This paper discusses the massively parallel solution of linear network programs. It integrates the general algorithmic framework of proximal minimization with D-functions (PMD) with primal-dual row-action algorithms. Three alternative algorithmic schemes are studied: quadratic proximal point, entropic proximal point, and least 2-norm perturbations. Each is solving a linear network problem by solving a sequence of nonlinear approximations. The nonlinear subproblems decompose for massively parallel computing. The three algorithms are implemented on a Connection Machine CM-2 with up to 32K processing elements, and problems with up to 16 million variables are solved. A comparison of the three algorithms establishes their relative efficiency. Numerical experiments also establish the best internal tactics which can be used when implementing proximal minimization algorithms. Finally, the new algorithms are compared with an implementation of the network simplex algorithm executing on a CRAY Y-MP vector supercomputer.
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D.P. Bertsekas, “Distributed asynchronous relaxation methods for linear network flow problems,” Report LIDS-P-1606, Massachusetts Institute of Technology, Cambridge, MA, 1986.
D.P. Bertsekas, “Distributed relaxation methods for linear network flow problems,” Proc. 25th IEEE Conf. Dec. & Contr., 1986, 2101–2106.
D.P. Bertsekas and D.El Baz, “Distributed asynchronous relaxation methods for convex network flow problems,” SIAM J. on Control and Optimization 25(1), (1987), 74–85.
D.P. Bertsekas, P. Hossein, and P. Tseng, “Relaxation methods for network flow problems with convex arc costs,” SIAM J. on Control and Optimization 25 (1987), 1219–1243.
D.P. Bertsekas and J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall: Englewood Cliffs, NJ, 1989.
W.J. Carolan, J.E. Hill, J.L. Kennington, S. Niemi, and S.J. Wichmann, “An empirical evaluation of the KORBX algorithms for military airlift applications,” Operations Res. 38 (1990), 240–248.
Y. Censor, “Parallel application of block-iterative methods in medical imaging and radiation therapy,” Math. Programming Series B, 42(2), (1988), 307–326.
Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” J. of Optimization Theory and Applications, 34 (1981), 321–353.
Y. Censor, A.R. De Pierro, T. Elfving, G.T. Herman, and A.N. Iusem, “On iterative methods for linearly constrained entropy maximization,” in Numerical Analysis and Mathematical Modelling, vol. 24, A. Wakulicz, ed., Banach Center Publications, PWN-Polish Scientific Publisher: Warsaw, Poland, (1990), 145–163.
Y. Censor and S.A. Zenios, “The proximal minimization algorithm with D-functions,” J. of Optimization Theory and Applications, 73(3), (1992), 455–468.
Y. Censor and S.A. Zenios, “On the use of D-functions in primal-dual algorithms and in the proximal minimization algorithm,” in Optimization and Nonlinear Analysis, N. 244 Pitman Research Notes in Mathematics Series, A. Ioffe, M. Marcus, and S. Reich, eds., Longman: London, England, 1992, 76–97.
J. Eckstein, “Splitting methods for monotone operators with applications to parallel optimization,” Technical report LIDS-TM-1877, Dept. of Civil Engineering, Massachusetts Inst. of Technology, Cambridge, MA, 1989.
J. Eckstein, “Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming,” Math. of Operations Res., 18 (1993), to appear.
J. Eckstein, “An alternating step method for monotropic programming on the Connection Machine CM-2,” ORSA J. on Computing, 5 (1993), to appear.
J.R. Eriksson, “An iterative primal-dual algorithm for linear programming,” Report LiTH-MAT-R-1985–10, Dept. of Mathematics, Linkøping Univ., S-581 83 Linkøping, Sweden, 1985.
D. Klingman, A. Napier, and J. Stutz, “NETGEN-A program for generating large-scale (un)-capacitated assignment, transportation, and minimum cost flow network problems,” Mgmt. Sci. 20 (1974), 814–822.
R. De Leone and O.L. Mangasarian, “Serial and parallel solution of large scale linear programs by augmented Lagrangian successive overrelaxation,” in Optimization, Parallel Processing and Applications, vol. 304 of Lecture Notes in Economics and Mathematical Systems, A. Kurzhanski, K. Neumann, and D. Pallaschke, eds., Springer-Verlag, Berlin, 1988, 103–124.
X. Li and S.A. Zenios, “A massively parallel δ-relaxation algorithm for linear transportation problems,” in Advances in Optimization and Parallel Computing, P. Pardalos,ed., Elsevier Science Publications, 1992, 164–176.
O.L. Mangasarian and R.R. Meyer, “Nonlinear perturbations on linear programs,” SIAM J. on Control and Optimization, 17 (1979), 745–752.
J.M. Mulvey and S.A. Zenios, “GENOS 1.0: A generalized network optimization system. User's guide.” Report 87–12–03, Decision Sciences Dept., the Wharton School, Univ. of Pennsylvania, Philadelphia, PA, 1987.
S.S. Nielsen and S.A. Zenios, “Massively parallel algorithms for singly constrained nonlinear programs,” ORSA J. on Computing, 4(2), (1992), 166–181.
S.S. Nielsen and S.A. Zenios, “Massively parallel algorithms for nonlinear stochastic network problems,” Operations Res. 43(2), (1993), to appear.
M.Ç. Pinar and S.A. Zenios, “Naval personnel assignment: An application of linear-quadratic penalty methods,” in Computer Science and Operations Research: New developments in their Interfaces, O. Balci, R. Sharda, and S.A. Zenios, eds., Pergamon Press, 1992, 43–58.
M.Ç. Pinar and S.A. Zenios, “Parallel decomposition of multicommodity network flows using a linear-quadratic penalty algorithm,” ORSA J. on Computing 4(3), (1992), 235–249.
R.T. Rockafellar, “Augmented Lagrangians and applications to proximal point algorithms in convex programming,” Math. of Operations Res. 1 (1976), 97–116.
R.T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. on Control and Optimization 14 (1976), 877–898.
G.L. Schultz and R.R. Meyer, “An interior point method for block angular optimization,” SIAM J. on Optimization, 1(4), (1991), 583–602.
R. Setiono, “Interior dual least 2-norm algorithms for linear programs,” SIAM J. on Control and Optimization, (1992), to appear.
Stephen J. Wright, “Implementing proximal point methods for linear programming,” J. of Optimization Theory and Applications 65 (1990), 531–554.
S.A. Zenios, “On the fine-grain decomposition of multicommodity transportation problems,” SIAM J. of Optimization 1 (1991), 643–669.
S.A. Zenios and Y. Censor, “Massively parallel row-action algorithms for some nonlinear transportation problems,” SIAM J. of Optimization 1 (1991), 373–400.
S.A. Zenios and R.A. Lasken, “Nonlinear network optimization on a massively parallel Connection Machine,” Annals of Operations Res. 14 (1988), 147–165.
S.A. Zenios and J.M. Mulvey, “A distributed algorithm for convex network optimization problems,” Parallel Computing 6 (1988), 45–56.
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Nielsen, S.S., Zenios, S.A. Proximal minimizations with D-functions and the massively parallel solution of linear network programs. Comput Optim Applic 1, 375–398 (1993). https://doi.org/10.1007/BF00248763
- network programs
- parallel computing
- nonlinear programming