B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening

Herzog, Roland; Meyer, Christian; Wachsmuth, Gerd

doi:10.1137/110821147

Cited by 45 publications

(46 citation statements)

References 31 publications

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“…In Herzog et al [2013], the authors considered an optimal control problem arising in elasto-plasticity and in Wachsmuth [2013b] the author studies a control constrained optimal control problem governed by the obstacle problem. We also mention that, except Herzog et al [2013], all these results in infinite dimensions involve polyhedral cones. The definition of polyhedral cones is recalled in Section 4.2.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Programs with Complementarity Constraints in Banach Spaces

Wachsmuth

2014

J Optim Theory Appl

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We consider optimization problems in Banach spaces involving a complementarity constraint defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. Under the additional assumption that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case that K is not a cone and apply the theory to two examples.

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Section: Introductionmentioning

confidence: 99%

Mathematical Programs with Complementarity Constraints in Banach Spaces

Wachsmuth

2014

J Optim Theory Appl

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“…To prove (10) observe that, by the same argument as before, x → max{x, 1} is Hadamard-differentiable. Thanks to the chain rule for Hadamard-derivatives, the mapping f : R → R, f (x) = max{|x|, 1} is thus Hadamard-differentiable, too.…”

Section: Differentiability For a Finite-dimensional VI Of The Second mentioning

confidence: 91%

“…In addition, different types of stationarity concepts have been investigated in that framework: C-, B-, M-and strong stationary points. The utilized proof techniques include regularization approaches as well as differentiability properties (directional, conic) of the solution map, or elements of set-valued analysis (see, e.g., [2][3][4][5][6][7][8][9][10][11][12]). …”

Section: Introductionmentioning

confidence: 99%

Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

Reyes

Meyer

2015

J Optim Theory Appl

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We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primaldual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

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“…The solution we choose, which was largely investigated in the framework of control theory for problems with hardening but not in the framework of shape optimization, is the use of a regularized penalized problem to get rid of the variational inequality. On this issue we mention, for the static case, [24], [26], [27], [11] (using a primal formulation), [28], [5] (for a second order optimality condition), for the quasi-static case [69] and for other plastic models [10] and [36].…”

Section: Introductionmentioning

confidence: 99%

Elasto-plastic Shape Optimization Using the Level Set Method

Maury¹,

Allaire²,

Jouve³

2018

SIAM J. Control Optim.

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This article is concerned with shape optimization of structures made of a material obeying Hencky's laws of plasticity, with the stress bound expressed by the von Mises effective stress. The ill-posedness of the model is circumvented by using two regularized versions of the mechanical problem. The first one is the classical Perzyna formulation which is regularized, the second one is a new regularized formulation proposed for the von Mises criterion. Shape gradients are calculated thanks to the adjoint method. The optimal shape is numerically computed by using the level set method. To illustrate the validity of the method, 2D examples are performed.

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B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening

Cited by 45 publications

References 31 publications

Mathematical Programs with Complementarity Constraints in Banach Spaces

Mathematical Programs with Complementarity Constraints in Banach Spaces

Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

Elasto-plastic Shape Optimization Using the Level Set Method

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