An interior-point algorithm for large-scale nonlinear optimization with inexact step computations

SIAM Journal on Scientific Computing

Volumn 32, Issue 6, 2010, Pages 3447-3475

  Curtis, Frank E ^a   Schenk, Olaf ^b   Wächter, Andreas ^c  

^a Lehigh University   (United States)

^b UNIVERSITY OF BASEL   (Switzerland)

^c IBM T J WATSON RESEARCH CENTER   (United States)

Author keywords

Constrained optimization; Inexact linear system solvers; Interior point methods; Krylov subspace methods; Large scale optimization; Nonconvex optimization; Trust regions

Indexed keywords

LINEAR PROGRAMMING; LINEAR SYSTEMS; MATRIX ALGEBRA; NEWTON-RAPHSON METHOD; NONLINEAR PROGRAMMING;

INEXACT LINEAR SYSTEM SOLVER; INTERIOR POINT ALGORITHM; INTERIOR-POINT METHOD; KRYLOV-SUBSPACE METHODS; LARGE SCALE NONLINEAR OPTIMIZATION; LARGE-SCALE OPTIMIZATION; LINEAR SYSTEM SOLVER; NONCONVEX OPTIMIZATION; NONCONVEX-OPTIMIZATION; TRUST REGION;

CONSTRAINED OPTIMIZATION;

EID: 79251473397     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090747634     Document Type: Article

References (48)

1. V. Arnautu and P. Neittaanmaki, Optimal Control from Theory to Computer Programs, Kluwer Academic Publishers, Dordrecht, Netherlands, 2003.
2. J. T. Betts, Practical Methods for Optimal Control using Nonlinear Programming, Adv. Des. Control, SIAM, Philadelphia, 2001.
3. L. T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. Van Bloemen Waanders, Real-time PDE-constrained Optimization, Comput. Sci. Eng., SIAM, Philadelphia, 2007.
4. L. T. Biegler, O. Ghattas, M. Heinkenschloss, and B. Van Bloemen Waanders, eds., Large-scale PDE-constrained Optimization, Lect. Notes Comput. Sci. Eng. 30, Springer, New York, 2003.
5. G. Biros and O. Ghattas, Inexactness issues in the Lagrange-Newton- Krylov-Schur method for PDE-constrained optimization, in Large-Scale PDE-Constrained Optimization, Lect. Notes Comput. Sci. Eng. 30, L. T. Biegler, O. Ghattas, M. Heinkenschloss, and B. Van Bloemen Waanders, eds., Springer, New York, 2003, pp. 93-114.
6. G. Biros and O. Ghattas, Parallel Lagrange-Newton-Krylov-Schur methods for PDEconstrained optimization. Part I: The Krylov-Schur solver, SIAM J. Sci. Comput., 27 (2005), pp. 687-713.
7. G. Biros and O. Ghattas, Parallel Lagrange-Newton-Krylov-Schur methods for PDEconstrained optimization. Part II: The Lagrange-Newton solver and its application to optimal control of steady viscous flows, SIAM J. Sci. Comput., 27 (2005), pp. 714-739.
8. M. Bollhöfer, M. J. Grote, and O. Schenk, Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media, SIAM J. Sci. Comput., 31 (2009), pp. 3781-3805.
9. M. Bollhöfer and Y. Saad, Multilevel preconditioners constructed from inverse-based ILUs, SIAM J. Sci. Comput., 27 (2006), pp. 1627-1650.
10. A. Borzi and V. Schulz, Multigrid methods for PDE optimization, SIAM Rev., 51 (2009), pp. 361-395.
11. R. H. Byrd, F. E. Curtis, and J. Nocedal, An inexact SQP method for equality constrained optimization, SIAM J. Optim., 19 (2008), pp. 351-369.
12. R. H. Byrd, F. E. Curtis, and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization, Math. Program., 122 (2010), pp. 273-299.
13. R. H. Byrd, J.-Ch. Gilbert, and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program., 89 (2000), pp. 149-185.
14. R. H. Byrd, M. E. Hribar, and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), pp. 877-900.
15. F. E. Curtis and J. Nocedal, Flexible penalty functions for nonlinear constrained optimization, IMA J. Numer. Anal., 28 (2008), pp. 749-769.
16. F. E. Curtis, J. Nocedal, and A. Wächter, A matrix-free algorithm for equality constrained optimization problems with rank-deficient Jacobians, SIAM J. Optim., 20 (2009), pp. 1224-1249.
17. R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp. 400-408.
18. E. D. Dolan, J. J. Moré, and T. S. Munson, Benchmarking Optimization Software with COPS 3.0, Argonne National Laboratory Research report, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 2004.
19. M. Fisher, J. Nocedal, Y. Trémolet, and S. J. Wright, Data assimilation in weather forecasting: A case study in PDE-constrained optimization, Optim. Eng., 10 (2009), pp. 409-426.
20. R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Brooks/Cole, Florence, KY, 2002.
21. R. W. Freund, Preconditioning of symmetric, but highly indefinite linear systems, in 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, A. Sydow, ed., Wissenschaft and Technik, Berlin, 1997, pp. 551-556.
22. R. W. Freund and F. Jarre, A QMR-based interior-point algorithm for solving linear programs, Math. Program., 76 (1997), pp. 183-210.
23. D. M. Gay, Hooking Your Solver to AMPL, Technical report, Computing Sciences Research Center, Bell Laboratories, Murray Hill, NJ, 1997.
24. N. I. M. Gould, I. Bongartz, A. R. Conn, and Ph. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Trans. Math. Software, 21 (1995), pp. 123-160.
25. N. I. M. Gould, D. Orban, and Ph. L. Toint, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Trans. Math. Software, 29 (2003), pp. 373-394.
26. E. Haber and U. M. Ascher, Preconditioned all-at-once methods for large, sparse parameter estimation problems, Inverse Problems, 17 (2001), pp. 1847-1864.
27. M. Heinkenschloss and D. Ridzal, An inexact trust-region SQP method with applications to PDE-constrained optimization, in Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, K. Kunisch, G. Of, and O. Steinbach, eds., Springer, 2008, pp. 613-620.
28. M. Heinkenschloss and L. N. Vicente, Analysis of inexact trust-region SQP algorithms, SIAM J. Optim., 12 (2002), pp. 283-302.
29. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theory Appl. 23, Springer, Dordrecht, Netherlands, 2009.
30. H. Jäger and E. W. Sachs, Global convergence of inexact reduced SQP methods, Optim. Methods Softw., 7 (1997), pp. 83-110.
31. D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, Englewood Cliffs, NJ, 1970.
32. H. Maurer and H. D. Mittelmann, Optimization techniques for solving elliptic control problems with control and state constraints. Part 1: Boundary control, Comput. Optim. Appl., 16 (2000), pp. 29-55.
33. E. Neufeld, High Resolution Hyperthermia Treatment Planning, Ph.D. thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, Hartung-Gorre Verlag, NR. 17947, 2008, DOI 10.3929/ethz-a-005743425.
34. J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res., 2nd ed., Springer, New York, 2006.
35. C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), pp. 43-71.
36. R. P. Pawlowski, J. P. Simonis, H. F. Walker, and J. N. Shadid, Inexact Newton dogleg methods, SIAM J. Numer. Anal., 46 (2008), pp. 2112-2132.
37. M. J. D. Powell, A hybrid method for nonlinear equations, in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Gordon and Breach, London, 1970, pp. 87-114.
38. E. E. Prudencio, R. H. Byrd, and X.-C. Cai, Parallel full space SQP Lagrange-Newton- Krylov-Schwarz algorithms for PDE-constrained optimization problems, SIAM J. Sci. Comput., 27 (2006), pp. 1305-1328.
39. D. Ridzal, Trust-region SQP Methods with Inexact Linear System Solves for Large-scale Optimization, Ph.D. thesis, Rice University, Houston, TX, 2006.
40. O. Schenk, M. Bollhöfer, and R. A. Römer, On large-scale diagonalization techniques for the Anderson model of localization, SIAM Rev., 50 (2008), pp. 91-112.
41. O. Schenk and K. Gärtner, On fast factorization pivoting methods for sparse symmetric indefinite systems, Electron. Trans. Numer. Anal., 23 (2006), pp. 158-179.
42. O. Schenk, A. Wächter, and M. Weiser, Inertia-revealing preconditioning for large-scale nonconvex constrained optimization, SIAM J. Sci. Comput., 31 (2008), pp. 939-960.
43. T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), pp. 626-637.
44. P. L. Toint, Nonlinear Stepsize Control, Trust Regions and Regularizations for Unconstrained Optimization, Technical report, Department of Mathematics, FUNDP, University of Namur, Namur, Belgium, 2008.
45. S. Ulbrich, Generalized SQP-methods with "Parareal" time-domain decomposition for timedependent PDE-constrained optimization, in Real-Time PDE-Constrained Optimization, L. T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, eds., SIAM, Philadelphia, 2007, pp. 145-168.
46. A. Wächter and L. T. Biegler, Failure of global convergence for a class of interior point methods for nonlinear programming, Math. Program., 88 (2000), pp. 565-574.
47. A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2005), pp. 25-57.
48. D. P. Young, W. P. Huffman, R. G. Melvin, C. L. Hilmes, and F. T. Johnson, Nonlinear elimination in aerodynamic analysis and design optimization, in Large-Scale PDEConstrained Optimization, L. T. Biegler, O. Ghattas, M. Heinkenschloss, and B. Van Bloemen Waanders, eds., Springer, New York, 2003, pp. 17-44.