Abstract
We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces. We study these problems without any convexity and compactness assumptions. Our results are different from any existence results of these types of problems in topological vector spaces.
Highlights
Let X, dX and Y, dY be two metric spaces, let Z be a real Hausdorff topological vector space ordered by a nonempty pointed closed convex cone K in Z with nonempty interior, and let W be a real Banach space ordered by a nonempty pointed closed convex cone C with nonempty interior
We find in the literatures that Luo et al 1, Stein and Still 2, Stein 3, Birbil et al 4, Liou et al 5, Lin and Still 6, Lin 7, as well asLin and Hsu 8 have studied mathematical program with equilibrium constraint and bilevel problem on topological vector spaces
To the best of our knowledge, there are no existence theorems in the literatures for loose saddle point and vector saddle point problems for functions defined on the product of metric spaces
Summary
Let X, dX and Y, dY be two metric spaces, let Z be a real Hausdorff topological vector space ordered by a nonempty pointed closed convex cone K in Z with nonempty interior, and let W be a real Banach space ordered by a nonempty pointed closed convex cone C with nonempty interior. The following vector mathematical programs with equilibrium constraint on metric spaces are considered. To the best of our knowledge, there are no existence theorems in the literatures for loose saddle point and vector saddle point problems for functions defined on the product of metric spaces. We study the loose saddle point problems and vector saddle point problems for functions defined on the product of metric spaces. As for applications of our existence theorems on saddle point problems, we study bilevel problems on metric spaces. Our results on mathematical programs with equilibrium constraint, bilevel problems, loose saddle point problems, and vector saddle point problems are different from any existence results of these types of problems in the literatures
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