5Gfrerer, Helmut
5Henrion, René
5Kruger, Alexander
4Mordukhovich, Boris
2Fabian, Marian
2Haslinger, Jaroslav
2Pathó, Róbert
2Červinka, Michal
1Adam, Lukas
1Adly, Samir
1Benko, Matúš
1Beremlijski, Petr
1Ferris, Michael
1Minchenko, Leonld
1Outrata, Michal
1Pištěk, M
1Ramirez, Hector
1Ramírez, Hector
1Sarabi, Ebrahim

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110102 Applied Mathematics
90101 Pure Mathematics
70103 Numerical and Computational Mathematics
4Aubin property
4Variational analysis
3Isolated calmness
3Solution map
20802 Computation Theory and Mathematics
20906 Electrical and Electronic Engineering
2Calmness
2Constant rank CQ
2Directional limiting coderivative
2Generalized differentiation
2Optimality conditions
2Parameterized generalized equation
2Qualification conditions
2Regular and limiting coderivative
2Second-order theory
2Stability
2Tilt stability

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Format Type

About errors bounds in metric spaces

- Fabian, Marian, Henrion, René, Kruger, Alexander, Outrata, Jiri

**Authors:**Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2011**Type:**Text , Conference paper**Relation:**International Conference Operations Research p. 33-38**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**The paper presents a general primal space classification scheme of necessary and suffficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects - slopes are used to characterize the error bound property of extended-real valued functions on metric sapces.

**Authors:**Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2011**Type:**Text , Conference paper**Relation:**International Conference Operations Research p. 33-38**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**The paper presents a general primal space classification scheme of necessary and suffficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects - slopes are used to characterize the error bound property of extended-real valued functions on metric sapces.

Boris Mordukhovich, the never tiring traveller, celebrates his sixtieth birthday

- Henrion, René, Kruger, Alexander, Outrata, Jiri

**Authors:**Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2008**Type:**Text , Journal article**Relation:**Set-Valued Analysis Vol. 16, no. 2-3 (2008), p. 125-127**Full Text:****Reviewed:**

**Authors:**Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2008**Type:**Text , Journal article**Relation:**Set-Valued Analysis Vol. 16, no. 2-3 (2008), p. 125-127**Full Text:****Reviewed:**

Calculus for directional limiting normal cones and subdifferentials

- Benko, Matúš, Gfrerer, Helmut, Outrata, Jiri

**Authors:**Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri**Date:**2019**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 27, no. 3 (2019), p. 713-745**Full Text:****Reviewed:****Description:**The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

**Authors:**Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri**Date:**2019**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 27, no. 3 (2019), p. 713-745**Full Text:****Reviewed:****Description:**The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

Error bounds : Necessary and sufficient conditions

- Fabian, Marian, Henrion, René, Kruger, Alexander, Outrata, Jiri

**Authors:**Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2010**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 18, no. 2 (2010), p. 121-149**Full Text:****Reviewed:****Description:**The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space. Â© 2010 Springer Science+Business Media B.V.

**Authors:**Fabian, Marian , Henrion, René , Kruger, Alexander , Outrata, Jiri**Date:**2010**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 18, no. 2 (2010), p. 121-149**Full Text:****Reviewed:****Description:**The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space. Â© 2010 Springer Science+Business Media B.V.

Full stability of locally optimal solutions in second-order cone programs

- Mordukhovich, Boris, Outrata, Jiri, Sarabi, Ebrahim

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

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

Graphical derivatives and stability analysis for parameterized equilibria with conic constraints

- Mordukhovich, Boris, Outrata, Jiri, Ramirez, Hector

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

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

On computation of generalized derivatives of the normal-cone mapping and their applications

- Gfrerer, Helmut, Outrata, Jiri

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2016**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 41, no. 4 (2016), p. 1535-1556**Full Text:**false**Reviewed:****Description:**The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the normal-cone mapping related to C2 inequality constraints under very weak qualification conditions. This enables us to provide the graphical derivative and the regular coderivative of the solution map to a class of parameterized generalized equations with the constraint set of the investigated type. On the basis of these results, we finally obtain a characterization of the isolated calmness property of the mentioned solution map and derive strong stationarity conditions for an MPEC with control constraints. © 2016 INFORMS.

- Gfrerer, Helmut, Outrata, Jiri

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2016**Type:**Text , Journal article**Relation:**Optimization Vol. 65, no. 4 (2016), p. 671-700**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**The paper concerns the computation of the limiting coderivative of the normal-cone mapping related to inequality constraints under weak qualification conditions. The obtained results are applied to verify the Aubin property of solution maps to a class of parameterized generalized equations.

On Cournot-Nash-Walras equilibria and their computation

- Outrata, Jiri, Ferris, Michael, Červinka, Michal, Outrata, Michal

**Authors:**Outrata, Jiri , Ferris, Michael , Červinka, Michal , Outrata, Michal**Date:**2016**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 24, no. 3 (2016), p. 387-402**Full Text:**false**Reviewed:****Description:**This paper concerns a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/firms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of CNW equilibria under reasonable conditions and examine their local stability with respect to small perturbations of problem data. In this way we show the uniqueness of CNW equilibria under mild additional requirements. Finally, we suggest some efficient numerical approaches and compute several instances of an illustrative test example. © 2016, Springer Science+Business Media Dordrecht.

On lipschitzian properties of implicit multifunctions

- Gfrerer, Helmut, Outrata, Jiri

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

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

On optimal control of a sweeping process coupled with an ordinary differential equation

**Authors:**Adam, Lukas , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series B Vol. 19, no. 9 (November 2014 2014), p. 2709-2738**Full Text:**false**Reviewed:****Description:**We study a special case of an optimal control problem governed by a differential equation and a differential rate{independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process. We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.

On regular coderivatives in parametric equilibria with non-unique multipliers

- Henrion, René, Outrata, Jiri, Surowiec, Thomas

**Authors:**Henrion, René , Outrata, Jiri , Surowiec, Thomas**Date:**2012**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 136, no. 1 (December 2012), p. 111-131**Full Text:**false**Reviewed:****Description:**This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations (GEs). The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such GEs. The advantages are illustrated by means of examples.**Description:**C1

On relaxing the Mangasarian-Fromovitz constraint qualification

- Kruger, Alexander, Minchenko, Leonld, Outrata, Jiri

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

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

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

On the Aubin property of a class of parameterized variational systems

- Gfrerer, Helmut, Outrata, Jiri

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

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

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

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.

- Haslinger, Jaroslav, Outrata, Jiri, Pathó, Róbert

**Authors:**Haslinger, Jaroslav , Outrata, Jiri , Pathó, Róbert**Date:**2012**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 20, no. 1 (2012), p. 31-59**Full Text:**false**Reviewed:****Description:**The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the twodimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite-element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions. Â© 2011 Springer Science+Business Media B.V.

- Beremlijski, Petr, Haslinger, Jaroslav, Outrata, Jiri, Pathó, Róbert

**Authors:**Beremlijski, Petr , Haslinger, Jaroslav , Outrata, Jiri , Pathó, Róbert**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 5 (2014), p. 3371-3400**Full Text:**false**Reviewed:****Description:**The present paper deals with shape optimization in discretized two-dimensional (2D) contact problems with Coulomb friction, where the coefficient of friction is assumed to depend on the unknown solution. Discretization of the continuous state problem leads to a system of finite-dimensional implicit variational inequalities, parametrized by the so-called design variable, that determines the shape of the underlying domain. It is shown that if the coefficient of friction is Lipschitz and sufficiently small in the C0,1 -norm, then the discrete state problems are uniquely solvable for all admissible values of the design variable (the admissible set is assumed to be compact), and the state variables are Lipschitzian functions of the design variable. This facilitates the numerical solution of the discretized shape optimization problem by the so-called implicit programming approach. Our main results concern sensitivity analysis, which is based on the well-developed generalized differential calculus of B. Mordukhovich and generalizes some of the results obtained in this context so far. The derived subgradient information is then combined with the bundle trust method to compute several model examples, demonstrating the applicability and efficiency of the presented approach. © 2014 Society for Industrial and Applied Mathematics

Some remarks on stability of generalized equations

- Henrion, René, Kruger, Alexander, Outrata, Jiri

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

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

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