Coderivatives of Frontier and Solution Maps in Parametric Multiobjective Optimization

Huy, Nguyễn Quang; Mordukhovich, Boris S.; Yao, Jen‐Chih

doi:10.11650/twjm/1500405137

Cited by 35 publications

(25 citation statements)

References 31 publications

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“…Recall that a different route is used in [25] to derive estimates of the coderivative of the solution mapping of a parametric multiobjective optimization problem. The results in the latter paper are written in terms of the frontier map, which is the extension of the notion of optimal value function to multiobjective programs.…”

Section: ∇ψ(Xȳz) (α β) = 0 and (α β) ∈ N (ψ(Xȳz))mentioning

confidence: 99%

Solving Ill-posed Bilevel Programs

Zemkoho

2016

Set-Valued Var. Anal

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This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain setvalued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper-and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.

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Section: ∇ψ(Xȳz) (α β) = 0 and (α β) ∈ N (ψ(Xȳz))mentioning

confidence: 99%

Solving Ill-posed Bilevel Programs

Zemkoho

2016

Set-Valued Var. Anal

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“…mization problems, sets of solutions of such problems are instances of such multimaps [14,19,30,33,34,36,43,65]. It has been well recognized that differential inclusions, which are certainly of independent interest, play a key role in optimal control theory ( [1-3, 7-10, 13, 15, 20, 21, 37, 40, 70, 73]...).…”

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confidence: 99%

Codifferential Calculus

Penot

Xue

2010

Set-Valued Anal

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In this paper, some exact calculus rules are obtained for calculating the coderivatives of the composition of two multivalued maps. Similar rules are displayed for sums. A crucial role is played by an intermediate set-valued map called the resolvent. We first establish inclusions for contingent, Fréchet and limiting coderivatives. Combining them, we get equality rules. The qualification conditions we present are natural and less exacting than classical conditions.

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“…Since C is Fréchet normally regular at (p, x) and  C admits a local upper Lipschitzian selection at (p, y, x), it follows from It is worth mentioning here that under suitable hypotheses Huy, Mordukhovich and Yao also established inclusions (3.14) and (3.15) (see [20,Theorems 3.3 and 3.4]) for the mixed coderivative. In order to obtain inclusion (3.15), they imposed an assumption (see [20,Theorem 3.4]) that the efficient solution map S : P ⇒ X defined by …”

Section: ) Around P and H Admits A Local Upper Lipschitzian Selectiomentioning

confidence: 94%

“…The proof is complete. Note that if G is order semicontinuous at (p, y), i.e., for any sequence {(p n , y n )} ⊂ epi G := {(p, y) ∈ P × Y | y ∈ G(p) + K } converging to (p, y) there is a sequence {(p n , v n )} ⊂ gph G with v n ≼ K y n such that {v n } contains a subsequence converging to y, then Bao and Mordukhovich also got estimate (3.1) in [40,Proposition 4.3] for the Mordukhovich/normal coderivative and the opposite inclusion of (3.1) in [40,Proposition 4.4] (also see [20,Lemma 3.2]) for the mixed coderivative.…”

Section: And H Admits a Local Upper Lipschitzian Selection Atmentioning

confidence: 98%

“…This approach can bring a new look to the behavior of the efficient point multifunction F . The recent paper [20] deals with the efficient point multifunction F by using the so-called Mordukhovich and mixed coderivatives. Namely, the authors gave formulae for computing and estimating the mixed coderivative of efficient point multifunctions in parametric vector optimization problems with respect to the so-called generalized order optimality.…”

Section: Introductionmentioning

confidence: 99%

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