Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type variational principle
Ekeland's principle is a maximal point theorem in a product space, the second component of which is partially ordered by a cone, that has to satisfy special conditions (f.i. bounded base, nuclearity, Daniel's property, inner points). The aim of the paper is to find a variational principle of Ekeland's type, where that cone is only convex, pointed, and closed. The authors succeed: For the case, that the mentioned product space is the product of two Banach spaces (that means, that the objective function has values in a Banach space), and the cone in the second space has only the permitted properties, they prove a theorem of Ekeland's type (their theorem 9), where only the third condition (the condition for approximate efficiency) has a slightly changed form (compared with the up to now known results). The proof uses essentially a helping cone, which is well-based. The new results are not only an essential contribution to make the connection between Ekeland's principle, Pareto efficiency, and nuclearity of cones clearer, but there are interesting applications, f.i. for vector valued approximation problems (generalized Kolmogorov conditions for approximate solutions) and, using a nonconvex separation theorem, for optimality conditions of approximate solutions, if the objective function satisfies differentiability conditions. Finally hints are given, that the results can be extended to Hausdorff locally convex spaces.
scientific article; zbMATH DE number 5070762
scientific article