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50101 Pure Mathematics
2Variational analysis
1Asplund space
1Boris Mordukhovich
1Calmness
1Composition operators
1Dual condition
1Error bounds
1Extremality
1Feasible set mapping
1Hölder metric subregularity
1Linear programming
1Local error bounds
1Mathematical Programs with Equilibrium Constraints
1Mathematics
1Maximal monotonicity
1Maximality of sum of two maximal monotone operators
1Metric regularity
1Metric subregularity
1Model with Given Friction

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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:**

About regularity of collections of sets

**Authors:**Kruger, Alexander**Date:**2006**Type:**Text , Journal article**Relation:**Set-Valued Analysis Vol. 14, no. 2 (Jun 2006), p. 187-206**Full Text:****Reviewed:****Description:**The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.**Description:**2003001526

A dual criterion for maximal monotonicity of composition operators

- Jeyakumar, Vaithilingam, Wu, Zhiyou

**Authors:**Jeyakumar, Vaithilingam , Wu, Zhiyou**Date:**2007**Type:**Text , Journal article**Relation:**Set-Valued Analysis Vol. 15, no. 3 (2007), p. 265-273**Full Text:**false**Reviewed:****Description:**In this paper we present a dual criterion for the maximal monotonicity of the composition operator T:=A* SA, where S:Y→→ Y is a maximal monotone (set-valued) operator and A: X→ Y is a continuous linear map with the adjoint A*, X and Y are reflexive Banach spaces, and the product notation indicates composition. The dual criterion is expressed in terms of the closure condition involving the epigraph of the conjugate of Fitzpatrick function associated with S, and the operator A. As an easy application, a dual criterion for the maximality of the sum of two maximal monotone operators is also given. © 2006 Springer Science+Business Media B.V.**Description:**C1

- 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.

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.

Error bounds and Hölder metric subregularity

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736**Full Text:****Reviewed:****Description:**The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736**Full Text:****Reviewed:****Description:**The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

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