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

Authors:
Roland Herzog;Ekkehard Sachs
Affiliations:
roland.herzog@mathematik.tu-chemnitz.de;sachs@uni-trier.de
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2010

Citing 11
Cited 5

Control of an elliptic problem with pointwise state constraints

SIAM Journal on Control and Optimization
On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization

SIAM Journal on Scientific Computing
The Primal-Dual Active Set Strategy as a Semismooth Newton Method

SIAM Journal on Optimization
Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver

SIAM Journal on Scientific Computing
Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems

SIAM Journal on Matrix Analysis and Applications
Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

SIAM Journal on Matrix Analysis and Applications
On two numerical methods for state-constrained elliptic control problems

Optimization Methods & Software
Elliptic optimal control problems with L1-control cost and applications for the placement of control devices

Computational Optimization and Applications
Preconditioning Saddle-Point Systems with Applications in Optimization

SIAM Journal on Scientific Computing
Optimal Solvers for PDE-Constrained Optimization

SIAM Journal on Scientific Computing
Approximate Nullspace Iterations for KKT Systems

SIAM Journal on Matrix Analysis and Applications

Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations

SIAM Journal on Scientific Computing
Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

Computational Optimization and Applications
Preconditioning for Allen-Cahn variational inequalities with non-local constraints

Journal of Computational Physics
All-at-once solution of time-dependent Stokes control

Journal of Computational Physics
Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems

Computational Optimization and Applications

Quantified Score

Hi-index	0.01

Visualization

Abstract

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 problems demonstrate the performance of the PCG iteration. Regularized state-constrained problems in three dimensions with more than 750,000 variables are solved.