Abstract
Serial and parallel successive overrelaxation (SOR) methods are proposed for the solution of the augmented Lagrangian formulation of the dual of a linear program. With the proposed serial version of the method we have solved linear programs with as many as 125,000 constraints and 500,000 variables in less than 72 hours on a MicroVax II. A parallel implementation of the method was carried out on a Sequent Balance 21000 multiprocessor with speedup efficiency of over 65% for problem sizes of up to 10,000 constraints, 40,000 variables and 1,400,000 nonzero matrix elements.
This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-86-0255.
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Abbreviations
- SOR:
-
Solution of Linear Programs
References
D. P. Bertsekas: “Necessary and sufficient conditions for a penalty method to be exact”, Mathematical Programming 9, 1975, 87–99.
D. P. Bertsekas: “Constrained optimization and Lagrange multiplier methods”, Academic Press, New York 1982.
E. G. Golshtein: “An iterative linear programming algorithm based on the use of modified Lagrange function”, US-USSR Seminar on Distribution and Production Problems of Industry and Enterprise, October 15–17, 1980, University of Maryland, College Park, Proceedings Published in College Park, Maryland 1982.
Y. Y. Lin, J. S. Pang: “Iterative methods for large convex quadratic programs: A survey”, SIAM Journal on Control and Optimization 25, 1987, 383–411.
O. L. Mangasarian: “Nonlinear programming”, McGraw-Hill, New York 1969.
O. L. Mangasarian: “Unconstrained Lagrangians in nonlinear programming”, SIAM Journal on Control 13, 1975, 772–791.
O. L. Mangasarian: “Solution of symmetric linear complementarity problems by iterative methods”, Journal of Optimization Theory and Applications 22, 1977, 465–485.
O. L. Mangasarian: “Iterative solution of linear programs”, SIAM Journal on Numerical Analysis 18, 1981, 606–614.
O. L. Mangasarian: “Sparsity-preserving SOR algorithms for separable quadratic and linear programs”, Computers and Operations Research 11, 1984, 105–112.
O. L. Mangasarian: “Normal solution of linear programs”, Mathematical Programming Study 22, 1984, 206–216.
O. L. Mangasarian, R. De Leone: “Parallel successive overrelaxation methods for symmetric linear complementarity problems and linear programs”, Journal of Optimization Theory and Applications 54 (3), September 1987.
O. L. Mangasarian, R. De Leone: “Error bounds for strongly convex programs and (super)linearly convergent iterative schemes for the least 2-norm solution of linear programs”, Applied Mathematics and Optimization, 1988.
O. L. Mangasarian, R. De Leone: “Parallel gradient projection successive overrelaxation for symmetric linear complementarity problems and linear programs”, University of Wisconsin Computer Sciences Department Tech. Rept. 659, August 1987, submitted to Annals of Operations Research.
O. L. Mangasarian, R. R. Meyer: “Nonlinear perturbation of linear programs”, SIAM Journal on Control and Optimization 17, 1979, 745–752.
B. A. Murtagh, M. A. Saunders: “MINOS 5.0 user’s guide”, Stanford University Technical Report SOL 83. 20, December 1983.
J. S. Pang: “More results on the convergence of iterative methods for the symmetric linear complementarity problem”, Journal of Optimization Theory and Applications 49, 1986, 107–134.
B. T. Polyak: “Introduction to optimization”, Optimization Software, Inc., New York 1987 (translated from Russian).
B. T. Polyak, N. V. Tretiyakov: “Concerning an iterative method for linear programming and its economic interpretation”, Economics and Mathematical Methods 8(5) 1972, 740–751 (Russian).
R. T. Rockafellar: “A dual approach to solving nonlinear programming problems by unconstrained optimization”, Mathematical Programming 5, 1973, 354–373.
R. T. Rockafellar: “Monotone operators and the proximal point algorithm”, SIAM Journal on Control and Optimization 14, 1976, 877–898.
R. T. Rockafellar: “Augmented Lagrangians and applications of the proximal point algorithm in convex programming”, Mathematics of Operations Research 1, 1976, 97–116.
Yu. P. Syrov, Sh. S. Churkreidze: “Questions on the optimization of inter-branch and inter-regional connections in the planning for the growth of a unified national-economic system”, Institute of National Economy of Irkutsk, Irkutsk 1970 ( Russian).
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De Leone, R., Mangasarian, O.L. (1988). Serial and Parallel Solution of Large Scale Linear Programs by Augmented Lagrangian Successive Overrelaxation. In: Kurzhanski, A., Neumann, K., Pallaschke, D. (eds) Optimization, Parallel Processing and Applications. Lecture Notes in Economics and Mathematical Systems, vol 304. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46631-1_11
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DOI: https://doi.org/10.1007/978-3-642-46631-1_11
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