On second-order Karush--Kuhn--Tucker optimality conditions for $C^{1,1}$ vector optimization problems
Nguyen Van Tuyen
TL;DR
This paper addresses second-order optimality for $C^{1,1}$ vector programs with inequality constraints. It develops a Taylor-type inequality based on the limiting second-order subdifferential and the second-order tangent set, and introduces an Abadie-type second-order constraint qualification (ASOCQ). Under ASOCQ, it derives a set of second-order KKT-type necessary conditions for local weak efficiency, and provides a strong second-order sufficiency condition for local efficiency. The results extend prior work by refining the second-order framework and reducing reliance on stronger differentiability assumptions, with implications for nonconvex vector optimization where gradients are Lipschitz. Overall, the work strengthens the theoretical toolkit for identifying and certifying optimality in $C^{1,1}$ constrained vector problems.
Abstract
This paper focuses on optimality conditions for $C^{1,1}$ vector optimization problems with inequality constraints. By employing the limiting second-order subdifferential and the second-order tangent set, we introduce a new type of second-order constraint qualification in the sense of Abadie. Then we establish some second-order necessary optimality conditions of Karush--Kuhn--Tucker-type for local (weak) efficient solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution of the such problem. The obtained results improve existing ones in the literature.