Abstract
Primal-dual interior methods for nonconvex nonlinear programming have recently been the subject of significant attention from the optimization community. Several different primal-dual methods have been suggested, along with some theoretical and numerical results. Although the underlying motivation for all of these methods is relatively uniform, there axe nonetheless substantive variations in crucial details, including the formulation of the nonlinear equations, the form of the associated linear system, the choice of linear algebraic procedures for solving this system, strategies for adjusting the barrier parameter and the Lagrange multiplier estimates, the merit function, and the treatment of indefiniteness. Not surprisingly, each of these choices can affect the theoretical properties and practical performance of the method. This paper discusses the approaches to these issues that we took in implementing a specific primal-dual method.
“Is there any point to which you would wish to draw my attention?” “The curious incident of the dog in the night-time.” “The dog did nothing in the night-time.” “The dog did nothing in the night-time” “That was the curious incident,” remarked Sherlock Holmes.1 Sir Arthur Conan Doyle, Silver Blaze
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Argaez and R. A. Tapia (1997). On the global convergence of a modified augmented Lagrangian linesearch interior point Newton method for nonlinear programming, Report 95-38 (modified February 1997 ), Department of Computational and Applied Mathematics, Rice University, Houston, Texas.
R. H. Byrd, J. C. Gilbert, and J. Nocedal (1996). A trust region method based on interior point techniques for nonlinear programming, Technical Report OTC 96/02, Argonne National Laboratory, Argonne, Illinois.
R. H. Byrd, M. E. Hribar, and J. Nocedal (1997). An interior point algorithm for large-scale nonlinear programming, Technical Report, Northwestern University, Evanston, Illinois.
R. M. Chamberlain, C. Lemaréchal, H. C. Pedersen, and M. J. D. Powell (1982). The watchdog technique for forcing convergence in algorithms for constrained optimization,Math. Prog. Study 16, 1–17.
S. H. Cheng and N. J. Higham (1996). A modified Cholesky algorithm based on a symmetric indefinite factorization, Report 289, Department of Mathematics, University of Manchester, Manchester, United Kingdom.
A. R. Conn, N. I. M. Gould, and P. L. Toint (1996). A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints, Report RC 20639, IBM T. J. Watson Research Center, Yorktown Heights, New York.
I. Bongartz, A. R. Conn, N. Gould, and Ph. Toint (1995). CUTE: Constrained and unconstrained testing environment,ACM Trans. Math. Software 21, 123–160.
A. S. El-Bakry, R. A. Tapia, T. Tsuchiya, and Y. Zhang (1996). On the formulation and theory of the Newton interior-point method for nonlinear programming,J. Opt. Theory Appl. 89, 507–541.
A. V. Fiacco and G. P. McCormick (1968),Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York. Republished by SIAM, Philadelphia, 1990.
R. Fletcher (1987).Practical Methods of Optimization ( second edition ), John Wiley and Sons, Chichester.
A. Forsgren añd P. E. Gill (1996). Primal-dual interior methods for non- convex nonlinear programming, Report NA 96-3, Department of Mathematics, University of California, San Diego.
A. Forsgren, P. E. Gill, and W. Murray (1995). Computing modified Newton directions using a partial Cholesky factorization,SIAM J. Sci. Comp. 16, 139–150.
A. Forsgren, P. E. Gill, and J. R. Shinnerl (1996). Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization,SIAM J. Matrix Anal. Appl. 17, 187–211.
D. M. Gay, M. L. Overton, and M. H. Wright. Strategies for treating indef- initeness in a primal-dual method for nonconvex optimization, Technical Report, Bell Laboratories, Murray Hill, New Jersey, to appear.
P. E. Gill and W. Murray (1974). Newton-type methods for unconstrained and linearly constrained optimization.Mathematical Programming 28, 311–350.
P. E. Gill, W. Murray and M. H. Wright (1981),Practical Optimization, Academic Press, London and New York.
C. C. Gonzaga (1992). Path following methods for linear programming,SIAM Review 34, 167–224.
W. Hock and K. Schittkowski (1981).Test Examples for Nonlinear Programming Codes, Springer-Verlag, Berlin.
G. P. McCormick (1991). The superlinear convergence of a nonlinear primal-dual algorithm, Report T-550/91, Department of Operations Research, George Washington University, Washington, DC.
S. M. Robinson (1974). Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear programming algorithms,Mathematical Programming 7, 1–16.
R. B. Schnabel and E. Eskow (1990). A new modified Cholesky factorization,SIAM J. Sci. Stat. Comp. 11, 1136–1158.
G. W. Stewart and J. Sun (1990).Matrix Perturbation Theory, Academic Press, Boston.
M. H. Wright (1992). Interior methods for constrained optimization, in A. Iserles (ed.),Acta Numerica 1992, 341–407, Cambridge University Press, New York.
M. H. Wright (1994). Some properties of the Hessian of the logarithmic barrier function,Mathematical Programming 67, 265–295.
M. H. Wright (1995). Why a pure primal Newton barrier step may be infeasible,SI AM Journal on Optimization 5, 1–12.
M. H. Wright (1997). Ill-conditioning and computational error in primal- dual methods for nonlinear programming, Technical Report 97-4-04, Computing Sciences Research Center, Bell Laboratories, Murray Hill, New Jersey.
S. J. Wright (1996). Modifying the Cholesky factorization in interior- point methods. Technical Report ANL/MCS-P600-0596, Argonne National Laboratory, Argonne, Illinois.
S. J. Wright (1997).Primal-Dual Interior-Point Methods, Society for Industrial and Applied Mathematics, Philadelphia.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this chapter
Cite this chapter
Gay, D.M., Overton, M.L., Wright, M.H. (1998). A Primal-dual Interior Method for Nonconvex Nonlinear Programming. In: Yuan, Yx. (eds) Advances in Nonlinear Programming. Applied Optimization, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3335-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3335-7_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3337-1
Online ISBN: 978-1-4613-3335-7
eBook Packages: Springer Book Archive