A necessary and sufficient condition for Altman's maximal element problem for constraint sets with many equalities

Abstract

IN [l], ALTMAN proposed a maximal element principle for the optimization problem with the same constraint set as in the Dubovitskii-Milyutin formalism. These sets contain only one equality constraint. In many applications, however, one equality constraint is insufficient to formulate the optimal control problems. It is the purpose of the present paper to further investigate the maximal element problem initiated in [l]. Concerning necessary conditions, such investigations can be developed in two directions. One direction is to generalize the results to constraint sets involving n equality constraints in the operator form, In this direction the maximal element principle can be treated as some generalization of results from [3] and [6]. The other direction is to include in the constraint set n equality constraints in the general nonoperator form. These kinds of problems were investigated in [9], using some properties of cones, in the form of problems of minimizing the functional under constraints. The maximal element principle enables us to consider these optimization problems in the wider sense, as problems of maximal element on the set of constraints. The sufficient condition for the maximal element problem was not discussed in [l]. In the present paper, the sufficient condition is obtained as a generalization of the DubovitskiiMilyutin sufficient condition from [2] and its extension for many equality constraints from [S]. After proving the mentioned results, an example of constraint sets in optimal control from [4] is discussed from the new point of view of maximal element problem and the maximal element principle for this problem is formulated.