This article deals with lexicographic equilibrium problems on Banach spaces. We first study the existence of solutions for such problems. Then, we investigate the Painlev-Kuratowski convergence of the solution sets for a family of perturbed problems in a such way that they are perturbed by sequences constrained sets and objective functions converging. Several illustrative examples are given which clarify the essentialness of imposed assumptions. As an application, we discuss various results on the Painlev-Kuratowski convergence for lexicographic variational inequalities.

In this paper, we deal with the inheritation of the semicontinuity of set-valued maps via general scalarization for sets, which is regarded as the framework of generalizations of results by Kuwano, Tanaka, and Yamada in 2010. Since the unified scalarization functions for sets satisfy certain desired semicontinuity, our main theorems can be reduced to the results in earlier study.

In this paper, we study a linearly p + -constrained least-squares problem.We develop the Henig conical regularization approach as a unified framework to deal with the lack of Slater-type constraint qualification.We establish some stability estimates for the regularized problems.For the separable case, p ∈ [1, ∞), we provide an explicit characterization of the Henig dilating cones associated with p + and the associated regularized KKT systems.For the non-separable case ∞ + , we give a condition which ensures that the solution of the least-squares problem can be approximated by regularized solutions whose dual solutions do not contain any finitely additive singular part.