Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints

  title={Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints},
  author={Roland Herzog and Ekkehard W. Sachs},
  journal={SIAM J. Matrix Anal. Appl.},
Optimality systems and their linearizations arising in optimal control of partial differential equations with pointwise control and (regularized) state constraints are considered. The preconditioned conjugate gradient (PCG) method in a nonstandard inner product is employed for their efficient solution. Preconditioned condition numbers are estimated for problems with pointwise control constraints, mixed control-state constraints, and of Moreau-Yosida penalty type. Numerical results for elliptic… 

Figures and Tables from this paper

Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems
Since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable Krylov subspace methods such as Minres, also in discretized settings.
Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems
Two new preconditioners based on a full block matrix factorization of the Schur complement of the Jacobian matrices, where the active-set blocks are merged into the constraint blocks are presented.
Preconditioners for state‐constrained optimal control problems with Moreau–Yosida penalty function
Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant
On the numerical solution of state- and control-constrained optimal control problems
We consider the iterative solution of algebraic systems, arising in optimal control problems, constrained by a partial differential equation, with additional box constraints on the state and the
A Robust Preconditioner for Distributed Optimal Control for Stokes Flow with Control Constraints
This work develops a block-diagonal preconditioner that is robust with respect to the discretization parameter as well as the active set for the distributed optimal control problem for the Stokes equations with inequality constraints on the control.
An efficient preconditioning method for state box-constrained optimal control problems
It is shown that there occur very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh for a wide range of the penalization and discretization parameters.
A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems
A block‐diagonal preconditioner is proposed for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter.
Iterative solution methods for mesh approximation of control and state constrained optimal control problem with observation in a part of the domain
Iterative solution methods for finite-dimensional constrained saddle point problems are investigated theoretically and numerically. These saddle point problems arise when approximating differential
A preconditioned MinRes solver for time‐periodic parabolic optimal control problems
This work proposes a new preconditioned MinRes solver for the frequency domain equations and shows that this solver is robust with respect to the space and time discretization parameters as well as the involved ‘bad’ model parameters of the state equation.
Preconditioning PDE-constrained optimization with L1-sparsity and control constraints


Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints.
Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its
Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems
In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in
On two numerical methods for state-constrained elliptic control problems
A linear-quadratic elliptic control problem with pointwise box constraints on the state with Lavrentiev type regularization is considered and the convergence of the regularized controls is proven for regularization parameter tending to zero.
Block Preconditioners for KKT Systems in PDE—Governed Optimal Control Problems
This paper is concerned with block preconditioners for linear KKT systems that arise in optimization problems governed by partial differential equations. The preconditioners exhibit, like the KKT
Elliptic optimal control problems with L1-control cost and applications for the placement of control devices
For solving the non-differentiable optimal control problem, a semismooth Newton method is proposed that can be stated and analyzed in function space and converges locally with a superlinear rate.
Semi-smooth Newton methods for state-constrained optimal control problems
Preconditioners for Karush-Kuhn-Tucker Matrices Arising in the Optimal Control of Distributed Systems
In this paper preconditioners for linear systems arising in interior-point methods for the solution of distributed control problems are derived and analyzed. The matrices K in these systems have a
Analysis of block matrix preconditioners for elliptic optimal control problems
Several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear‐quadratic elliptic control problem with Neumann boundary conditions are described and a regularization term with a parameter α is included to ensure that the problem is well posed.
A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems
This paper provides a preconditioned iterative technique for the solution of saddle point problems by reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration.
Optimal Solvers for PDE-Constrained Optimization
Two optimal preconditioners are introduced for large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type, and the theoretical proof indicates that these approaches may have much broader applicability for other PDEs.