Existence theorems for Pareto optimization; multivalued and Banach space valued functionals

Abstract

Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.