Abstract
The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
Similar content being viewed by others
References
M.P. Bendsøe, A. Ben-Tal and J. Zowe, “Optimization methods for truss geometry and topology design,” Structural Optimization, vol. 7, pp. 141-159, 1994.
A. Brooke, D. Kendrick and A. Meeraus, GAMS: A User's Guide, Scientific Press, 1988.
R.H. Byrd, M.E. Hribar and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” Technical Report OTC 97/05, Optimization Technology Center, Northwestern University, 1997.
A.R. Conn, N. Gould and Ph.L. Toint, “A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds,” Math. of Computation, vol. 66, pp. 261-288, 1997.
A.R. Conn, N. Gould and Ph.L. Toint, “A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints,” Technical Report Report 96/9, Dept of Mathematics, FUNDP, Namur (B), 1997.
A.R. Conn, N.I.M. Gould and Ph.L. Toint, LANCELOT: a Fortran Package for Large-Scale Nonlinear Optimization (Release A). Springer Verlag, Heidelberg, New York, 1992.
A. El-Bakry, R. Tapia, T. Tsuchiya and Y. Zhang, “On the formulation and theory of the Newton interior-point method for nonlinear programming,” J. of Optimization Theory and Appl., vol. 89, pp. 507-541, 1996.
A.V. Fiacco and G.P. McCormick, “Nonlinear Programming: Sequential Unconstrainted Minimization Techniques,” Research Analysis Corporation, McLean Virginia, 1968 (Republished in 1990 by SIAM, Philadelphia).
R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” Technical Report NA/171, University of Dundee, Dept. of Mathematics, Dundee, Scotland, 1997.
A. Forsgren and P.E. Gill, “Primal-dual interior methods for nonconvex nonlinear programming,” Technical Report NA 96-3, Department of Mathematics, University of California, San Diego, 1996.
R. Fourer, D.M. Gay and B.W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Scientific Press, 1993.
R.S. Gajulapalli and L.S. Lasdon, “Computational experience with a safeguarded barrier algorithm for sparse nonlinear programming,” Technical report, University Texas at Austin, 1997.
D.M. Gay, M. L. Overton and M.H. Wright, “A primal-dual interior method for nonconvex nonlinear programming,” in Proceedings of the 1996 International Conference on Nonlinear Programming, Kluwer: Boston, 1998, to appear.
P.E. Gill, W. Murray, D.B. Ponceleón and M.A. Saunders. “Solving reduced KKT systems in barrier methods for linear and quadratic programming,” Technical Report SOL 91-7, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1991.
R.L. Graham, “The largest small hexagon,” Journal of Combinatorial Theory, vol. 18, pp. 165-170, 1975.
W. Hock and K. Schittkowski, “Test examples for nonlinear programming codes,” volume 187 of Lecture Notes in Economics and Mathematical Systems, Springer Verlag: Heidelberg, 1981.
H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Transactions on Signal Processing, vol. 45, pp. 526-532, 1997.
I.J. Lustig, R.E. Marsten and D.F. Shanno, “Interior point methods for linear programming: computational state of the art,” ORSA J. on Computing, vol. 6, pp. 1-14, 1994.
H. Mittelmann, Benchmarks for optimization software, http://plato.la.asu.edu/bench.html.
B.A. Murtagh and M.A. Saunders, “MINOS 5.4 user's guide,” Technical Report SOL 83-20R, Systems Optimization Laboratory, Stanford University, 1983 (Revised February, 1995).
D.F. Shanno and E.M. Simantiraki, “Interior-point methods for linear and nonlinear programming,” in The State of the Art in Numerical Analysis, Oxford University Press: New York, pp. 339-362, 1997.
R.J. Vanderbei, Large-scale nonlinear AMPL models, http://www.sor.princeton.edu/~rvdb/ampl/nlmodels/.
R.J. Vanderbei, “LOQO: An interior point code for quadratic programming,” Technical Report SOR 94-15, Princeton University, 1994.
R.J. Vanderbei, “Symmetric quasi-definite matrices,” SIAM Journal on Optimization, vol. 5,no. 1, pp. 100-113, 1995.
R.J. Vanderbei, Linear Programming: Foundations and Extensions, Kluwer Academic Publishers: Boston, MA, 1996.
T. Wang, R.D.C. Monteiro and J.S. Pang, “An interior-point potential reduction method for constrained equations,” Mathematical Programming, vol. 74, pp. 159-195, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vanderbei, R.J., Shanno, D.F. An Interior-Point Algorithm for Nonconvex Nonlinear Programming. Computational Optimization and Applications 13, 231–252 (1999). https://doi.org/10.1023/A:1008677427361
Issue Date:
DOI: https://doi.org/10.1023/A:1008677427361