Scalarization approaches for set-valued vector optimization problems and vector variational inequalities (Q1025036)

Let \(C \subset \mathbb R^n\) be a closed, convex and pointed cone with \(int C \neq \emptyset\), and \(K \neq \emptyset\) be a convex subset of \(\mathbb R^n\). Let \(F: K \mapsto 2^{\mathbb R^n}\). In this paper, the authors consider the following set-valued (respectively weak) vector optimization problems \[ \min_{C \setminus {0}} F(x), \text{ subject to } x \in K, \] \[ \min_{intC} \text{ subject to } x \in K, \] as well as set-valued (respectively weak) vector variational inequalities. The purpose of the paper is to present some scalarization methods for these problems. For this, the authors establish some equivalence relationships among set-valued (scalar) optimization problems and set-valued (scalar) quasi-optimization problems under convexity assumptions of the objective functions. Finally, they give some examples to illustrate the scalarization techniques.