Vector variational inequality as a tool for studying vector optimization problems

Proper Efficiency for Set-Valued Optimization Problems and Vector Variational-Like Inequalities

“…Very recently, Long et al [19] generalized the results of [1] to nondifferential pseudoinvexity. For other results on this topic, we refer readers to [2,3,4,17,20,22,23] and the references therein.…”

Generalized Vector Minty's Lemma

“…In particular, Minty's lemma [2,6,7,10,14,15,18] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, free-boundary problems, elasticity problems and stochastic optimal control problems when the operator is monotone and the domain is convex. The classical Minty's inequality and Minty's lemma offered the regularity results of the solution for a generalized nonhomogeneous boundary value problem [2] and, when the operator is a gradient, also a minimum principle for convex optimization problems [6].…”

“…Giannessi [7] established the equivalence between a differentiable convex vector optimization problem and a vector variational inequality. Lee et al [15] showed that vector variational inequality can be an efficient tool for studying vector optimization problems. Moreover, using a vector variational-like inequality, Lee et al [14] proved existence theorems for solutions of nondifferentiable invex optimization problems.…”

Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces

“…Among its topological properties, the connectedness is of interest. Recently, Lee et al [25], Cheng [26] have studied the connectedness of weak efficient solutions set for vector variational inequalities in finite dimensional Euclidean space. Gong [27][28][29] has studied the connectedness of the various solutions set for VEPs in infinite dimension space.…”