Error bounds : Necessary and sufficient conditions
- Authors: Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri
- Date: 2010
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 18, no. 2 (2010), p. 121-149
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- Description: The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space. © 2010 Springer Science+Business Media B.V.
Error bounds and metric subregularity
- Authors: Kruger, Alexander
- Date: 2015
- Type: Text , Journal article
- Relation: Optimization Vol. 64, no. 1 (2015), p. 49-79
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: Necessary and sufficient criteria for metric subregularity (or calmness) of set-valued mappings between general metric or Banach spaces are treated in the framework of the theory of error bounds for a special family of extended real-valued functions of two variables. A classification scheme for the general error bound and metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
From convergence principles to stability and optimality conditions
- Authors: Klatte, Diethard , Kruger, Alexander , Kummer, Bernd
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 19, no. 4 (2012), p. 1043-1072
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- Description: We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures. © Heldermann Verlag.
- Description: 2003010677
Calmness of the feasible set mapping for linear inequality systems
- Authors: Cánovas, Maria , López, Marco , Parra, Juan , Toledo, Javier
- Date: 2014
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 22, no. 2 (2014), p. 375-389
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system's data.
On Hölder calmness of solution mappings in parametric equilibrium problems
- Authors: Anh, Lam Quoc , Kruger, Alexander , Thao, Nguyen
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 22, no. 1 (2012), p. 331-342
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- Description: We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces.
Calmness of efficient solution maps in parametric vector optimization
- Authors: Chuong, Thai Doan , Kruger, Alexander , Yao, J. C.
- Date: 2011
- Type: Journal article
- Relation: Journal of Global Optimization Vol. 51, no. 4 (2011), p. 677-688
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. © 2011 Springer Science+Business Media, LLC.
On lipschitzian properties of implicit multifunctions
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2016
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 26, no. 4 (2016), p. 2160-2189
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool we employ the directional limiting coderivative which, together with the graphical derivative, enables a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [A. L. Dontchev and R. T. Rockafellar, SIAM J. Optim., 6 (1996), pp. 1087{1105]. All important statements are illustrated by examples. © 2016 Society for Industrial and Applied Mathematics.
Nonlinear metric subregularity
- Authors: Kruger, Alexander
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 820-855
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this article, we investigate nonlinear metric subregularity properties of set-valued mappings between general metric or Banach spaces. We demonstrate that these properties can be treated in the framework of the theory of (linear) error bounds for extended real-valued functions of two variables developed in Kruger (Error bounds and metric subregularity. Optimization 64(1):49-79, 2015). Several primal and dual space local quantitative and qualitative criteria of nonlinear metric subregularity are formulated. The relationships between the criteria are established and illustrated.
Calmness of partially perturbed linear systems with an application to the central path
- Authors: Cánovas, Maria , Hall, Julian , López, Marco , Parra, Juan
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Vol. 68, no. 2-3 (2019), p. 465-483
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- Description: In this paper we develop point-based formulas for the calmness modulus of the feasible set mapping in the context of linear inequality systems with a fixed abstract constraint and (partially) perturbed linear constraints. The case of totally perturbed linear systems was previously analyzed in [Canovas MJ, Lopez MA, Parra J, et al. Calmness of the feasible set mapping for linear inequality systems. Set-Valued Var Anal. 2014;22:375-389, Section 5]. We point out that the presence of such an abstract constraint yields the current paper to appeal to a notable different methodology with respect to previous works on the calmness modulus in linear programming. The interest of this model comes from the fact that partially perturbed systems naturally appear in many applications. As an illustration, the paper includes an example related to the classical central path construction. In this example we consider a certain feasible set mapping whose calmness modulus provides a measure of the convergence of the central path. Finally, we underline the fact that the expression for the calmness modulus obtained in this paper is (conceptually) implementable as far as it only involves the nominal data.
Robust and continuous metric subregularity for linear inequality systems
- Authors: Camacho, J. , Cánovas, Maria , López, Marco , Parra, Juan
- Date: 2023
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 967-988
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- Description: This paper introduces two new variational properties, robust and continuous metric subregularity, for finite linear inequality systems under data perturbations. The motivation of this study goes back to the seminal work by Dontchev, Lewis, and Rockafellar (2003) on the radius of metric regularity. In contrast to the metric regularity, the unstable continuity behavoir of the (always finite) metric subregularity modulus leads us to consider the aforementioned properties. After characterizing both of them, the radius of robust metric subregularity is computed and some insights on the radius of continuous metric subregularity are provided. © 2022, The Author(s).