Second-order variational analysis in conic programming with applications to optimality and stability
- 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
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- 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
- 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
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- 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.
Full stability of locally optimal solutions in second-order cone programs
- Authors: Mordukhovich, Boris , Outrata, Jiri , Sarabi, Ebrahim
- Date: 2014
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 24, no. 4 (2014), p. 1581-1613
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- Description: The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.
Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems
- Authors: Cánovas, Maria , López, Marco , Mordukhovich, Borris , Parra, Juan
- Date: 2012
- Type: Text , Journal article
- Relation: TOP Vol. 20, no. 2 (2012), p. 310-327
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.