Minimax-maximin relations for the problem of vector-valued criteria optimization

The minimax-maximin relations for vector-valued functionals over the real field are studied. An increase in the dimensionality of criteria may result in a violation of some basic relations, for example, in an inequality between maximin and minimax that is always true for classic problems. Accordingly, the conditions for its correctness or violation need to be established. This paper introduces the definitions of set-valued minimax and maximin for multidimensional criteria and with an analogue in the classic minimax inequality. Necessary and sufficient conditions for its correctness and violation are described for two particular types of vector-valued functionals: the bilinear ones and those with separated variables.