Lagrange multiplier theorem of multiobjective programming problems with set functions (Q1176838)

The authors consider a class of multiobjective programming problems with set functions in which the domination structure does not necessarily contain the nonnegative orthant. First, a generalized Farkas-Minkowski theorem for set functions is proved. As a consequence, a Lagrange multipliers theorem is obtained for the case of single objective programming problems with set functions. Next, with a scalarization proposition, the result is extended to the case of multiobjective programming problems with set functions under the hypotheses of a generalized Slater's constraint qualification and of a restriction to proper efficiency. The authors also relate proper efficient solutions to the zero-like functions associated with the subdifferentials of the set functions involved, which presents a new perspective for a proper efficient solution.