Abstract
We present a matrix-free line search algorithm for large-scale equality constrained optimization that allows for inexact step computations. For strictly convex problems, the method reduces to the inexact sequential quadratic programming approach proposed by Byrd et al. [SIAM J. Optim. 19(1) 351–369, 2008]. For nonconvex problems, the methodology developed in this paper allows for the presence of negative curvature without requiring information about the inertia of the primal–dual iteration matrix. Negative curvature may arise from second-order information of the problem functions, but in fact exact second derivatives are not required in the approach. The complete algorithm is characterized by its emphasis on sufficient reductions in a model of an exact penalty function. We analyze the global behavior of the algorithm and present numerical results on a collection of test problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bongartz I., Conn A.R., Gould N.I.M., Toint Ph.L.: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21(1), 123–160 (1995)
Byrd R.H., Curtis F.E., Nocedal J.: An inexact SQP method for equality constrained optimization. SIAM J. Optim. 19(1), 351–369 (2008)
Conn A.R., Gould N.I.M., Toint Ph.: Trust-Region Methods. MPS-SIAM Series on Optimization. SIAM Publications, Philadelphia (2000)
Curtis, F.E., Haber, E.: Numerical experience with an inexact SQP method for PDE-constrained optimization. (In preparation)
Dennis J.E., Vicente L.N.: On the convergence theory of trust-region based algorithms for equality-constrained optimization. SIAM J. Optim. 7(4), 927–950 (1997)
Di-Pillo J., Lucidi S., Palagi L.: Convergence to second-order stationary points of a primal–dual algorithm model for nonlinear programming. Math. Oper. Res. 30(4), 879–915 (2005)
Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. A 91, 201–213 (2002)
El-Alem M.: Convergence to a 2nd order point of a trust-region algorithm with nonmonotonic penalty parameter for constrained optimization. J. Optim. Theory Appl. 91(1), 61–79 (1996)
Facchinei F., Lucidi S.: Convergence to second order stationary points in inequality constrained optimization. Math. Oper. Res. 23, 746–766 (1998)
Freund, R.W., Nachtigal, N.M.: A new Krylov-subspace method for symmetric indefinite linear systems. In: Ames, W.F. (ed.) Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, pp. 1253–1256. IMACS (1994)
Gould N.I.M., Orban D., Toint Ph.L.: CUTEr and sifdec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)
Gould N.I.M., Toint Ph.L.: A note on the second-order convergence of optimization algorithms using barrier functions. Math. Program. 85(2), 433–438 (1999)
Heinkenschloss, M., Ridzal, D.: An inexact trust-region SQP method with applications to PDE-constrained optimization. In: Steinbach, O., Of, G. (eds.) Numerical Mathematics and Advance Applications: Proceedings of Enumath 2007, the 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007, Springer, Heidelberg (2008, submitted). doi:10.1007/978-3-540-69777-0_73
Heinkenschloss M., Vicente L.N.: Analysis of inexact trust-region SQP algorithms. SIAM J. Optim. 12, 283–302 (2001)
Kelley, C.T.: Iterative methods for linear and nonlinear equations: Matlab codes (1994). http://www4.ncsu.edu/~ctk/matlab_roots.html
Moguerza J.M., Prieto F.J.: An augmented Lagrangian interior point method using directions of negative curvature. Math. Program. 95(3), 573–616 (2003)
Paige C.C., Saunders M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975)
Paige, C.C., Saunders, M.A.: MINRES: sparse symmetric equations (2003). http://www.stanford.edu/group/SOL/software/minres.html
Ridzal, D.: Trust region SQP methods with inexact linear system solves for large-scale optimization. Ph.D. thesis, Rice University (2006)
Saad Y., Schultz M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
Ulbrich S.: Generalized SQP-methods with “parareal” time-domain decomposition for time-dependent PDE-constrained optimization. In: Biegler, L., Ghattas, O., Heinkenschloss, M., Keyes, D., Bloemen Waanders, B. (eds) Real-Time PDE-Constrained Optimization, pp. 145–168. SIAM, Philadelphia (2008)
Vanderbei R.J., Shanno D.F.: An interior point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Richard H. Byrd was supported by National Science Foundation grants CCR-0219190 and CHE-0205170.
Frank E. Curtis was supported by Department of Energy grant DE-FG02-87ER25047-A004.
Jorge Nocedal was supported by National Science Foundation grant CCF-0514772 and by a grant from the Intel Corporation.
Rights and permissions
About this article
Cite this article
Byrd, R.H., Curtis, F.E. & Nocedal, J. An inexact Newton method for nonconvex equality constrained optimization. Math. Program. 122, 273–299 (2010). https://doi.org/10.1007/s10107-008-0248-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-008-0248-3
Keywords
- Large-scale optimization
- Constrained optimization
- Nonconvex programming
- Inexact linear system solvers
- Krylov subspace methods