On a semismooth* Newton method for solving generalized equations
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2021
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 489-517
- Relation: https://doi.org/10.1137/19M1257408
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- Description: In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models. © 2021 Society for Industrial and Applied Mathematics
On computation of optimal strategies in oligopolistic markets respecting the cost of change
- Authors: Outrata, Jiri , Valdman, Jan
- Date: 2020
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 92, no. 3 (2020), p. 489-509
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
On the Aubin property of solution maps to parameterized variational systems with implicit constraints
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2020
- Type: Text , Journal article
- Relation: Optimization Vol. 69, no. 7-8 (2020), p. 1681-1701
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: In the paper, a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems, the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g. parameter-dependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explicit computation of these coderivatives. The procedure is illustrated by an example. © 2019, © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
- Description: The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854.
On the Aubin property of a class of parameterized variational systems
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2017
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 86, no. 3 (2017), p. 443-467
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples. © 2017, Springer-Verlag GmbH Germany.
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.
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.
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.
On relaxing the Mangasarian-Fromovitz constraint qualification
- Authors: Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri
- Date: 2014
- Type: Text , Journal article
- Relation: Positivity Vol. 18, no. 1 (2014), p. 171-189
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.
Some remarks on stability of generalized equations
- Authors: Henrion, René , Kruger, Alexander , Outrata, Jiri
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 159, no. 3 (2012), p. 681-697
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multivalued term amounts to the regular normal cone to a (possibly nonconvex) set given by C 2 inequalities. Instead of the linear independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian-Fromovitz and the constant rank qualification conditions. Based on the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constraints are derived, and a workable characterization of the isolated calmness of the considered solution map is provided. © 2012 Springer Science+Business Media, LLC.