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20102 Applied Mathematics
20103 Numerical and Computational Mathematics
2Linear programming
2Variational analysis
10101 Pure Mathematics
1Calmness
1Calmness modulus
1Computer science
1Cone-constraint programming
1Constrained problem
1DC functions
1Feasible set mapping
1Isolated calmness
1Linear semi-infinite optimizations
1Local error bounds
1Mapping
1Objective functions
1Semi infinite programming
1Semi-infinite programs

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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.

Calmness of the feasible set mapping for linear inequality systems

- Cánovas, Maria, López, Marco, Parra, Juan, Toledo, Javier

**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**Full Text:**false**Reviewed:****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.

Necessary and sufficient optimality conditions in DC semi-infinite programming

- Correa, Rafael, López, Marco, Pérez-Aros, Pedro

**Authors:**Correa, Rafael , López, Marco , Pérez-Aros, Pedro**Date:**2021**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 837-865**Full Text:****Reviewed:****Description:**This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section. © 2021 Society for Industrial and Applied Mathematics

**Authors:**Correa, Rafael , López, Marco , Pérez-Aros, Pedro**Date:**2021**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 837-865**Full Text:****Reviewed:****Description:**This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section. © 2021 Society for Industrial and Applied Mathematics

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