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20101 Pure Mathematics
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2Aubin property
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1Linear semi-infinite optimizations
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Second-order variational analysis in conic programming with applications to optimality and stability

- Mordukhovich, Boris, Outrata, Jiri, Ramírez, Hector

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.

Calmness modulus of linear semi-infinite programs

- Cánovas, Maria, Kruger, Alexander, López, Marco, Parra, Juan, Théra, Michel

**Authors:**Cánovas, Maria , Kruger, Alexander , López, Marco , Parra, Juan , Théra, Michel**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 29-48**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**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.

**Authors:**Cánovas, Maria , Kruger, Alexander , López, Marco , Parra, Juan , Théra, Michel**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 29-48**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**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.

Qualitative stability of a class of non-monotone variational inclusions. Application in electronics

**Authors:**Adly, Samir , Outrata, Jiri**Date:**2013**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 20, no. 1 (2013), p. 43-66**Full Text:**false**Reviewed:****Description:**The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated cahnness) of a special non-monotone variational inclusion. We provide a characterization of these properties in terms of the problem data and show their importance for the design of electrical circuits involving nonsmooth and non-monotone electronic devices Uke DIAC (Diode Alternating Current). Circuits with other devices like SCR (Silicon Controlled Rectifiers), Zener diodes, thyristors, varactors and transistors can be analyzed in the same way. © Heldermann Verlag.**Description:**2003011029

On stability of M-stationary points in MPCCs

- Červinka, Michal, Outrata, Jiri, Pištěk, M

**Authors:**Červinka, Michal , Outrata, Jiri , Pištěk, M**Date:**2014**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 22, no. 3 (2014), p. 575-595**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**We consider parameterized Mathematical Programs with Complementarity Constraints arising, e.g., in modeling of deregulated electricity markets. Using the standard rules of the generalized differential calculus we analyze qualitative stability of solutions to the respective M-stationarity conditions. In particular, we provide characterizations and criteria for the isolated calmness and the Aubin properties of the stationarity map. To this end, we introduce the second-order limiting coderivative of mappings and provide formulas for this notion and for the graphical derivative of the limiting coderivative in the case of the normal cone mapping to ℝn Funding ARC- Australian Research Council

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