An ADMM numerical approach to linear parabolic state constrained optimal control problems

Glowinski, Roland; Song, Yongcun; Yuan, Xiaoming

doi:10.1007/s00211-020-01104-4

Cited by 19 publications

(13 citation statements)

References 43 publications

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“…The discrete problems (4.4) have been solved by reformulating the necessary and sufficient optimality condition (4.5) as a system of equations by means of the complementarity function Φ(a, b) := min(a, b), and by subsequently applying a semismooth Newton method with tolerance 10 −10 . For an alternative to this solution approach, we refer to [31], where an ADMM-scheme for state-constrained parabolic optimal control problems is developed. We would like to point out that the calculation of the y D -integral in (4.5) was carried out with a subdivided three-point Gauss-rule in space and time in our numerical experiments (i.e., overall twelve nodes per triangle and 6 nodes per time interval).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For additional comments on this topic, see also the discussions in the introductions of [31,37,49,50]. The strategy that is most commonly used in the literature to circumvent the above difficulties surrounding the Slater condition in the analysis of parabolic distributed optimal control problems of the type (P) is to employ regularization or penalization techniques that either incorporate the constraint y ≥ ψ into the objective function or introduce artificial bounds on the controls u which ensure a higher regularity of the attainable states y.…”

Section: Introductionmentioning

confidence: 99%

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New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

Christof

Vexler

2021

ESAIM: COCV

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We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfill a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0)-cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

Christof

Vexler

2021

ESAIM: COCV

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“…One can extend our algorithm to solve state constrained parabolic optimal control problem without much more effort. We compared the extended algorithm with ADMM described in [19].…”

Section: ≤ Tolmentioning

confidence: 99%

“…The initial values are set as u = 0, z = 0 and λ = 0. For more detail, we refer to [19]. ≤ tol where g k = {g k n } N −1 n=0 denotes gradient at k-th iteration.…”

Section: ≤ Tolmentioning

confidence: 99%

A duality-based approach for solving linear parabolic control constrained optimal control problems

Wang¹,

Yu²,

Wu³

2022

Preprint

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This paper concerns with the optimal control problem governed by linear parabolic equations subject to bound constraints on control variable. This type of problem has important applications in the heating and cooling systems. By applying the scheme of Fenchel duality, we derive the dual problem explicitly where the control constraints in primal problem are embedded in the dual objective functional. The existence and uniqueness of the solution to dual problem are proved and the first-order optimality condition is also derived. In addition, we discuss the saddle point property between solution of the primal problem and the dual problem. The solution of primal problem can be readily obtained by the solution of the dual problem. To solve the dual problem, we design an implementable conjugate gradient method. Three example problems are solved, numerical results show that the proposed method is highly efficient.

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“…In particular, the ADMM and its variants have been applied to solve some optimal control problems constrained by time-independent PDEs in, e.g., [2,27,56]. In [26], the ADMM was applied to parabolic optimal control problems with state constraints, and its convergence is proved without any assumption on the existence and regularity of the Lagrange multiplier. In [24], the Peaceman-Rachford splitting method (see [46]) which is closely related to the ADMM was suggested to solve approximate controllability problems of parabolic equations numerically.…”

Section: Remarks On the Direct Application Of Admmmentioning

confidence: 99%

Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond

Song¹,

Yuan²,

Yue³

2020

Preprint

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Optimal control problems subject to both parabolic partial differential equation (PDE) constraints and additional constraints on the control variables are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can be directly applied to such a problem. An attractive advantage of this direct ADMM application is that the additional constraint on the control variable can be untied from the parabolic PDE constraint; these two inherently different constraints thus can be treated individually in iterations. At each iteration of the ADMM, the main computation is for solving an optimal control problem with a parabolic PDE constraint while it is not interacted with the constraint on the control variable. Because of its inevitably high dimensionality after the space-time discretization, the parabolic optimal control problem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative scheme is required. It then becomes important to find an easily implementable and efficient inexactness criterion to execute the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative scheme. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and it can be executed automatically with no need to set empirically perceived constant accuracy a prior. The inexactness criterion turns out to allow us to solve the resulting optimal control problems with the only parabolic PDE constraints to medium or even low accuracy and thus saves computation significantly, yet convergence of the overall two-layer nested iterative scheme can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by preliminary numerical results. Our methodology can also be extended to a range of optimal control problems constrained by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations and fractional parabolic equations.

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An ADMM numerical approach to linear parabolic state constrained optimal control problems

Cited by 19 publications

References 43 publications

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

A duality-based approach for solving linear parabolic control constrained optimal control problems

Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond

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