Automatic Fine-tuning Xpress-MP to Solve MIP

1. © 2008 Fair Isaac Corporation.
Automatic Fine-tuning Xpress-MP to Solve MIP
by Gabriel Tavares
Horia Tipi
Alkis Vazacopoulos
INFORMS Annual Meeting
Washington, DC, October, 2008

2. 2 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Agenda
 Heuristics in Xpress-MP
 Xpress-Tuner
New Features
Process
Creating Specialized Tuning Strategies
 Conclusions

3. 3 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Classes of Heuristics in Xpress-MP
 Branch-&-Bound Heuristics (Integral Nodes)
 Feasibility Pump Heuristics
 Diving Heuristics
 Local Search Heuristics
Node BestSoln BestBound Sols Active Depth Gap GInf Time
11500 56138.00000 55821.97518 11 29 48 0.56% 0 29
+ 11516 56138.00000 55828.26120 12 12 5 0.55% 0 29
+ 11516 56138.00000 55828.26120 13 12 5 0.55% 0 29
+ 11516 56137.00000 55828.26120 14 12 5 0.55% 0 29
* 11572 56137.00000 55852.82171 15 33 29 0.51% 0 40
11600 56137.00000 55932.33767 15 32 8 0.36% 253 43
* 11636 56137.00000 56137.00000 16 20 29 7.3e-12 0 45
The * Xpress Heuristics
The + Xpress Heuristics

4. 4 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
CORAL Benchmark for MIP
 Publicly available from
http://coral.ie.lehigh.edu/mip-instances/
 Consists of 372 MIPs from various sources (mostly from NEOS)
4 instances are infeasible
 On an average problem, Xpress 2008A out-of-the-box found:
4.1 (+) heuristics at the root node
2.5 (+) heuristics during B&B
1.6 (*) heuristics

5. 5 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress 2008A Serial vs 2008A 2-Threads
on CORAL Set
 The Xpress parallel search is now deterministic (by default)
2008A
2 Threads
Speedup
Total Time to Find the Optimal
Solutions found by Both Solvers
7 h 57 m 5 h 9 m 1.54x
Within 30 Minutes:
2008A
Serial
2008A
2 Threads
Number of Optimal Solutions 218 220
Deterministic
Algorithm

6. 6 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Number of Improved Solutions Xpress
2008A Serial vs 2008A 2-Threads
Number of Improved
Solutions by
2008A
2008A
2 Threads
(+) heuristics at root node 4.1 4.1
(+) heuristics during B&B 2.5 2.7
(*) heuristics 1.6 1.4

7. 7 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Using another Local Search (LS)
Heuristic Families on CORAL Set
 Set controls
HEURSEARCHROOTSELECT=5
HEURSEARCHTREESELECT=5
HEURSEARCHEFFORT=1.0
 Xpress 2 Threads + LS is better in about 1/3 of the cases
Number of Improved
Solutions by
2008A
2 Threads
2008A 2
Threads + LS
(+) heuristics at root node 4.1 6.5
(+) heuristics during B&B 2.7 2.2
(*) heuristics 1.4 0.9

8. 8 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Fine-Tuning Threads in Xpress-MP
 Each Individual Thread of the
Parallel Search can be Specialized:
To improve the Best Objective by
Applying local search Heuristics
Applying diving Heuristics
Using Depth-First-Search etc. OR
To improve the Best Bound by
Using Best-First-Search (BFS)
Increasing the effort on strong branching
Applying in-Tree Cuts, etc.

9. 9 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Multiple Worker Paradigm
Many Workers
Same Tasks
Few Specialized
Workers

10. 10 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Using a Specialized Local Search
(LS) Heuristic Thread on CORAL Set
 Set control MIPTHREADSELECT=2
Activates other Local Search Heuristics to be run on 1 Thread
 1 Thread + 1 LS Thread is better in 38% of the cases
Number of Improved
Solutions by
2008A
2 Threads
2008A 1 Thread
+ 1 LS Thread
(+) heuristics at root node 4.1 4.1
(+) heuristics during B&B 2.7 4.3
(*) heuristics 1.4 0.9

11. 11 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
The MIPLIB 2003 Experience
Problem
Old Best Known Obj.
Value (*)
Xpress Improved Obj.
Value (**)
GAIN
(|1-(**)/(*)|)
atlanta-ip 95.009549704 90.00987861 5.3%
msc98-ip 20980991.006 19839497.006 5.4%
protfold -30 -31 3.3%
rd-rplusc-21 171182 165395.2753 3.4%
sp97ar 664565103.76 660705646.5 0.6%
stp3d unknown 500.736 N/A
ds 283.4425 116.59 58.9%
momentum3 370177.036 236426.335 36.1%
t1717 193221 170195 11.9%
liu 1172 1102 5.9%
dano3mip 691.2 687.733333 0.5%
OptimalUnsolved
Solving Hard Mixed Integer Programming Problems with Xpress-MP:
A MIPLIB 2003 Case Study, Informs Journal on Computing, to appear
by Richard Laundy, Michael Perregaard, Gabriel Tavares, Horia Tipi, and Alkis Vazacopoulos

12. 12 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Node BestSoln BestBound Sols Active Depth Gap GInf Time
100000 1098.000000 560.000000 1 64916 79 49.00% 207 15576
100500 1098.000000 560.000000 1 65228 70 49.00% 229 15659
101000 1098.000000 560.000000 1 65581 56 49.00% 267 15734
101500 1098.000000 560.000000 1 65918 102 49.00% 147 15808
+ 101653 1090.000000 560.000000 2 66033 23 48.62% 0 15834
liu
 Is a MIPLIB 2003 model that involves the floor plan and placement
problem in the physical design of VLSI circuits
The best known solution is 1102
 Consider the algorithm:
1 standard Thread
1 specialized Thread (Breadth-First Search and heavy 2 LS Heuristics)
NODESELECTION=2
HEURSEARCHROOTSELECT=5
HEURSEARCHTREESELECT=5
HEURSEARCHFREQ=1
HEURSEARCHEFFORT=10000
liu’s best solution is now 1090

13. 13 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Improved Branching & Heuristics in
2008A
Computing Time to the N-th Solution
(Geometric Mean )
0 sec
5 sec
10 sec
15 sec
20 sec
25 sec
30 sec
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
Solution Number
2008A
2007B
2008A 2 threads

14. 14 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Improving Out-of-the-Box
Performance using the Xpress-Tuner
 Xpress-Tuner allows the user to automatically find a set of control
parameters and their values that can improve the solve time of
the default settings for a single or a group of problems
 Xpress-Tuner 2008A New Features:
Tuning Process is Parallelized
Allows to Warm Start a Problem with a Known Solution
Improved Tuning Methods
Cloning of Tuning Methods
 Xpress-Tuner usually finds settings that can improve the
computing times by a factor between 2x and 10x

15. 15 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress-Tuner: How to Tune
(Automatically) an Optimization Problem?

16. 16 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress-Tuner in Action
 Mittelmann Benchmark for MIP
 Advertising Up-Front Market Sales Plan Generation
 Planogram Optimization
 Set Covering

17. 17 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Mittelmann results
 Considered subset of 9 MIPs from Mittelmann for which the best
reported solve time was larger than 2 minutes (as of 6/4/08)
 Computer used in Mittelmann tests
2.667 GHz Intel Core 2 (4GB, Linux, 64 bit)
 Computer used in Xpress tests
2 GHz Intel Core 2 (2GB, Windows, 32 bit)

18. 18 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Mittelmann Tuning Results
MIP
Fastest Time in
Mittelmann’s
Website
Xpress-MP
Defaults
Xpress-MP
Tuned
Speedup
to Best
neos5 188 s 4142 s 1218 s 0.2x
seymour1 165 s 676 s 193 s 0.8x
dano3_5 214 s 212 s 161 s 1.3x
neos9 180 s > 2h 91 s 2.0x
bienst2 161 s 330 s 75 s 2.1x
neos818918 3069 s 5944 s 1105 s 2.8x
neos823206 552 s 5984 s 123 s 4.5x
neos11 153 s 40 s 15 s 10x
neos808444 481 s 125 s 10 s 48x

19. 19 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Advertising Up-Front Market Sales
Plan Generation
 Three harder MIPs obtained from a TV Network
 Goal was to get a 0.01% solution gap within 1 minute, for each
problem
 Computer used
2 GHz Intel Core 2 (2GB, Windows, 32 bit)

20. 20 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
A solution for a Single Advertising
Plan
start
date
6351 6352 6354 6355 6356 10084
1/1/2007 10 / 966 10 / 572 10 / 310
1/8/2007 11 / 454 10 / 1006 10 / 510 10 / 308
1/15/2007 8 / 462
1/22/2007 10 / 770 10 / 308
4/30/2007 10 / 308
6/11/2007 10 / 770 5 / 308
6/18/2007 2 / 770 10 / 308
6/25/2007 3 / 1100 7 / 440 7 / 770 10 / 462 10 / 308 1 / 770
7/2/2007 10 / 770 10 / 462 10 / 308
7/9/2007 10 / 770 7 / 462 10 / 308

21. 21 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Advertising Plan Generation:
Out-of-the-Box vs Tuned
MIP
Xpress Default
Settings
Xpress-Tuned
Plan A no solution 60 s
(0.3% gap)
Plan B no solution 60 s
(0.7% gap)
Plan C 60 s
(1.4%)
60 s
(0.4%)

22. 22 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress-MP for Planogram
Optimization
 Five “hard” MIPs obtained from a Large Retailer
 Goal was to get a 1.5% solution gap within 5 minutes, for each
problem
 Computer used
2 GHz Intel Core 2 (2GB, Windows, 32 bit)

23. 23 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.

24. 24 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress-MP for Planogram Optimization:
Out-of-the-Box vs Tuned
MIP
Xpress Default
Settings
Xpress-Tuned
P2871 300 s
(2.0% gap)
187 s
(1.3% gap)
P2879 300 s
(1.7% gap)
35 s
(1.4% gap)
P2750 31 s
(1.3% gap)
31 s
(1.3% gap)
P2757 33 s
(1.1% gap)
10 s
(0.7% gap)
P2864 300 s
(8.8% gap)
300 s
(3.5% gap)

25. 25 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
A Greedy Tuning Strategy
 Candidate Algorithms:
LP
Branching
Cutting
Diving Heuristics
Local Search Heuristics
Fixing strategy T
Tune Algorithm As
Are Results
Satisfactory?
s ← s + 1 T ← strategy(As)
A Greedy Tuning Strategy
s ← 1
Define T
Strategy
T
INPUT OUTPUT

26. 26 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Xpress-Tuner in Action
 Show New Features
 Illustrates how to use a Greedy Tuning Strategy
Show How to tune the LS heuristics

27. 27 © 2007 Fair Isaac Corporation.© 2008 Fair Isaac Corporation.
Conclusions
 Xpress-MP
Is a top-notch solver for MIP heuristics
Many options available
 Xpress-Tuner
It automatic tunes Xpress, resulting in solve time gains
of 2x to 10x on a typical case
Thank You