A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

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“…Let z ∈ L 2 ( ) and let u * , u * h denote the solutions of problems (9) and (11), respectively. Then there holds…”

Section: Application To Control Of Elliptic Equationsmentioning

confidence: 99%

“…The approach presented in the previous section applies to a large class of control problems for partial differential equations [6,9]. In the present section its application to control problems for linear elliptic partial differential equations is shown.…”

Section: Application To Control Of Elliptic Equationsmentioning

confidence: 99%

A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Hinze

2005

Comput Optim Applic

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459

411

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“…As a consequence, the effort for solving the nonlinear and linearized Navier-Stokes system become essentially the same. For a discussion of the SQP method in the case of time-dependent problems, we refer, e.g., to [19].…”

Section: Wwwgamm-mitteilungenorgmentioning

confidence: 99%

Algorithms for PDE‐constrained optimization

2010

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MSC (2000)49-M05, 49-M37, 76-D55, 90-C06, Some first and second order algorithmic approaches for the solution of PDE-constrained optimization problems are reviewed. An optimal control problem for the stationary Navier-Stokes system with pointwise control constraints serves as an illustrative example. Some issues in treating inequality constraints for the state variable and alternative objective functions are also discussed.

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“…Among the optimize-then-discretize approaches, the SQP methods dominate the published material [1,17,22,23,27,32,33]. Here, Robinson's theory of generalized equations [29] can be used to analyze the function space methods, which leaves, however, the question of how to solve the infinite-dimensional linear-quadratic programs.…”

Section: Introductionmentioning

confidence: 99%

Interior Point Methods in Function Space

Weiser¹

2005

SIAM J. Control Optim.

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Abstract.A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite-dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short-step path-following method is established.Key words. interior point methods in function space, optimal control, complementarity functions AMS subject classifications. 49M15, 90C48, 90C51 DOI. 10.1137/S03630129034372771. Introduction. Numerical methods for solving optimal control problems governed by ODEs fall into two categories, the indirect methods [2,3,4,6,14,15,31] relying on Pontryagin's maximum principle, and the direct methods [7,17,21,30,37] based on the Karush-Kuhn-Tucker necessary conditions. Direct methods can be characterized by several features. Among them are the following:(i) Position of discretization: Discretize-then-optimize approaches use an a priori parameterization of the control and possibly the state variables to reduce the optimal control problem to a finite-dimensional nonlinear program. These large nonlinear programs can then be solved by standard NLP solvers. Adaptive mesh refinement can be performed after the finite-dimensional optimum has been reached. On the other hand, optimize-then-discretize approaches formulate the optimization algorithms directly in the infinite-dimensional function space, employing discretization only for solving linear operator equations. Adaptive mesh refinement is used to meet the accuracy requirements imposed on the solution of the linear equations by the optimization algorithm.Somewhere in between are function space sequential quadratic programming (SQP) methods where linear-quadratic programs are discretized. (ii) Type of optimization algorithm: Among the most popular algorithms employed for solving the optimization problems arising in optimal control are SQP and interior point methods. A recent alternative are semismooth Newton methods [5,34]. Discretize-then-optimize methods are covered by a vast amount of published literature using almost any available algorithm for solving the finite-dimensional NLPs. Solutions on consecutive mesh refinement levels or in consecutive SQP steps often exhibit pronounced similarities. This redundancy can be directly exploited by active set-type methods. In contrast, interior point methods are considered to benefit less from this redundancy [20,40]. Nevertheless, interior point methods are reported to be very efficient for solving optimal control problems-a fact that is not well explained by straightforward application of finite-dimensional interior point convergence theory to

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A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

Cited by 46 publications

References 28 publications

A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Algorithms for PDE‐constrained optimization

Interior Point Methods in Function Space

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