Parallel Implementation of a Central Decomposition Method for Solving Large-Scale Planning Problems

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.

[1]  A. A. Goldstein,et al.  Newton's method for convex programming and Tchebycheff approximation , 1959, Numerische Mathematik.

[2]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[3]  G. Dantzig,et al.  THE DECOMPOSITION ALGORITHM FOR LINEAR PROGRAMS , 1961 .

[4]  G. Dantzig,et al.  The decomposition algorithm for linear programming: notes on linear programming and extensions-part 57. , 1961 .

[5]  M. Minoux.  Optimum Synthesis of a Network with Non-Simultaneous Multicommodity Flow Requirements* , 1981 .

[6]  John R. Beaumont,et al.  Studies on Graphs and Discrete Programming , 1982 .

[7]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[8]  V. F. Dem'yanov,et al.  Nondifferentiable Optimization , 1985 .

[9]  David K. Smith,et al.  Mathematical Programming: Theory and Algorithms , 1986 .

[10]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[11]  G. Sonnevend.  New Algorithms in Convex Programming Based on a Notion of “Centre” (for Systems of Analytic Inequalities) and on Rational Extrapolation , 1988 .

[12]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[13]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[14]  J. Filar,et al.  Algorithms for singularly perturbed limiting average Markov control problems , 1990, 29th IEEE Conference on Decision and Control.

[15]  Yinyu Ye,et al.  A Potential Reduction Algorithm Allowing Column Generation , 1992, SIAM J. Optim..

[16]  Michael J. Todd,et al.  Solving combinatorial optimization problems using Karmarkar's algorithm , 1992, Math. Program..

[17]  J. Goffin,et al.  Decomposition and nondifferentiable optimization with the projective algorithm , 1992 .

[18]  Warren B. Powell,et al.  Multicommodity network flows: The impact of formulation on decomposition , 1993, Math. Program..

[19]  N. Nakicenovic,et al.  Long-term strategies for mitigating global warming☆ , 1993 .

[20]  Andrzej Ruszczynski,et al.  Parallel decomposition of multistage stochastic programming problems , 1993, Math. Program..

[21]  Jack J. Dongarra,et al.  The PVM Concurrent Computing System: Evolution, Experiences, and Trends , 1994, Parallel Comput..

[22]  Message Passing Interface Forum.  MPI: A message - passing interface standard , 1994 .

[23]  P. Ghosh,et al.  Joint implementation of climate change commitments opportunities and apprehensions , 1994 .

[24]  J. Gondzio.  HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method , 1995 .

[25]  Yurii Nesterov,et al.  New variants of bundle methods , 1995, Math. Program..

[26]  Jean-Philippe Vial,et al.  A cutting plane method from analytic centers for stochastic programming , 1995, Math. Program..

[27]  Thomas L. Sterling,et al.  BEOWULF: A Parallel Workstation for Scientific Computation , 1995, ICPP.

[28]  Abdel Lisser,et al.  Survivability in Telecommunication Networks , 1995 .

[29]  Jacek Gondzio,et al.  Solving nonlinear multicommodity flow problems by the analytic center cutting plane method , 1997, Math. Program..

[30]  Jack Dongarra,et al.  MPI: The Complete Reference , 1996 .

[31]  William Gropp,et al.  Users guide for mpich, a portable implementation of MPI , 1996 .

[32]  William Gropp,et al.  User''s Guide for mpich , 1996 .

[33]  Olivier du Merle,et al.  A Short Note on the Comparative Behaviour of Kelley's Cutting Plane Method and the Analytic Center Cutting Plane Method , 1996 .

[34]  Jacek Gondzio,et al.  ACCPM — A library for convex optimization based on an analytic center cutting plane method☆ , 1996 .

[35]  M. Pernice,et al.  PVM: Parallel Virtual Machine - A User's Guide and Tutorial for Networked Parallel Computing [Book Review] , 1996, IEEE Parallel & Distributed Technology: Systems & Applications.

[36]  Martin Grötschel,et al.  Capacity and Survivability Models for Telecommunication Networks , 1997 .

[37]  J. Filar,et al.  Optimal Ergodic Control of Singularly Perturbed Hybrid Stochastic Systems , 1997 .

[38]  P. Merkey,et al.  Beowulf: harnessing the power of parallelism in a pile-of-PCs , 1997, 1997 IEEE Aerospace Conference.

[39]  Jean-Philippe Vial,et al.  Interior Point Methods for Nondifferentiable Optimization , 1998 .

[40]  Robert Sarkissian.  Telecommunications networks routing and survivability optimization using a central cutting plane method , 1998 .

[41]  J. Vial.  Interior Point Methods for Nondiierentiable Optimization , 1998 .

[42]  J. Gondizo,et al.  Warm Start and ε-Subgradients in a Cutting Plane Scheme for Block-Angular Linear Programs , 1999, Comput. Optim. Appl..