Abstract
We use a decomposition approach to solve three types of realistic problems: block-angular linear programs arising in energy planning, Markov decision problems arising in production planning and multicommodity network problems arising in capacity planning for survivable telecommunication networks. Decomposition is an algorithmic device that breaks down computations into several independent subproblems. It is thus ideally suited to parallel implementation. To achieve robustness and greater reliability in the performance of the decomposition algorithm, we use the Analytic Center Cutting Plane Method (ACCPM) to handle the master program. We run the algorithm on two different parallel computing platforms: a network of PC's running under Linux and a genuine parallel machine, the IBM SP2. The approach is well adapted for this coarse grain parallelism and the results display good speed-up's for the classes of problems we have treated.
Similar content being viewed by others
References
M. Abbad and J. Filar, “Algorithms for singularly perturbed limiting average Markov control problems, ” IEEE Transactions on Automatic Control, vol. 9, pp. 153–168, 1992.
D. Alevras, M. Grötschel, and R. Wesséli, “Capacity and survivability models for telecommunication networks, ” Tech. Report, Konrad-Zuse-Zentrum für Information stechnik, Takustrasse 7, D-14195 Berlin, Germany, June 1997.
O. Bahn, O. du Merle, J.-L. Goffin, and J.-P. Vial, “A cutting plane method from analytic centers for stochastic programming, ” Mathematical Programming, vol. 69, pp. 45–73, 1995.
O. Bahn, A. Haurie, S. Kypreos, and J.-P. Vial, “A multinational MARKAL model to study joint implementation of carbon dioxide emission reduction measures, ” in Joint Implementation of Climate Change Commitments: Opportunities and Apprehensions, P. Ghosh and J. Puri (Eds.), Tata Energy Research Institute: New Delhi, 1994, pp. 43–50.
D.J. Becker, T. Sterling, D. Savarese, J.E. Dorband, U.A. Ranawake, and C.V. Packer, “Beowulf: A parallel workstation for scientific computation, ” in Proceedings of the International Conference on Parallel Processing (ICPP), 1995, pp. 11–14.
J.F. Benders, “Partitioning procedures for solving mixed-variables programming problems, ” Numerische Mathematik, vol. 4, pp. 238–252, 1962.
D.P. Bertsekas and J.N. Tsitsiklis, “Parallel an Distributed Computations, ” Prentice-Hall: Englewood Cliffs, 1989.
E. Cheney and A. Goldstein, “Newton's method for convex programming and Tchebycheff approximation, ” Numerische Mathematik, vol. 1, pp. 253–268, 1959.
G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programming, ” Econometrica, vol. 29, pp. 767–778, 1961.
O. Du Merle, J.-L. Goffin, and J.-P. Vial, “On the comparative behavior of Kelley's cutting plane method and the analytic center cutting plane method, ” Tech. Report, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, March 1996. To appear in Computational Optimization and Applications.
J. Eckstein, “Large-scale parallel computing, optimization, and operations research: A survey, ” ORSA CSTS Newsletter, vol. 14, pp. 1–28, 1993.
J. Filar and A. Haurie, “Optimal ergodic control of singularly perturbed hybrid stochastic systems, ” Lectures in Applied Mathematics, vol. 33, pp. 101–126, 1997.
M.P.I. Forum, “MPI: A message-passing interface standard, ” International Journal of Supercomputer Applications, vol. 8, 1994.
A. Geist, A. Begelin, J. Dongarra, W. Jiang, and R. Mancheck, “PVM: Parallel virtual machine—A user's guide and tutorial for networked parallel computing, ” MIT Press: Cambridge, 1994.
G. Ghellinck and J.-P. Vial, “A polynomial Newton method for linear programming, ” Algorithmica, vol. 1, pp. 425–453, 1986.
J.-L. Goffin, J. Gondzio, R. Sarkissian, and J.-P. Vial, “Solving nonlinear multicommodity flow problems by the analytic center cutting plane method, ” Mathematical Programming, vol. 76, pp. 131–154, 1997.
J.-L. Goffin, A. Haurie, and J.-P. Vial, “Decomposition and nondifferentiable optimization with the projective algorithm, ” Management Science, vol. 38, pp. 284–302, 1992.
J.-L. Goffin and J.-P. Vial, “Interior point methods for nondifferentiable optimization, ” in Operations Research Proceedings 1997, P. Kischka, H.-W. Lorenz, U. Derigs, W. Domaschke, P. Kleinschmidt, and R. Möhring (Eds.), Springer Verlag: Berlin, Germany, 1998, pp. 35–49.
J. Gondzio, “HOPDM(version 2.12)—a fast LP solver based on a primal-dual interior point method, ” European Journal of Operational Research, vol. 85, pp. 221–225, 1995.
J. Gondzio, O. Du Merle, R. Sarkissian, and J.-P. Vial, “ACCPM—a library for convex optimization based on an analytic center cutting plane method, ” European Journal of Operational Research, vol. 94, pp. 206–211, 1996.
J. Gondzio and J.-P. Vial, “Warm start and "-subgradients in cutting plane scheme for block-angular linear programs, ” Computational Optimization and Applications, vol. 14, pp.17–36, 1999.
W. Gropp and E. Lusk, “User's guide to MPICH, a portable implementation of MPI, ” Tech. Report ANL/MCSTM-ANL-96/6, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, USA, 1996.
P. Huard, “Resolution of mathematical programming with nonlinear constraints by the methods of centers, ” in Nonlinear Programming, J. Abadie (Ed.), North-Holland: Amsterdam, 1967, pp. 209–222.
K.L. Jones, I.J. Lustig, J.M. Farvolden, and W.B. Powell, “Multicommodity network flows: The impact of formulation on decomposition, ” Mathematical Programming, vol. 62, pp. 95–117, 1993.
N.K. Karmarkar, “Anewpolynomial-time algorithm for linear programming, ” Combinatorica, vol. 4, pp. 373–395, 1984.
J.E. Kelley, “The cutting plane method for solving convex programs, ” Journal of the IAM, vol. 8, pp. 703–712, 1960.
K.C. Kiwiel, A survey of bundle methods for nondifferentiable optimization, ” in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe (Eds.), Kluwer Academic Publishers, 1989, pp. 262–282.
C. Lemaréchal, “Nondifferentiable optimization, ” in Handbooks in Operations Research and Management Science, G. Nemhauser, A.R. Kan, and M. Todd (Eds.), vol. 1 of Optimization, North-Holland, 1989, pp. 529–572.
C. Lemaréchal, A. Nemirovskii, and Y. Nesterov, “New variants of bundle methods, ” Mathematical Programming, vol. 69, pp. 111–147, 1995.
A. Lisser, R. Sarkissian, and J.-P. Vial, “Survivability in telecommunication networks, ” Tech. Report 1995.3, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, March 1995.
M. Minoux, “Optimum synthesis of a network with non-simultaneous multicommodity flow requirements, ” in Studies on Graphs and Discrete Programming, P. Hansen (Eds.), Noth-Holland, 1981, pp. 269–277.
M. Minoux, “Mathematical Programming: Theory and Algorithms, ” Wiley: New York, 1986.
J.E. Mitchell and M.J. Todd, “Solving combinatorial optimization problems using Karmarkar's algorithm, ” Mathematical Programming, vol. 56, pp. 245–284, 1992.
N. Nakicenovic, A. Gruebler, A. Inaba, S. Messner, S. Nilsson, Y. Nishimura, H-H. Rogner, A. Schaefer, L. Schrattenholzer, M. Strubegger, J. Swisher, D. Victor, and D. Wilson, “Long-term strategies for mitigating global warming, ” Energy—The International Journal, vol. 18, pp. 409–601, 1993.
A. Nemirovsky and D. Yudin, “nformational complexity and efficient methods for solution of convex extremal problems, ” Wiley & Sons: New York, 1983.
J. Renegar, “Apolynomial-time algorithm, based on Newton's method, for linear programming, ” Mathematical Programming, vol. 40, pp. 59–93, 1988.
D. Ridge, D. Becker, P. Merkey, and T. Sterling, “Beowulf: Harnessing the power of parallelism in a pile-of-PCs, ” in Proceedings, IEEE Aerospace, 1997.
A. Ruszczyński, “A regularized decomposition method for minimizing a sum of polyhedral functions, ” Mathematical Programming, vol. 33, pp. 309–333, 1985.
A. Ruszczyński, “Parallel decomposition of multistage stochastic programs, ” Mathematical Programming, vol. 58, pp. 201–228, 1993.
R. Sarkissian, “Telecommunications networks: Routing and survivability optimization using a central cutting plane method, ” PhD thesis, École Polytechnique Fédérale de Lausanne, CH-1205 Ecublens, November 1997.
M. Snir, S.W. Otto, S. Huss-Ledernan, D.W. Walker, and J. Dongarra, “MPI: the complete reference, ” MIT Press, Cambridge, 1996.
G. Sonnevend, “New algorithms in convex programming based on a notion of “centre” (for systems of analytic inequalities) and on rational extrapolation, ” in Trends in Mathematical Optimization: Proceedings of the 4th French–German Conference on Optimization in Irsee, Germany, April 1986, K.H. Hoffmann, J.B. Hiriart-Urruty, C. Lemaréchal, and J. Zowe (Eds.), vol. 84 of International Series of Numerical Mathematics, Birkhäuser Verlag: Basel, Switzerland, 1988, pp. 311–327.
Y. Ye, “A potential reduction algorithm allowing column generation, ” SIAM Journal on Optimization, vol. 2, pp. 7–20, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gondzio, J., Sarkissian, R. & Vial, JP. Parallel Implementation of a Central Decomposition Method for Solving Large-Scale Planning Problems. Computational Optimization and Applications 19, 5–29 (2001). https://doi.org/10.1023/A:1011298218729
Issue Date:
DOI: https://doi.org/10.1023/A:1011298218729