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FATCOP: A Mixed Integer Program Solver

FATCOP: A Mixed Integer Program Solver. Michael Ferris Qun Chen University of Wisconsin-Madison Jeffrey Linderoth Argonne National Laboratories. MIP formulation. minimize c T x subject to Ax  b l  x  u and some x j integer.

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FATCOP: A Mixed Integer Program Solver

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  1. FATCOP: A Mixed Integer Program Solver Michael FerrisQun Chen University of Wisconsin-Madison Jeffrey Linderoth Argonne National Laboratories

  2. MIP formulation minimize cTx subject to Ax  b l  x  u and some xj integer Problems are specified by application convenient format - GAMS, AMPL, or MPS

  3. Top node 0 2 1 3 4 xf 0 xf 1 xg 0 xg 1 Branch-and-Bound Algorithm Integer infeasible Integer feasible incumbent = Z LP infeasible LP relaxation Zlp > Z

  4. The “Seymour Problem” • Set covering problem used in proof of four color theorem • CPLEX 6.0 and Condor (2 option files) • Running since June 23, 1999 • Currently >340 days CPU time per job • (7.9 million nodes; 1.5 million nodes)

  5. FAT COP • FAT - large # of processors • opportunistic environment (Condor) • COP - Master Worker control • fault tolerant: task exit, host suspend • portable parallel programming • Mixed Integer Program Solver • Branch and Bound: LP relaxations • MPS file, AMPL or GAMS input

  6. condor condor FATCOP worker FATCOP shadow (global RM) worker pvmd Master pvmd starter (local RM) startd startd startd startd startd startd startd startd startd startd INTERNET master worker Condor Pool

  7. CPLEX OSL SOPLEX MINOS ... FATCOP Internet Protocol Condor-PVM PVM MW GAMS AMPL MPS Application Problem LPSOLVER INTERFACE

  8. FATCOP Daily Log

  9. MIP Technology • Each task is a subtree, time limit • Diving heuristic • Cutting planes (global) • Pseudocosts • Preprocessing • Master checkpoint • Worker has state, how to share info?

  10. TEST on MIPLIB problems MIPLIB: 57 problems, varying size and difficulty (several not proven optimal) Solved by FATCOP sequential solver: 41 (72%) Solved by FATCOP: 46 (80%)

  11. FATCOP: sequential/parallel

  12. FATCOP on real problems

  13. FATCOP vs CPLEX

  14. Machine utilization • For one run of “NSA” problem, used 122 machines (linux, intel/solaris, sun), 26 from UNM, 11 from NCSA. • Average number of machines in use 111 • Wall clock time - 31.8 hrs • Total worker cpu time - 3014 hrs • Total worker suspension time - 40 hrs

  15. Simple Interfaces • Write the heuristic code in C/C++ • Perform heuristics at root node • Rounding/reformulation at any node • Searching heuristics at integer nodes • Compile and copy the object file to the solver directory • Turn on option “perform heuristics”

  16. Product Design Problem • Maximize market share by choosing a product profile (or other objectives) • Product has K attributes, and each attribute has L levels, N customers • Important marketing problem • NP hard, but good heuristics • How good are these solutions?

  17. MIP Solution for Product Design using FATCOP • Run FATCOP parallel solver • Incorporate following user defined heuristics: • Upper bound (NP/GA heuristics) • Priority Branching (Greedy heuristic) • Reformulation (both in root and subtrees)

  18. Product design - optimality • K=12, L=3, N=200 • Use FATCOP parallel solver • Turn on all user defined heuristics • Provable optimal solution found • The FATCOP job lasted 10.1 hours, evaluated 18,115,883 nodes and utilized 75 machines on average

  19. Product design - practicality • K=12, L=3, N=200 • Use FATCOP sequential solver • Turn on all user defined heuristics • Prove the solution found by heuristics is within 20% gap in about 90,000 nodes. • Run CPLEX MIP solver for comparison

  20. Back to Seymour • Schmieta, Pataki, Linderoth and Ferris • explored to depth 8 in tree • applied cuts at each of these 256 nodes • solved in parallel, using whatever resources available (CPLEX, FATCOP,...) • Problem solved with over 1 year CPU • over 10 million nodes, 11,000 hours

  21. Seymour Node 319 • FATCOP • 47.0 hrs with 2,887,808 nodes • average number of machine used is 108 • CPLEX • 12 days, 10 hrs with 356,600 nodes • single machine, clique cuts useful

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