Pareto-Fenchel ε-Subdifferential Composition Rule and ε-Efficiency

Maghri, Mounir El

doi:10.1080/01630563.2013.812110

Cited by 4 publications

(7 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This concept reduces to the well-known of (exact) Benson (C, ε)-proper solution of (P), denoted by Be ( f, S), whenever C = D\{0} or ε = 0 and cone C = D. Also, it is not difficult to prove that, under generalized convexity assumptions, the notion of Benson (q + D, ε)-proper solution of (P), for q / ∈ −D\{0}, coincides with the notion of properly ε-efficient solution with respect to q given by El Maghri in [22] (see [22, Section 2 and Theorem 3.1] and Theorems 2.4 and 2.5).…”

Section: Preliminariesmentioning

confidence: 92%

Duality related to approximate proper solutions of vector optimization problems

et al. 2015

View full text Add to dashboard Cite

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).

show abstract

Section: Preliminariesmentioning

confidence: 92%

Duality related to approximate proper solutions of vector optimization problems

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Remark 2.3 Definition 2.7 reduces to the strong ε-subdifferential concept given by Kutateladze [15] by considering C = {q}, q ∈ D (see also [7]). …”

Section: Definition 27mentioning

confidence: 99%

“…Next, we recall the notion of p-regular ε-subdifferentiability given by El Maghri in [7] (see also [9]), which will be used along the paper.…”

Section: Definition 27mentioning

confidence: 99%

“…Remark 3.1 In [7,9], the author considered a proper ε-subdifferential, defined in terms of proper approximate solutions with respect to a vector in the sense of Henig. Indeed,…”

Section: Moreover If K Has a Compact Base Thenmentioning

confidence: 99%

“…On the other hand, by [7,Lemma 5.3], there exists q 2 ∈ R p + , μ 2 , q 2 = ε 2 , and M 2 ∈ M p×n such that λ 2 = μ 2 M 2 and …”

mentioning

confidence: 97%

See 2 more Smart Citations

Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings

Gutiérrez

Huerga

Novo

et al. 2015

J Optim Theory Appl

View full text Add to dashboard Cite

In this paper, we derive exact chain rules for a proper epsilon-subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-subdifferential generalizes and improves several of the most important proper epsilon-subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values.

show abstract