Numerical solution of KKT systems in PDE-constrained optimization problems via the affine scaling trust-region approach†

†Work supported by MIUR, Rome, Italy, through ‘Cofinanziamenti Programmi di Ricerca Scientifica di Interesse Nazionale’ and ‘Gruppo Nazionale per il Calcolo Scientifico’, Florence, Italy

Abstract

A recently proposed trust-region approach for bound-constrained nonlinear equations is applied to the Karush-Kuhn-Tucker (KKT) system arising from the discretization of a class of partial differential equation (PDE)-constrained optimization problems. Two different implementations are developed that take into account the large dimension and the special structure of the problems. The linear algebra phase is analysed considering the possibility of solving the arising linear systems by either direct methods or short-recurrence iterative linear solvers. Viability of the approach is proved through several numerical experiments on large KKT systems arising from the discretization of control problems.

Keywords:

• box-constrained nonlinear equations
• trust-region methods
• large-scale problem
• KKT systems
• PDE-constrained optimization

2000 AMS Subject Classification :

• 65H10
• 65K05

Notes

†Work supported by MIUR, Rome, Italy, through ‘Cofinanziamenti Programmi di Ricerca Scientifica di Interesse Nazionale’ and ‘Gruppo Nazionale per il Calcolo Scientifico’, Florence, Italy