Abstract
The relaxation method for linear inequalities is studied and new bounds on convergence obtained. An asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order. An improvement of the estimate by an order of magnitude takes place if strong underrelaxation is used. Bounds on convergence usually involve the so-called condition number of a system of linear inequalities, which we estimate in terms of their coefficient matrix.
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References
S. Agmon, “The relaxation method for linear inequalities”,Canadian Journal of Mathematics 6 (1954) 382–392.
Y. Censor, P.P.B. Eggermont and D. Gordon “Strong underrelaxation in Kaczmarz's method for inconsistent systems”, Technical Report MIPG62, Department of Radiology, University of Pennsylvania, Philadelphia, PA, December, 1981.
R.W. Cottle and J.-S. Pang, “On solving linear complementarity problems as linear programs”,Mathematical Programming Study 7 (1978) 88–107.
J.L. Goffin, “On the nonpolynomiality of the relaxation method for systems of linear inequalities”,Mathematical Programming 22 (1982) 93–103.
J.L. Goffin, “Acceleration in the relaxation method for linear inequalities and subgradient optimization”, in: E.A. Nurminski, ed., Progress in Nondifferentiable Optimization (IIASA, Laxenburg, 1982) pp. 29–59.
J.L. Goffin, “The relaxation method for solving system of linear inequalities”,Mathematics of Operations Research 5 (1980) 388–414.
G.T. herman,Image reconstructions from projections, the fundamentals of computerized tomography (Academic Press, New York, 1980).
G.T. Herman, “A relaxation method for reconstructing objects from noisy X-rays”,Mathematical Programming 8 (1975) 1–19.
A.S. Householder and F.L. Bauer, “On certain iterative methods for solving linear systems”,Numerische Mathematik 2 (1960) 55–59.
L.G. Khachian, “A polynomial algorithm in linear programming”,Doklady Akademii Nauk SSSR 244 (1979) 1093–1095 [translated inSoviet Mathematics Doklady 20 (1979) 191–194].
Th. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities”,Canadian Journal of Mathematics 6 (1954) 393–404.
W. Oettli, “An iterative method, having a linear rate of convergence, for solving a pair of dual linear programs”,Mathematical Programming 3 (1972) 302–311.
M.J. Todd, “Some results on the relaxation methods for linear inequalities”, Technical Report 419, SORIE Cornell University, Ithaca, NY, April, 1979.
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Mandel, J. Convergence of the cyclical relaxation method for linear inequalities. Mathematical Programming 30, 218–228 (1984). https://doi.org/10.1007/BF02591886
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DOI: https://doi.org/10.1007/BF02591886