Abstract.
The paper considers an example of Wächter and Biegler which is shown to converge to a nonstationary point for the standard primal–dual interior-point method for nonlinear programming. The reason for this failure is analyzed and a heuristic resolution is discussed. The paper then characterizes the performance of LOQO, a line-search interior-point code, on a large test set of nonlinear programming problems. Specific types of problems which can cause LOQO to fail are identified.
Similar content being viewed by others
References
Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: Benchmarks comparing LOQO with SNOPT and NITRO on most problems in the cute/schittkowski test set.
Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: LOQO parameter tuning on problems in the cute/schittkowski test set.
Bongartz, I., Conn, A.R., Gould, N., Toint, Ph.L.: Constrained and unconstrained testing environment. ACM Trans. MAth. Soft. 21(1), 123–160 (1995)
Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large scale nonlinear programming. SIAM J. Opt. 9(4), 877–900 (1999)
El-Bakry, A., Tapia, R., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior-point method for nonlinear programming. J. Optim. Theory and Appl. 89, 507–541 (1996)
Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrainted Minimization Techniques. Research Analysis Corporation, McLean Virginia, 1968. Republished in 1990 by SIAM, Philadelphia.
Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Scientific Press, 1993
Lustig, I.J., Marsten, R.E., Shanno, D.F.: Interior point methods for linear programming: computational state of the art. ORSA J. on Computing 6, 1–14 (1994)
Marazzi, M., Nocedal, J.: Feasibility control in nonlinear optimization. In A. DeVore, A. Iserles, and E. Suli, editors, Foundations of Computational Mathematics. London Mathematical Society Lecture Note Series 284, pages 125–154. Cambridge University Press, 2001
Polyak, R.A.: Modified barrier functions (theory and methods). Math. Prog. 54, 177–222 (1992)
Schittkowski, K.: More Test Samples for Nonlinear Programming codes. Springer Verlag, Berlin-Heidelberg-New York, 1987
Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods. Math. Prog. 87(2), 303–316 (2000)
Simantiraki, E.M., Shanno, D.F.: An infeasible-interior-point method for linear complimentarity problems. SIAM J. Optimization 7, 620–640 (1997)
Simantiraki, E.M., Shanno, D.F.: An infeasible-interior-point method for solving mixed complimentarity problems. In M.C. Ferris and J.S. Pang, editors, Complementarity and Variational Problems: State of the Art, pages 366–400. SIAM, Philadelphia, 1997
Vanderbei, R.J.: AMPL models. www.sor.princeton.edu/∼rvdb/ampl/nlmodels.
Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)
Wächter, A., Biegler, L.: Failure of global convergence for a class of interior point methods for nonlinear programming. Math. Prog. Series A 88(3), 565–574 (2000)
Ye, Y., Todd, M.J., Mizuno, S.: An \(o(\sqrt{n}l)\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)
Author information
Authors and Affiliations
Additional information
Research of the first and third authors supported by NSF grant DMS-9870317, ONR grant N00014-98-1-0036.
Research of the second author supported by NSF grant DMS-9805495.
Rights and permissions
About this article
Cite this article
Benson, H., Shanno, D. & Vanderbei, R. Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing. Math. Program., Ser. A 99, 35–48 (2004). https://doi.org/10.1007/s10107-003-0418-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-003-0418-2