On the Problems of Minmax–Maxmin Type Under Vector-Valued Criteria

Abstract

This paper is devoted to methods of solving problems of dynamic optimization under multivalued criteria. Such problems require a full description of the related Pareto boundary for the set of all values of the vector criteria and also an investigation of the dynamics of such set. Of special interest are problems for systems that also include a bounded disturbance in the system equation. Hence, it appears useful to develop methods of calculating guaranteed estimates for possible realizations of related solution dynamics. Such estimates are introduced in this paper. Introduced here are the notions of vector values for minmax and maxmin with basic properties of such items. In the second part of this work, there given are some sufficient conditions for the fulfillment of an analogy of classical scalar inequalities that involve relations between minmax and maxmin. An illustrative example is worked out for a linear-quadratic type of vector-valued optimization with bounded disturbance in the system equations.