Abstract
By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach.
Similar content being viewed by others
References
D.P. Bertsekas (1977) ArticleTitleApproximation procedures based on the method of multipliers Journal of Optimization Theory and Applications 23 487–510 Occurrence Handle10.1007/BF00933293
J. Burke S. Xu (1998) ArticleTitleThe global linear convergence of a non-interior path-following algorithm for linear complementarity problem Mathematics of Operations Reseasch 23 719–734
P.L. Chang (1980) A minimax approach to nonlinear programming Department of Mathematics, University of Washington Seattle, WA
B. Chen X. Chen (2000) ArticleTitleA global linear and local quadratic continuation smoothing method for variational inequalities with box constrains Computational Optimization and Applications 13 131–158
R.W. Cottle G.B. Dantzig (1970) ArticleTitleA generalization of the linear complementarity problem Journal of Combinatorial Theory 8 79–90
R.W. Cottle J.S. Pang R.E. Stone (1992) The Linear Complementarity Problem Academic Press Boston, MA
X. Chen Y. Ye (1999) ArticleTitleOn homotopy-smoothing methods for variational inequalities SIAM Journal on Control and Optimization 37 589–616 Occurrence Handle10.1137/S0363012997318602
A.A. Ebiefung (1995) ArticleTitleNonlinear mappings associated with the generalized linear complementarity problem Mathematical Programming 69 255–268
A.A. Ebiefung M.M. Kostreva (1993) ArticleTitleThe generalized Leontief input–output model and its application to the choice of the new technology Annals of Operations Research 44 161–172 Occurrence Handle10.1007/BF02061065
Engelke, S. and Kanzow, C. (2000) Predictor–corrector smoothing methods for the solution of linear programming, Preprint, Department of Mathematics, University of Hamburg, Germany, March, 2000.
S.-C. Fang H.-S.J. Tsao (1996) ArticleTitleOn the entropic perturbation and exponential penalty methods for linear programming Journal of Optimization Theory and Applications 89 461–466 Occurrence Handle10.1007/BF02192539
S.-C. Fang J.R. Rajasekera H.-S. J. Tsao (1997) Entropy Optimization and Mathematical Programming Kluwer Academic Publishers Boston/London/Dordrecht
A. Fischer C. Kanzow (1996) ArticleTitleOn the finite termination of an iterative method for linear complementarity problems Mathematical Programming 74 279–292 Occurrence Handle10.1016/0025-5610(96)00008-1
T. Fujisawa E.S. Kuh (1972) ArticleTitlePiecewise-linear theory of nonlinear networks SIAM Journal on Applied Mathematics 22 307–328 Occurrence Handle10.1137/0122030
A.A. Goldstein (1997) Chebyshev approximation and linear inequalities via exponentials Department of Mathematics, University of Washington Seattle, WA
M.S. Gowda R. Sznajder (1994) ArticleTitleThe generalized order linear complementarity problem SIAM Journal of Matrix Analysis and Applications 15 779–795 Occurrence Handle10.1137/S0895479892237859
M.S. Gowda R. Sznajder (1996) ArticleTitleA generalization of the Nash equilibrium theorem on bimatrix games International Journal of Game Theory 25 1–12
A.J. Hoffman (1952) ArticleTitleOn approximate solutions of systems of linear equalities Journal of Research of the National Bureau of Standards 49 263–265
Z.H. Huang J. Han (2003) ArticleTitleNon-interior continuation method for solving the monotone semidefinite complementarity problem Applied Mathematics and Optimization 47 195–211 Occurrence Handle10.1007/s00245-003-0765-7
Z.H. Huang J. Han Z. Chen (2003) ArticleTitleA predictor–corrector smoothing Newton algorithm, based on a new smoothing function, for solving the nonlinear complementarity problem with a P 0 function Journal of Optimization Theory and Applications 117 39–68 Occurrence Handle10.1023/A:1023648305969
Z.H. Huang L. Qi D. Sun (2004) ArticleTitleSub-quadratic convergence of a smoothing Newton algorithm for the P 0- and monotone LCP Mathematical Programming 99 423–441 Occurrence Handle10.1007/s10107-003-0457-8
Z.H. Huang L. Zhang J. Han (2004) ArticleTitleA hybrid smoothing–nonsmooth Newtontype algorithm yielding an exact solution of the P 0-LCP Journal of Computational Mathematics 22 797–806
T. Illés J.M. Peng C. Roos T. Terlaky (2001) ArticleTitleA strongly polynomial procedure yielding a maximally complementarity solution for P *(κ) linear complementarity problems SIAM Journal of Optimization 11 320–340
Kort, B.W. and Bertsekas, D.P. (1972), A new penalty function for constrained minimization, Proceedings of the 1972 IEEE Conference on Decision and Control, New Orleans, Louisiana.
X.S. Li (1991) ArticleTitleAn aggregate function method for nonlinear programming Science in China (Ser. A) 34 1467–1473
X.S. Li S.-C. Fang (1997) ArticleTitleOn the entropic regularization method for solving min–max problems with applications Mathematical Methods of Operations Research 46 119–130 Occurrence Handle10.1007/BF01199466
O.L. Mangasarian (1979) ArticleTitleGeneralized linear complementarity problems as linear programming Opemtions Research Verfahren 31 393–402
S.R. Mohan S.K. Neogy R. Sridhar (1996) ArticleTitleThe generalized linear complementarity problem revisited Mathematical Programming 74 197–218 Occurrence Handle10.1016/0025-5610(96)82467-1
S. Mehrotra Y. Ye (1993) ArticleTitleOn finding the optimal facet of linear programs Mathematical Programming 62 497–515 Occurrence Handle10.1007/BF01585180
P.J. Peng Z. Lin (1999) ArticleTitleA non-interior continuation method for generalized linear complementarity problems Mathematical Programming 86 533–563
H.D. Qi L.Z. Liao (1999) ArticleTitleA smoothing Newton method for extended vertical linear complementarity problems SIAM Journal on Matrix Analysis and Applications 21 45–66 Occurrence Handle10.1137/S0895479897329837
H.D. Qi L.Z. Liao Z. Lin (1999) ArticleTitleRegularized smoothing approximations to vertical nonlinear complementarity problems Journal of Mathematical Analysis and Applications 230 261–276 Occurrence Handle10.1006/jmaa.1998.6205
L. Qi D. Sun (2000) ArticleTitleImproving the convergence of non-interior point algorithm for nonlinear complementarity problems Mathematics of Computation 69 283–304 Occurrence Handle10.1090/S0025-5718-99-01082-0
A. Schrijver (1986) Theory of Linear and Integer Programming Wiley New York
M. Sun (1987) ArticleTitleSingular control problems in bounded intervals Stochastics 21 303–344
M. Sun (1989) ArticleTitleMonotonicity of Mangasarian’s iterative algorithm for the generalized linear complementarity problem Journal of Mathematical Analysis and Applications 144 473–485 Occurrence Handle10.1016/0022-247X(89)90347-8
D. Sun J. Han Y.B. Zhao (1998) ArticleTitleOn the finite termination of the damped-Newton algorithm for the linear complementarity problem Acta Mathematica Applicatae Sinica 21 148–154
R. Sznajder M.S. Gowda (1995) ArticleTitleGeneralizations of P 0 and P-properties, extended vertical and horizontal LCP’s Linear Algebra and its Applications 223 IssueID224 695–716
P. Tseng (1999) Analysis of a non-interior continuation method based on Chen– Mangasarian smoothing functions for complementarity problems M. Fukushima L. Qi (Eds) Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods Kluwer Academic Publishers Boston 381–404
P. Tseng (2000) Error bounds and superlinear convergence analysis of some Newton-type methods in optimization G. Di Pillo F. Giannessi (Eds) Nonlinear Optimization and Related Topics Kluwer Academic Publishers Boston 445–462
P. Tseng D.P. Bertsekas (1993) ArticleTitleOn the convergence of the exponential multiplier method for convex programming Mathematical Programming 60 1–19 Occurrence Handle10.1007/BF01580598
Y. Ye (1992) ArticleTitleOn the finite convergence of interior-point algorithms for linear programming Mathematical Programming 57 325–335 Occurrence Handle10.1007/BF01581087
Author information
Authors and Affiliations
Corresponding author
Additional information
This author’s work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10271002 and 10401038).
This author’s work was partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.
Rights and permissions
About this article
Cite this article
Fang, SC., Han, J., Huang, ZH. et al. On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems. J Glob Optim 33, 369–391 (2005). https://doi.org/10.1007/s10898-004-6098-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-004-6098-5
Keywords
- Entropy function
- Finite termination
- Non-interior continuation method
- Vertical linear complementarity problems
- Smoothing approximation