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.
Special Issue on recent advances in continuous optimization on the occasion of the 25th European conference on Operational Research (EURO XXV 2012)
- Authors: Weber, Gerhard-Wilhelm , Kruger, Alexander , Martinez-Legaz, Juan , Mordukhovich, Boris , Sakalauskas, Leonidas
- Date: 2014
- Type: Text , Journal article
- Relation: Optimization Vol. 63, no. 1 (2014), p. 1-5
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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.
Tilt stability in nonlinear programming under mangasarian-fromovitz constraint qualification
- Authors: Mordukhovich, Boris , Outrata, Jiri
- Date: 2013
- Type: Text , Journal article
- Relation: Kybernetika Vol. 49, no. 3 (2013), p. 446-464
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving further the well-known Constant Rank Constraint Qualification, we derive new necessary and sufficient conditions for tilt-stable local minimizers.
- Description: The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving further the well-known Constant Rank Constraint Qualification, we derive new necessary and sufficient conditions for tilt-stable local minimizers. Funded: ARC
Graphical derivatives and stability analysis for parameterized equilibria with conic constraints
- Authors: Mordukhovich, Boris , Outrata, Jiri , Ramirez, Hector
- Date: 2015
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 687-704
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- Description: The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.
Subdifferentials and stability analysis of feasible set and pareto front mappings in linear multiobjective optimization
- Authors: Cánovas, Maria , López, Marco , Mordukhovich, Boris , Parra, Juan
- Date: 2020
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 315-334
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: The paper concerns multiobjective linear optimization problems in
- Description: Funding details: European Commission, EC Funding details: European Regional Development Fund, FEDER Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP180100602 Funding details: Australian Research Council, ARC, DP-190100555 Funding details: Air Force Office of Scientific Research, AFOSR, 15RT04 Funding details: DMS-1512846, DMS-1808978 Funding text 1: This research has been partially supported by grants MTM2014-59179-C2-(1,2)-P and PGC2018-097960-B-C2(1,2) from MINECO/MICINN, Spain, and ERDF, “A way to make Europe”, European Union. Funding text 2: Research of the second author is also partially supported by the Australian Research Council (ARC) Discovery Grants Scheme (Project Grant # DP180100602). Funding text 3: Research of third author was partially supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant #15RT04, and by Australian Research Council under grant DP-190100555.