Generalized Farkas' theorem and optimization of infinitely constrained problems

This note is concerned with the generalization of Farkas' theorem of the alternative and its application to derive the necessary optimality conditions for min-max problems with satisfaction conditions. Farkas' theorem is generalized to a system of linear inequalities with max operations. The problems studied require a solution at which the worst objective value attains its minimum over a set of solutions fulfilling satisfaction conditions. The satisfaction conditions claim that plural performance criteria should be kept below the permissible level, whatever disturbances may happen or whatever opponents' decisions may be taken. We present a generalized Farkas' theorem in order to derive the necessary optimality conditions for the problems of this class.