Pareto optimality in multiobjective problems

In this study, the optimization theory of Dubovitskii and Milyutin is extended to multiobjective optimization problems, producing new necessary conditions for local Pareto optima. Cones of directions of decrease, cones of feasible directions and a cone of tangent directions, as well as, a new cone of directions of nonincrease play an important role here. The dual cones to the cones of direction of decrease and to the cones of directions of nonincrease are characterized for convex functionals without differentiability, with the aid of their subdifferential, making the optimality theorems applicable. The theory is applied to vector mathematical programming, giving a generalized Fritz John theorem, and other applications are mentioned. It turns out that, under suitable convexity and regularity assumptions, the necessary conditions for local Pareto optima are also necessary and sufficient for global Pareto optimum. With the aid of the theory presented here, a result is obtained for the, so-called, "scalarization" problem of multiobjective optimization.