)}{\text{Max}} \underset {\omega^2\in W_2(u)}{\text{Min}} \underset{u^2\in U^2(\omega^2)}{\text{Max}}\dots \underset{\omega^T \in W^T(u^{T-1})}{\text{Min}} \underset{u^T \in U^T(\omega^T)}{\text{Max}} \Phi(u,\omega),\tag{1} \] where the vector \(\Phi(u, \omega)\) is called the criteria vector, \(u^t\) is the monitoring factor, and \(\omega^t\) is the indeterminate factor. This problem can be formally reduced to the following: \(\Phi^* =\operatorname {Max} \Psi\), where \(\Psi\) is the set of guaranteed estimations of accessible values of the vector criteria. By means of the convolution \(\min_{i \in I(\mu)} \mu_i^{-1}\phi_i\), \(\mu \in M\), where \(M=\{\mu\geq 0 \mid \sum_{i=1}^Q\mu_i = 1 \) is the standard simplex in \(R^Q\}\) the problem can be reduced to the parametric family of problems: \[ \Theta (\mu) =\min_{\omega^1 \in W^1} \max_{u^1 \in U^1(\omega^1)}\dots \min_{\omega^T}\in W^T(u^{T-1})\;\max_{u^T}\in U^T(\omega^T)\;\min_{i \in I(\mu)} \mu^{-1}_i\phi_i(u, \omega), \] where \(I(\mu)= \{ i=1,2,\dots , Q\mid \mu_i \neq 0\}.\)