A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations
- First Online:
This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447–3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhancements, such as functionality to avoid the computation of a normal step during every iteration. The implementation is included in the IPOPT software package paired with an iterative linear system solver and preconditioner provided in PARDISO. Numerical results on a large nonlinear optimization test set and two PDE-constrained optimization problems with control and state constraints are presented to illustrate that the implementation is robust and efficient for large-scale applications.
KeywordsLarge-scale optimization PDE-constrained optimization Interior-point methods Nonconvex programming Line search Trust regions Inexact linear system solvers Krylov subspace methods
Mathematics Subject Classification49M05 49M15 49M37 65F10 65K05 65N22 90C06 90C26 90C30 90C51 90C90
Unable to display preview. Download preview PDF.
- 2.Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11—Revision 3.1, Argonne National Laboratory (2010)Google Scholar
- 3.Betts J.T.: Practical Methods for Optimal Control Using Nonlinear Programming Advances in Design and Control. SIAM, Philadelphia (2001)Google Scholar
- 5.Biegler, L.T., Ghattas, O., Heinkenschloss, M., Van Bloemen Waanders, B. (eds.): Large-Scale PDE-Constrained Optimization. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2003)Google Scholar
- 10.Curtis, F.E., Huber, J., Schenk, O., Wächter, A.: On the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations. Technical report, Lehigh ISE 12T-006, Optimization Online ID: 2011-04-2992 (2012)Google Scholar
- 15.Fourer R., Gay D.M., Kernighan B.W.: AMPL: A Modeling Language for Mathematical Programming. Brooks/Cole, Belmont (2002)Google Scholar
- 16.Freund, R.W.: Preconditioning of symmetric, but highly indefinite linear systems. In: Sydow, A. (ed.) 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, pp. 551–556. Wissenschaft & Technik, Berlin (1997)Google Scholar
- 20.Hinze M., Pinnau R., Ulbrich M., Ulbrich S.: Optimization with PDE Constraints, Volume 23 of Mathematical Modeling: Theory and Applications. Springer, Dordrecht (2009)Google Scholar
- 23.Kirk D.E.: Optimal Control Theory: An Introduction. Prentice-Hall, Englewood Cliffs (1970)Google Scholar
- 24.Powell M.J.D: A hybrid method for nonlinear equations. In: Rabinowitz, P. (ed.) Numerical methods for nonlinear algebraic equations, pp. 87–114. Gordon and Breach, London (1970)Google Scholar
- 27.Tröltzsch F.: Optimal control of partial differential equations: theory, methods, and applications, Volume 112. American Mathematical Society, Providence (2010)Google Scholar