Abstract
The Karush–Kuhn–Tucker (KKT) optimality conditions are necessary and sufficient for a convex programming problem under suitable constraint qualification. Recently, several papers (Dutta and Lalitha in Optim Lett 7(2):221–229, 2013; Lasserre in Optim Lett 4(1):1–5, 2010; Suneja et al. Am J Oper Res 3(6):536–541, 2013) have appeared wherein the convexity of constraint function has been replaced by convexity of the feasible set. Further, Ho (Optim Lett 11(1):41–46, 2017) studied nonlinear programming problem with non-convex feasible set. We have used this modified approach in the present paper to study vector optimization problem over cones. The KKT optimality conditions are proved by replacing the convexity of the objective function with convexity of strict level set, convexity of feasible set is replaced by a weaker condition and no condition is assumed on the constraint function. We have also formulated a Mond–Weir type dual and proved duality results in the modified setting. Our results directly extend the work of Ho (2017) Suneja et al. (2013) and Lasserre (2010).
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Notes
Slaters’s condition holds for S if there exists \(x_0 \in S, g_i (x_0) < 0\) for every \(i= 1,\ldots ,m\).
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Suneja, S.K., Sharma, S. & Yadav, P. Optimality and duality for vector optimization problem with non-convex feasible set. OPSEARCH 57, 1–12 (2020). https://doi.org/10.1007/s12597-019-00401-3
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DOI: https://doi.org/10.1007/s12597-019-00401-3