Duality related to approximate proper solutions of vector optimization problems

Abstract

In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering (C, ε)-proper efficient solutions of a recently introduced setvalued vector Lagrangian. In both approaches we obtain weak and strong duality results for (C, ε)-proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the (C, ε)-proper efficient solutions when the error ε tends to zero, the obtained duality results extend and improve several others in the literature.Mathematics Subject Classification 90C48 · 90C25 · 90C46 · 49N15 L. Huerga: Researcher of Spanish FPI Fellowship Programme (BES-2010-033742).

The linear scalarizations and Lagrange multipliers for vector optimization

“…近几十年来, 对于向量优化问题的理论研究除了各类解的概念外, 还包括各类解之间的关系, 特别 是多种真有效解之间的关系 (参见文献 [20][21][22]); 一定条件下 (弱) 有效解与各类真有效解的存在性研 究 (参见文献 [23][24][25][26]); 在凸性或广义凸性条件下的择一定理及其各类解的线性标量化和非凸条件下 的非线性标量化 (参见文献 [27][28][29][30][31][32][33]); 各类解的 Lagrange 乘子存在性定理、鞍点定理和对偶定理 (参 见文献 [34][35][36][37][38][39][40][41]) 等. 这些理论成果越来越多地发表在国际一流的重要期刊上.…”

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

“…Some well-known generalizations of the mentioned proper efficiency concepts are given by Benson [8], Borwein [9], Borwein and Zhuang [12], Hartley [29], Henig [30], and Hurwicz [33]. These concepts and corresponding generalizations are discussed, among others, by Durea, Florea and Strugariu [15], Eichfelder and Kasimbeyli [16], Gutiérrez et al [28], Hernández, Jiménez and Novo [31], Jahn [34,Ch. 4], and Luc [38,Def.…”