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

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

References

• Battermann , A. and Heikenschloss , M. 1998 . “ Reduce with Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed system ” . In Optimal Control of partial Differential Equations , Edited by: Desch , W. , Kappel , F. and Kunisch , K. 15 – 32 . Berlin : Int. Ser. Num. Math. Birkhauser Verlag .
   Google Scholar
• Battermann , A. and Sachs , E. W. 2001 . “ Block preconditioners for KKT systems in PDE-governed optimal control problems ” . In Fast Solution of Discretized Optimization Problems , Edited by: Hoffmann , K.-H. , Hoppe , R. H.W. and Schulz , V. Vol. 138 , 1 – 18 . Basel : Int. Ser. Numer. Math. Birkhauser .
   Google Scholar
• Bellavia , S. , Macconi , M. and Morini , B. 2003 . An affine scaling trust-region approach to bound-constrained nonlinear systems . Appl. Numer. Math. , 44 : 257 – 280 .
   Web of Science ®Google Scholar
• Bellavia , S. , Macconi , M. and Morini , B. 2004 . STRSCNE: a scaled trust-region solver for constrained nonlinear equations . Comput. Optim. Appl. , 28 : 31 – 50 .
   Web of Science ®Google Scholar
• Bellavia , S. and Morini , B. 2005 . An interior global method for nonlinear systems with simple bounds . Optim. Methods Softw. , 20 : 1 – 22 .
   Web of Science ®Google Scholar
• Bellavia , S. and Morini , B. 2006 . Subspace trust-region methods for large bound constrained nonlinear equations . SIAM J. Numer. Anal. , 44 : 1535 – 1555 .
   Web of Science ®Google Scholar
• Benzi , M. , Golub , G. H. and Liesen , J. 2005 . Numerical solution of saddle point problems . Acta Numer. , 14 : 1 – 137 .
   Google Scholar
• Bergamaschi , L. , Gondzio , J. , Venturin , M. and Zilli , G. 2007 . Inexact constraint preconditioners for linear systems arising in interior point methods . Comput. Optim. Appl. , 36 : 137 – 147 .
   Google Scholar
• Bergamaschi , L. , Gondzio , J. and Zilli , G. 2004 . Preconditioning indefinite systems in interior point methods for optimization . Comput. Optim. Appl. , 28 : 149 – 171 .
   Web of Science ®Google Scholar
• Bergounioux , M. , Haddou , M. , Hintermüller , M. and Kunisch , K. 2000 . A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems . SIAM J. Optim. , 11 : 495 – 521 .
   Web of Science ®Google Scholar
• Bergounioux , M. and Kunisch , K. 1997 . Augmented lagrangian techniques for elliptic state constrained optimal control problems . SIAM J. Control Optim. , 35 : 1524 – 1543 .
   Web of Science ®Google Scholar
• Dolan , E. D. and Moré , J. J. 2002 . Benchmarking optimization software with performance profiles . Math. Program. , 91 : 201 – 213 .
   Web of Science ®Google Scholar
• Francisco , J. B. , Krejic , N. and Martinez , J. M. 2005 . An interior-point method for solving box-constrained underdetermined nonlinear systems . J. Comput. Appl. Math. , 177 : 67 – 88 .
   Google Scholar
• Freund , R. W. and Nachtigal , N. M. Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics . A new Krylov-subspace method for symmetric indefinite linear system , Edited by: Ames , W. F. pp. 1253 – 1256 . IMACS .
   Google Scholar
• HSL . 2004 . HSL 2004 library , Numerical Analysis Group, CCLRC's Computational Science and Engineering Department . Available at http://www.cse.clrc.ac.uk/nag/hsl/
   Google Scholar
• Kanzow , C. and Klug , A. 2006 . An interior-point affine-scaling trust-region method for semismooth equations with box constraints . Comput. Optim. and Appl. , 35 : 177 – 197 .
   Web of Science ®Google Scholar
• Maurer , H. and Mittelmann , H. D. 2000 . Optimization techniques for solving elliptic control problems with control and state constraints. Part 1: boundary control . Comput. Optim. Appl. , 16 : 29 – 55 .
   Google Scholar
• Maurer , H. and Mittelmann , H. D. 2001 . Optimization techniques for solving elliptic control problems with control and state constraints. Part 2: distributed control . Comput. Optim. Appl. , 18 : 141 – 160 .
   Google Scholar
• Mittelmann , H. D. and Maurer , H. 2000 . Solving elliptic control problems with interior point and SQP methods: control and state constraints . J. Comput. Appl. Math. , 120 : 175 – 195 .
   Google Scholar
• Gould , N. I.M. and Scott , J. A. 2004 . A numerical evaluation of HSL packages for the direct solution of large sparse, symmetric linear systems of equations . ACM Trans. Math. Softw. , 30 ( 3 ) : 300 – 325 .
   Google Scholar
• Nocedal , J. and Wright , S. J. 1999 . “ Numerical Optimization ” . Springer, Springer Series in Operations Research .
   Google Scholar
• Paige , C. C. and Saunders , M. A. 1975 . Solution of sparse indefinite systems of linear equations . SIAM J. Numer. Anal. , 12 : 617 – 629 .
   Web of Science ®Google Scholar
• Rozložník , M. and Simoncini , V. 2002 . Krylov subspace methods for saddle point problems with indefinite preconditioning . SIAM J. Matrix Anal. Appl. , 24 : 368 – 391 .
   Web of Science ®Google Scholar
• Zhu , D. 2005 . An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems . J. Comput. Appl. Math. , 184 : 343 – 361 .
   Google Scholar