FATCOP: A Mixed Integer Program Solver

FATCOP: A Mixed Integer Program Solver

Michael Ferris Qun ChenUniversity of Wisconsin-Madison

Jeffrey Linderoth Argonne National Laboratories

MIP formulationMIP formulation

minimize cTxsubject to Ax b

l x uand some xj integer

Problems are specified by application convenient format - GAMS, AMPL, or MPS

Branch-and-Bound Branch-and-Bound AlgorithmAlgorithm

0 Top node

Integer infeasible

Integer feasibleincumbent = Z

LP relaxationZlp > Z

LP infeasible

21

xf 1xf 0

3 4

xg 0 xg 1

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)

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

condor

master worker

Condor Pool

startd

startdstartd

startd

startd startd

startd

startd

startd

startd

condorINTERNET

shadow(global RM)

FATCOP

Master pvmd

FATCOP worker

worker pvmd

starter(local RM)

GAMS AMPLMPS

FATCOPFATCOP

MW

Condor-PVM

CPLEXOSL

SOPLEXMINOS

...

Application Problem

PVM

Internet Protocol

LPSO

LVER

INTER

FAC

E

FATCOP Daily Log

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?

TEST on MIPLIB TEST on MIPLIB problemsproblems

MIPLIB: 57 problems, varying size and difficulty (several not proven optimal)

Solved by FATCOP sequential solver: 41 (72%)

Solved by FATCOP: 46 (80%)

EXECUTION TIME TREE SIZESEQUENTIAL PARALLEL SPEEDUP SEQUENTIAL PARALLEL

10TEAMSAIR04AIR05DANOINTL1521AVPK1PP08ACUTSQIUROUTVPM2

3.2 8.716.3-

0.683.66.4

0.51-

0.55

0.20 0.73 0.4216.8

0.06 0.78 2.8

0.0811.7

0.18

16.011.939.1 -11.9 4.6 2.3 6.1 - 3.2

10,130 3,606 18,512 - 4,0463,564,0813,469,870 13,842 - 626,358

9,340 3,666 14,755 686,680 4,7023,047,9815,001,600 9,6874,510,6701,088,824

Time in hours (“-“ means time limit of 48 hours exceeded)

FATCOP: sequential/parallel

Name Executiontime

Tree size Avg. # ofMachines

SolutionGap

Applicationarea

MCSCHED

NSA

SP97IC

SP98AR

SP98IC

1.3 hrs

31.8 hrs

2.1 hrs

1.1 hrs

0.7 hrs

1,331,176

255,854,364

22,588

147,597

130,600

41

111

35

33

23

0%

0%

3%

3%

3%

ConferenceScheduling

NationalSecurity

Dutch Railway

Dutch Railway

Dutch Railway

FATCOP on real problems

5% 3% 1% 0%Name

FATCOP CPLEX FATCOP CPLEX FATCOP CPLEX FATCOP CPLEX

Applicationarea

MCSCHED

NSA

SP97IC

SP98AR

SP98IC

1.2

18.6

0.7

0.4

0.3

-

-

3.2

2.4

1.4

1.3

24.5

2.1

1.1

0.7

-

-

-

3.8

1.4

2.8

29.2

-

-

-

-

-

-

-

-

1.3

31.8

-

-

-

-

-

-

-

-

ConferenceSchedulingNationalSecurityDutchRailwayDutchRailwayDutchRailway

Time unit is hour.“-“ indicates reached time limit of 48 hours.

FATCOP vs CPLEX

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

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”

Product Design Product Design ProblemProblem

• 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?

MIP Solution for MIP Solution for Product Design using Product Design using

FATCOPFATCOP• Run FATCOP parallel solver

• Incorporate following user defined heuristics:– Upper bound (NP/GA heuristics)– Priority Branching (Greedy heuristic)– Reformulation (both in root and

subtrees)

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 10.110.1 hours,

evaluated 18,115,88318,115,883 nodes and utilized 7575 machines on average

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

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

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