On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

Cánovas, M. J.; Gómez-Senent, F. J.; Parra, J.

doi:10.1007/s11228-007-0052-x

Cited by 18 publications

(13 citation statements)

References 25 publications

(37 reference statements)

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“…which contradicts (20) and thus completes the proof of the theorem. Now we are ready to establish a precise formula for computing the Lipschitz modulus at b, x ∈ gphE F of the epigraphical feasible set mapping from (5). In the next theorem we employ the l 1 -norm · 1 on R T , which is dual to the primal supremum norm (8) used above.…”

Section: Reaching In This Waymentioning

confidence: 99%

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

et al. 2020

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The paper concerns multiobjective linear optimization problems in R n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdifferentials and employ them to verifying Lipschitzian stability of the perturbed mappings with computing the associated Lipschitz moduli. The particular case of ordinary linear programs is analyzed, where we show that the subdifferentials of both multifunctions are proportional subsets. We also provide a method for computing the optimal value of linear programs without knowing any optimal solution. Some illustrative examples are also given in the paper.

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Section: Reaching In This Waymentioning

confidence: 99%

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

et al. 2020

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“…taking also into account (13) and the equivalence between the calmness of L and the metric subregularity of L −1 (see (5)).…”

Section: Preliminariesmentioning

confidence: 99%

“…and equality occurs in (17) provided that either T is finite or p ≤ 3. In [5], a general sufficient condition -called (H) therein-ensuring equality in (17) was introduced. Condition (H) always holds when either T is finite or p ≤ 3.…”

Section: Preliminariesmentioning

confidence: 99%

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Calmness Modulus of Linear Semi-infinite Programs

Cánovas¹,

Kruger²,

López³

et al. 2014

SIAM J. Optim.

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Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand-side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

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“…With respect to the finite linear optimization setting, we quote Li [15], dealing with sharp Lipschitz constants for the feasible and the optimal set under perturbations of the RHS of the constraints. In the semi-infinite context, [1] and [2] are concerned with the regularity modulus for constraint systems and optimization problems, respectively, with continuous data and canonical perturbations (RHS perturbations of the constraints and linear perturbations of the objective function).…”

Section: Introductionmentioning

confidence: 99%

Regularity modulus of arbitrarily perturbed linear inequality systems

Cánovas

Gómez-Senent

Parra

2008

Journal of Mathematical Analysis and Applications

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We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mapping. The main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system's data. In our context, both, the right-and the left-hand side of the system are subject to possible perturbations. This fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right-hand side. In these previous studies, the feasible set mapping is sublinear (which is not our current case) and the wellknown Radius Theorem constitutes a useful tool for determining the modulus. In our current setting we do not have an explicit expression for the radius of metric regularity, and we have to tackle the modulus directly. As an application we approach, under appropriate assumptions, the regularity modulus for a semi-infinite system associated with the Lagrangian dual of an ordinary nonlinear programming problem.

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On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

Cited by 18 publications

References 25 publications

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

Calmness Modulus of Linear Semi-infinite Programs

Regularity modulus of arbitrarily perturbed linear inequality systems

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