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Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems

Margherita Porcelli (margherita.porcelli***at***unibo.it)
Valeria Simoncini (valeria.simoncini***at***unibo.it)
Mattia Tani (mattia.tani2***at***unibo.it)

Abstract: We address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of PDE-constrained optimal control problems via active-set Newton methods, with control and (regularized) state constraints. We present 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. We discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems. Numerical experiments on 3D problems are presented, including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product.

Keywords: optimal control problems, Newton's method, active-set, saddle point matrices

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 3: Applications -- Science and Engineering (Control Applications )

Citation: SIAM Journal on Scientific Computing, 37:5 (2015), pp. S472-S502.

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Entry Submitted: 08/16/2014
Entry Accepted: 08/16/2014
Entry Last Modified: 12/24/2015

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