Pareto Subdifferential Calculus for Convex Vector Mappings and Applications to Vector Optimization

Maghri, Mounir El; Laghdir, Mohamed

doi:10.1137/070704046

Cited by 21 publications

(19 citation statements)

References 16 publications

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“…(c) On the other hand, for the case = 0, -regular -subdifferentiability assumption (on H ) in Theorem 3.1 becomes useless for > 0, and the formulas (for ∈ p, w ) of this theorem then reduces, as corollary, to the composition rule for the efficient subdifferentials we established in [9] 3 :…”

Section: Downloaded By [Unam Ciudad Universitaria] At 18:57 20 Decembmentioning

confidence: 91%

“…As mentioned in our previous works [9,10], that unlike the strong concept ( = s) which is a like-scalar notion, in fact, with Pareto concepts, the subdifferential calculus rules cannot be homogeneous in a sense that only Pareto subdifferentials are involved. A counterexample in [10] for the sum rule, indeed, shows with very regular mappings (linear operators in two dimensions) that for = s and for all , 1 and 2 nonnegative vectors,…”

Section: Pareto -Subdifferential Composition Rulesmentioning

confidence: 95%

“…The properties (a) to (c) are obtained exactly as for = 0 (see [9] to this purpose). Property (d) follows by contradiction using the fact that…”

Section: Pareto -Subdifferentialsmentioning

confidence: 98%

“…This kind of vector ( -)subdifferential knows these last decades a growing progress from a point of view of theoretical background (see, e.g., [9,10,18,[24][25][26]). Recently in [10], several properties and calculus rules have been established for weakly and properly Pareto -subdifferentials.…”

Section: Introductionmentioning

confidence: 99%

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

Maghri

2013

Numerical Functional Analysis and Optimization

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We are mainly concerned in this article with a first rule for the efficient (Pareto) -subdifferential concerning the composition of two convex vector mappings taking values in finite or infinite-dimensional pre-ordered spaces. The obtained formula is exact and holds under MoreauRockafellar or Attouch-Brézis qualification conditions. In fact, beyond regularity, the rule would not be accurate without convex (Fenchel) -subdifferential of one of the two composed mappings. But also, suitable fields of variation for vector parameters play crucial roles in the operation. These, moreover, really enable to derive for cone-constrained convex vector optimization problems, a complete -efficiency criterion of the Kuhn-Tucker type given in terms of a vector Lagrangian.

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Section: Downloaded By [Unam Ciudad Universitaria] At 18:57 20 Decembmentioning

confidence: 91%

Section: Pareto -Subdifferential Composition Rulesmentioning

confidence: 95%

“…The properties (a) to (c) are obtained exactly as for = 0 (see [9] to this purpose). Property (d) follows by contradiction using the fact that…”

Section: Pareto -Subdifferentialsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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

Maghri

2013

Numerical Functional Analysis and Optimization

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“…The usual exact strong subdifferential of f at x 0 ∈ dom f is denoted by ∂ s f (x 0 ), i.e., (see, for instance, [19]),…”

Section: Definition 27mentioning

confidence: 99%

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

Gutiérrez

Huerga

Novo

et al. 2015

J Optim Theory Appl

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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.

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