Nonsmooth multiobjective optimization problems and weak vector quasi-variational inequalities

Abstract

In this article, we investigate a nonsmooth multiobjective optimization problem (MOP) under generalized invexity. First, the Kuhn–Tucker type optimality conditions for MOP are obtained. Furthermore, the relationships between weakly efficient solutions of MOP and vector valued saddle points of its Lagrange function are established. Last but not the least, the relations between weakly efficient solutions of MOP and solutions of Hartman–Stampacchia weak vector quasi-variational inequalities and Hartman–Stampacchia nonlinear weak vector quasi-variational inequalities are also derived under some suitable assumptions. These results extend and improve some known results in the literature.

Appendix

FKKM Lemma Guu and Li (2009), Wu et al. (2011): Let $K$ be a nonempty subset of $R^{m}$, $G:K\rightarrow 2^{R^{m}}$ be a KKM mapping, i.e., for every finite subset $\{x_{1},x_{2},\ldots ,x_{m}\}$ of $K,\text{ co }\{x_{1},x_{2},\ldots ,x_{m}\}$ is contained in $\bigcup _{i=1}^{m}G(x_{i})$ where $\text{ co }$ denotes the convex hull, such that for any $x\in K,G(x)$ is closed and $G(x^{*})$ is bounded for some $x^{*}\in K$. Then there exists $y^{*}\in K$ such that $y^{*}\in G(x)$ for all $x\in K$, i.e., $\bigcap _{x\in K}G(x)\ne \emptyset$.