Liping Tang, Dongshan Lang, Xinmin Yang, Approximate solutions to vector variational inequalities based on improvement sets: Existence, characterization, and optimality conditions
Full Text: PDF
DOI: 10.23952/jnva.10.2026.4.02
Volume 10, Issue 4, 1 August 2026, Pages 735-751
Abstract. In this paper, we use the improvement set as a key tool to formulate new approximate vector variational inequality models and introduce novel concepts of approximate solutions to vector variational inequalities in Banach spaces. By using the KKM-Fan theorem and Brouwer’s fixed-point theorem, two existence results of such approximate solutions are established, respectively. In the convex setting, linear scalarization characterizations of the proposed approximate solutions are derived through the classical convex separation theorem, and optimality conditions for these approximate solutions are obtained via Ekeland’s variational principle and the Fermat rule. In the nonconvex setting, we refine a key property of the Clarke subdifferential of a specialized nonlinear scalarization function, namely the oriented distance function. By utilizing this improved property together with Ekeland’s variational principle, we establish optimality conditions for the proposed approximate solutions through the nonlinear scalarization approach.
How to Cite this Article:
L. Tang, D. Lang, X. Yang, Approximate solutions to vector variational inequalities based on improvement sets: Existence, characterization, and optimality conditions, J. Nonlinear Var. Anal. 10 (2026), 735-751.