Weak Pareto Optimality of Multiobjective Problem in a Banach Space

In the previous paper ([5]), we studied the ordinary multiobjective convex program on a locally convex linear topological space in the case that the objective functions and the constraint functions were continuous and convex, but not always Gateaux differ entiable. In the case, we showed that the generalized Kuhn-Tucker conditions given by a subdifferential formula were necessary and sufficient for weak Pareto optimum. In this paper, we consider the ordinary multiobjective program on a Banach space in the case that objective functions and constraint functions are locally Lipschitzian but not always convex, and derive Kuhn-Tucker forms given by Clarke's generalized gradients ([1]) as necessary conditions for weak Pareto optimum. Theorem 2.1 is a generalization of Theorem 1.1 of Schechter ([6]) which is concerned to ordinary pro gram with a scalar-valued objective function. In this paper, X and X* are a real Banach space and its continuous dual, whose origins are denoted by 0 and 0*, respectively. By 0 we denote the empty set.