Abstract
The Mizuno-Todd-Ye predictor-corrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity depends on the quality of the starting point. If the starting points are large enough, then the algorithm hasO(nL) iteration complexity. If a certain measure of feasibility at the starting point is small enough, then the algorithm has\(O(\sqrt {nL} )\) iteration complexity. At each iteration, both “feasibility” and “optimality” are reduced exactly at the same rate. The algorithm is quadratically convergent for problems having a strictly complementary solution, and therefore its asymptotic efficiency index is\(\sqrt 2\). A variant of the algorithm can be used to detect whether solutions with norm less than a given constant exist.
Similar content being viewed by others
References
J.F. Bonnans and C.C. Gonzaga, Convergence of interior point algorithms for the monotone linear complementarity problem, Research Report, INRIA, P.O. Box 105, 78153 Rocquencourt, France (1993), to appear in Mathematics of Operations Research.
R.M. Freund, An infeasible-start algorithm for linear programming whose complexity depends on the distance from the starting point to the optimal solution, Working Paper 3559-93-MSA, Sloan School of Management, Massachusetts Institute of Technology, Cambridge MA 02139, USA (1993).
C.C. Gonzaga and R.A. Tapia, On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set, Technical Reports TR92-41, Department of Mathematical Sciences, Rice University, Houston, Texas 77251-1892, USA (1992).
O. Güler, Generalized linear complementarity problems, Research Reports, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21228-5398, USA (1993), to appear in Mathematics of Operations Research.
J. Ji, F.A. Potra and S. Huang, A predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence, Reports on Computational Mathematics 18, Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA (1991).
M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Mathematical Programming 61 (1993) 263–280.
M. Kojima, S. Mizuno and M.J. Todd, Infeasible-interior-point primal-dual potential-reduction algorithms for linear programming, Technical Report 1023, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA, (1992), to appear in SIAM Journal on Optimization.
I.J. Lustig, R.E. Marsten and D.F. Shanno, Computational experience with a globally convergent primal-dual predictor-corrector algorithm for linear programming, Technical Report, School of Engineering and Applied Science, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA (1992).
J. Miao, Two infeasible interior-point predictor-corrector algorithms for linear programming, Research Report RRR 20-93, RUTCOR, Rutgers Center for Operations Research, Rutgers University, P.O. Box 5063, New Brunswick, NJ, USA (1993).
S. Mizuno, AnO(n 3 L) algorithm using a sequence for linear complementarity problems, Journal of the Operations Research Society of Japan 33 (1990)66–75.
S. Mizuno, Polynomiality of the Kojima-Megiddo-Mizuno infeasible interior point algorithm for linear programming, Mathematical Programming 67 (1994) 109–120.
S. Mizuno, M.J. Todd and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research, 18(1993)964–981.
R.D.C. Monteiro and S.J. Wright, Local convergence of interior-point algorithm for degenerate monotone LCP, Preprint MCS-P357-0493, Mathematics, and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (1993).
R.D.C. Monteiro and S.J. Wright, Superlinear primal-dual affine scaling algorithms for LCP, Technical Report 93-9, Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA (1993).
A.M. Ostrowski,Solution of Equations in Euclidian, and Banach Spaces (Academic Press, New York, 1973).
F.A. Potra, An infeasible interior-point predictor-corrector algorithm for linear programming, Reports on Computational Mathematics 26, Department of Mathematics The University of Iowa, Iowa City, IA 52242, USA (1992).
F.A. Potra, On a predictor-corrector method for solving linear programming from infeasible starting points, Reports on Computational Mathematics 34, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA (1992).
F.A. Potra, A quadratically convergent infeasible interior-point algorithm for linear programming, Reports on Computational Mathematics 28, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA (1992).
R.H. Tütüncü and M.J. Todd, Reducing, horizontal linear complementarity problems, Technical Report 1023, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, USA (1992), to appear in Linear Algebra and its Applications.
S.J. Wright, A path-following infeasible-interior-point algorithm for linear complementarity problems, Optimization Methods and Software 2 (1993) 79–106.
S.J. Wright, A path-following interior-point algorithm, for linear and quadratic problems, Preprint MCS-P401-1293, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (1993).
S.J. Wright, An infeasible interior point algorithm for linear complementarity problems, Mathematical Programming 67(1994)29–52.
F. Wu, S. Wu and Y. Ye, On quadratic convergence of the homogeneous and self-dual linear programming algorithm, Working paper, Department of Management Sciences, The University of Iowa, Iowa City, IA 52242, USA (1993).
Y. Ye and K. Anstreicher, On quadratic and\(O(\sqrt {nL} )\) convergence of predictor-corrector algorithm for LCP, Mathematical Programming 62(1993)537–551.
Y. Ye, O. Güler, R.A. Tapia and Y. Zhang, A quadratically convergent\(O(\sqrt {nL} )\)-iteration algorithm for linear programming, Mathematical Programming 59(1993)151–162.
Y. Ye, R. A. Tapia and Y. Zhang, A superlinearly convergent\(O(\sqrt {nL} )\)-iteration algorithm for linear programming, Technical Reports TR91-22, Department of Mathematical Sciences, Rice University, Houston, Texas 77251-1892, USA (1991).
Y. Ye, M.J. Todd and S. Mizuno, An\(O(\sqrt {nL} )\)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research 19(1994)53–67.
Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM J. Optimization 4(1994)208–227.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation under grant DMS-9305760.
Rights and permissions
About this article
Cite this article
Potra, F.A. AnO(nL) infeasible-interior-point algorithm for LCP with quadratic convergence. Ann Oper Res 62, 81–102 (1996). https://doi.org/10.1007/BF02206812
Issue Date:
DOI: https://doi.org/10.1007/BF02206812