Primal-Dual Strategy for Constrained Optimal Control Problems

Bergounioux, Maı̈tine; Ito, Kazufumi; Kunisch, Karl

doi:10.1137/s0363012997328609

Cited by 268 publications

(269 citation statements)

References 5 publications

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“…This algorithm is based on a generalized Moreau-Yosida approximation of the indicator function of the set U ad of admissible controls. For more details we refer to [3].…”

Section: A Primal-dual Active Set Algorithmmentioning

confidence: 99%

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The SQP method for control constrained optimal control of the Burgers equation

Tröltzsch¹,

Volkwein²

2001

ESAIM: COCV

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Abstract.A Lagrange-Newton-SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are included.Mathematics Subject Classification. 49J20, 49K20, 65Kxx.

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“…This algorithm is based on a generalized Moreau-Yosida approximation of the indicator function of the set U ad of admissible controls. For more details we refer to [3].…”

Section: A Primal-dual Active Set Algorithmmentioning

confidence: 99%

“…This is done by a primal-dual active set algorithm, which is based on a generalized Moreau-Yosida approximation of the indicator function of the admissible controls. The method was developed due to [3] and was extended in [9]. Let us also mention [12], where the primal-dual active set algorithm was applied to parabolic optimal control problems.…”

mentioning

confidence: 99%

The SQP method for control constrained optimal control of the Burgers equation

Tröltzsch¹,

Volkwein²

2001

ESAIM: COCV

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“…The discretized contact problem was solved numerically using the Augmented Lagrangian algorithm described in [1,2]. …”

Section: Numerical Resultsmentioning

confidence: 99%

Rolling Contact Problem with the Generalized Coulomb Friction

Chudzikiewicz

Myśliński

2006

Proc Appl Math and Mech

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This paper deals with the numerical solution of the wheel -rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. Problem FormulationConsider deformations of a viscoelastic strip Ω of height h lying on a rigid foundation due to the wheel rolling with the linear velocity V (see Fig. 1). Denote by, a stress tensor of the strip. We consider viscoelastic bodies obeying KelvinVoigt law [8]:where the strainand c 1 ijkl (x) are components of Hook's tensor satysfiying usual symmetry, boundedness and ellipticity conditions [8]. We use here and throughout the paper the summation convention over repeated indices [3]. The contact problem consists in finding the displacement field u satisfying [10]whereThe following initial and boundary conditions are imposedwhere u 0 , u 1 are given functions. On the boundary (0, T ) × Γ 2 a surface traction vector F = σ ij (u), i, j = 1, 2, is a priori unknown and is given by the conditions of contact and friction. Under the assumptions, that the strip displacement is small, the contact conditions on the boundary (0, T ) × Γ 2 take a form [3]:where r 0 is the wheel radii, h 0 is a distance between wheel axis and the rail, u N and F N = σ N denote normal components [3] of the displacement u an stress σ on the boundary (0, T ) × Γ 2 respectively. z + N denotes positive part of z N . Moreoverwhere u T and F T = σ T denote tangential components [3] of the displacement u and stress σ on the boundary (0, T ) × Γ 2 respectively, F = (F N , F T ). The choice of functions h N and h T allows to formulate different contact models. Function µ denotes a friction coefficient. Usually this coefficient is assumed to be constant. Numeruous experiments indicate [4,9] that this coefficient is dependent on sliding velocity or temperature. Therefore in this paper the friction coefficient is assumed to depend on the sliding velocityu T . Function w = w(t, x) in (5) denotes the distance between the bodies due to wear and satisfies the Archard law [2],This function is an internal state variable to model the wear process taking place at the contact interface [2]. Assuming that there is no wear and the friction coefficient µ is Lipschitz continuous function of the sliding velocityu T there exists [5,7,8,10] the solution to the contact problem (1) -(6).

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“…For the numerical approximation of solutions u of (P m ) we introduce a primaldual active set method or equivalently a semi-smooth Newton method [7,27]. Both are well known in the context of optimization with partial differential equations as constraints.…”

Section: Primal-dual Active Set Approachmentioning

confidence: 99%

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

Blank

Garcke

Sarbu

et al. 2013

IMA Journal of Numerical Analysis

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We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

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Primal-Dual Strategy for Constrained Optimal Control Problems

Cited by 268 publications

References 5 publications

The SQP method for control constrained optimal control of the Burgers equation

The SQP method for control constrained optimal control of the Burgers equation

Rolling Contact Problem with the Generalized Coulomb Friction

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

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