Abstract
We introduce a class of algorithms for the solution of linear programs. This class is motivated by some recent methods suggested for the solution of complementarity problems. It reformulates the optimality conditions of a linear program as a nonlinear system of equations and applies a Newton-type method to this system of equations. We investigate the global and local convergence properties and present some numerical results. The algorithms introduced here are somewhat related to the class of primal–dual interior-point methods. Although, at this stage of our research, the theoretical results and the numerical performance of our method are not as good as for interior-point methods, our approach seems to have some advantages which will also be discussed in detail.
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References
D. Bertsimas and J.N. Tsitsiklis, Introduction to Linear Optimization (Athena Scientific, Belmont, MA, 1997).
J.V. Burke and S. Xu, Preliminary numerical experience with non-interior path following methods for LCP, Talk presented at the International Conference on Nonlinear Programming and Variational Inequalities, Hongkong (December 1998).
J.V. Burke and S. Xu, A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem, Mathematical Programming 87 (2000) 113–130.
B. Chen and P.T. Harker, A non-interior-point continuation method for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications 14 (1993) 1168–1190.
X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Mathematics of Computation 67 (1998) 519–540.
J.E. Dennis, Jr., and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983). Reprinted by (SIAM, Philadelphia, PA, 1996).
M.C. Ferris and C. Kanzow, Complementarity and related problems, in: Handbook on Applied Optimization, eds. P.M. Pardalos and M.G.C. Resende (Oxford University Press) to appear.
A. Fischer, A special Newton-type optimization method, Optimization 24 (1992) 269–284.
A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems, Journal of Optimization Theory and Applications 86 (1995) 85–608.
C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications 17 (1996) 851–868.
C. Kanzow, A new approach to continuation methods for complementarity problems with uniform P-functions, Operations Research Letters 20 (1997) 85–92.
C. Kanzow and H. Pieper, Jacobian smoothing methods for nonlinear complementarity problems, SIAM Journal on Optimization 9 (1999) 342–373.
B. Kummer, Newton's method for nondifferentiable functions, in: Advances in Mathematical Optimization, eds. J. Guddat et al. (Akademie-Verlag, Berlin, Germany, 1988) pp. 114–125.
L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research 18 (1993) 227–244.
L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming 58 (1993) 353–367.
S. Smale, Algorithms for solving equations, in: Proceedings of the International Congress of Mathematicians (Amer. Math. Soc., Providence, 1987) pp. 172–195.
P. Tseng, Error bounds and superlinear convergence analysis of some Newton-type methods in optimization, in: Nonlinear Optimization and Related Topics, eds. G. Di Pillo and F. Giannessi (Kluwer Academic, 2000) pp. 445-462.
S.J. Wright, Primal-Dual Interior-Point Method (SIAM, Philadelphia, PA, 1997).
N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming 76 (1997) 469–491.
Y. Zhang, User's guide to LIPSOL: Linear programming interior point solver v0.4, Optimization Methods and Software 11 & 12 (1999) 385–396.
Y. Zhang, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optimization Methods and Software 10 (1998) 1–31.
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Engelke, S., Kanzow, C. On the Solution of Linear Programs by Jacobian Smoothing Methods. Annals of Operations Research 103, 49–70 (2001). https://doi.org/10.1023/A:1012934518595
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DOI: https://doi.org/10.1023/A:1012934518595