Set-Valued Quasi-Metrics and a General Ekeland's Variational Principle in Vector Optimization
Jinghui Qiu · SIAM Journal on Control and Optimization · 2013
We introduce the notion of compatibility between a set-valued quasi-metric and the original metric. By using this notion we prove a general set-valued Ekeland's variational principle (EVP), where the perturbation contains a set-valued quasi-metric which is compatible with the original metric. Here, we need not assume that the order cone is $w$-normal. By using the unified approach for approximate solutions introduced by Gutiérrez, Jiménez, and Novo, we deduce a general version of set-valued EVP based on $(C,\epsilon )$-efficient solutions, where $C$ is a coradiant set contained in the order cone. By choosing two specific versions of the coradiant set $C$ in the general version of EVP, we obtain several particular set-valued EVPs for $\epsilon$-efficient solutions in the sense of Németh and of Dentcheva and Helbig, respectively. These set-valued EVPs improve and generalize the related interesting results in [C. Gutiérrez, B. Jiménez, and V. Novo, SIAM J. Control Optim., 47 (2008), pp. 883--903].