recent mip performance improvements in ibm ilog cplex optimization studio

© 2013 IBM Corporation

IBM Software GroupILOG CPLEX Optimization

Recent MIP Performance Improvements in

IBM ILOG CPLEX Optimization Studio

Andrea TramontaniILOG CPLEX Optimization Studio, IBM

January 22, 2014

© 2013 IBM Corporation2


Outline

Main components of CPLEX MIP Solver

– Review of Branch and Bound (B&B)

– Main components of a (fast) B&B

– Performance impact of the main components

CPLEX 12.5.1 and CPLEX 12.6 performance improvements

– Benchmark results

– Overview of the main ingredients and estimated performance impact

– Concurrent root cut loops

– Lift and Project cuts



Mixed Integer Programming and Branch and Bound (B&B)

A Mixed Integer (linear) Program (MIP) is a problem of the form

State of the art MIP solvers tackle a MIP through a Branch and Bound algorithm based on

the underlying Linear Programming (LP) relaxation:

integer allor some

)(

jx

uxl

bAxtoSubject

xczMinimizeMIP T

uxl

bAxtoSubject

xczMinimizeLP T

)(



z* = 5.2Fathomed

Review of Branch and Bound (B&B)

Root;

x1=3.5

x2=2.3

Integer

z = 7x4=0.6

x3=0.3

Integer

z = 5

Infeas

x3=0.1

z* = 4.6z* = 5.4Fathomed

B&B algorithm:

– Enumerative solution scheme

based on the LP relaxation of MIP

Bounding:

– Nodes with z* > UB can be

fathomed without further

ramification.

Key points to avoid exponential

explosion of B&B tree:

– Strong LP relaxation

– Effective branching rules

– Primal heuristics

Upper Bound:

UB = 5



Main components of a (fast) B&B: Overview

Presolve and probing

– Clean up the MIP formulation by removing redundant constraints/variables

– Strengthen the MIP formulation by tightening variable bounds, lifting constraints,

aggregating variables

– Derive cliques and implications to be used for cutting planes and/or branching

Cutting planes

– Valid inequalities for the MIP that cut off integer infeasible points of the LP relaxation

– Iteratively separated on the fly to strengthen the LP relaxation

– Cut separation must be combined with clever cut filtering and cut purging to avoid

• Numerical difficulties

• Node throughput slowdown



Main components of a (fast) B&B: Overview (Cont.d)

Branching

– Divides the feasible region in a manner that all integer feasible solutions belong to one of

the branches

– Standard B&B: up and down branch on integer variables, chosen according to several

criteria:

• Strong branching, pseudo costs, pseudo reduced costs, conflicts, inferences

Primal heuristics

– Help finding quickly high quality feasible solutions

– Constructive heuristics (e.g., rounding heuristics)

– Improving heuristics (e.g., RINS heuristics)

Root node vs. B&B tree:

– Presolve, probing, cuts and primal heuristics heavily applied at the root node, before

resorting to branching

– Also applied at subsequent B&B nodes, but much less aggressively



Main components of a (fast) B&B: Measuring performance impact

How important is each component?

Compare runs with feature turned on and off

– Solution time degradation (geometric mean)

– # of solved models

• Essential or just speedup?

– Number of affected models

• General of problem specific?

More detailed analysis in:

T. Achterberg and R. Wunderling, “Mixed Integer Programming: Analyzing 12 Years of

Progress”, in: Jünger and Reinelt (eds.) Facets of Combinatorial Optimization, Festschrift for

Martin Grötschel, pp.449-481, Springer, Berlin-Heidelberg (2013)



Component Impact CPLEX 12.5.0 – Summary

Benchmarking setup

• 1769 models

• 12 core Intel Xenon 2.66 GHz

• Unbiased: At least one of all the

test runs took at least 10sec

99% 82% 91% 26%93%91% 46%83% 65%% affected



Component Impact CPLEX 12.5.0 – Presolve

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Component Impact CPLEX 12.5.0 – Cutting Planes

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Outline



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CPLEX MIP Performance Evolution

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v12.1 v12.2

v12.4v12.5

v12.6

Date: 28 September 2013

Testset: 3147 models (1792 in 10sec, 1554 in 100sec, 1384 in 1000sec)

Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads (deterministic since CPLEX 11.0)

Timelimit: 10,000 sec



Date: 28 September 2013

Testset: 3116 models

Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads for parallel


MIP Performance Improvement – CPLEX 12.5.0 to CPLEX 12.6

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CPLEX 12.5.0

CPLEX 12.5.1

CPLEX 12.6

1968

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1210

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678

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Time limits:

96 / 49 / 35

Deterministic parallel MIP (12 threads)

1.23x 1.37x 1.61x



Cuts

– Lift-and-project cuts

– Parallel cut loops to exploit variability

– Improvements in cut separation and purging

– Improvements in MIR cut aggregator

Presolve

– Probing enhancements

– Modified how to use multiple probing/presolve rounds

Branching

– Improvements in branching rule tie breaking

– Fixed a typo in branching rule

Other improvements

– Improvements in symmetry detection and exploitation

– Incremented default work memory from 128 MB to 2 GB

(t ≥ 100 sec)

8-13% speed-up

3-6% speed-up

2-10% speed-up

3-7% speed-up

3-8% speed-up

2-3% speed-up

4-7% speed-up

2-4% speed-up

1-10% speed-up

2-3% speed-up

Performance Improvements in CPLEX 12.5.1 and 12.6 – Summary












Outline



Root Cut Loop: Impact of the Initial LP Basis

Presolve/Probing + LP solve

Branch-and-cut

LP solution x* CutSEP (x*), Heur (x*)

Cut filtering

Add cuts to LP and resolve LP

Cut purging

Progress?YES

NO

LPs may have many alternative

optimal solutions/bases

Equivalent (but different) optimal

solutions/bases of the initial LP

relaxation may lead to

– Different cutting planes

– Different heuristic solutions

Reoptimizing with different cuts

amplifies the diversification

A small change in the initial LP basis

may have a big impact on the final

root node in terms of:

– Fractional solution x*

– Dual and primal bounds

– Cuts active in the LP



M. Fischetti, A. Lodi, M. Monaci, D. Salvagnin, and A. Tramontani: ”Tree Search Stabilization

by Random Sampling”, Tech. Rep. DEI, University of Padova, 2013 (submitted for publication)

Exploit multi-threading and MIP performance variability at root node:

– Run K = 2 root cut loops in parallel, each starting from a different optimal LP basis

– Along the process, share cuts and feasible solutions among the K cut loops

Then do regular parallel branch-and-cut

Concurrent Root Cut Loops

Shake

Presolve/Probing + LP solve

Cut Loop #1 Cut Loop #2

Parellel Branch-and-cut



Concurrent Root Cut Loops: Why K = 2?

Multi-threading is not for free and the overhead increases with K

CPLEX also applies other techniques in parallel at the root node

More cut loops means more cutting planes available,

– More cuts typically means better dual bound,

– Better dual bound does not mean less time to solve

– With the current cut selection strategy

• K = 3 cut loops is roughly a 5% performance degradation w.r.t. K = 2

Implementation details:

– Create diversification in the second parallel cut loop by changing

• The basis

• The random seed

• Other parameters

– Be conservative with the cuts:

• Sharing cuts between the two cut loops helps,

• But the cuts to be shared must be filtered aggressively.



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Concurrent Root Cut Loops: Performance Impact in CPLEX 12.5.10.9

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All Models

Affected Models

≈1864

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≈ 1089

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≈ 565

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≈ 1100 (59%)

models

≈ 686 (63%)

models

≈ 372 (66%)

models

1.04x 1.06x 1.09x

1.02x 1.04x 1.05x

Date: 23 May 2013


Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads













Outline



Lift and Project Cuts: Some references

Invented by Balas in the ‟70s:

– E. Balas, Disjunctive programming. Ann. Discrete. Math. 5, 3–51, 1979.

Successfully exploited in practice in the ‟90s:

– E. Balas, S. Ceria, and G. Cornuejols, A lift-and-project cutting plane algorithm for mixed-

integer programs. Mathematical Programming 58, 295-324, 1993.

– E. Balas, S. Ceria, and G. Cornuejols: Mixed 0-1 programming by lift-and-project in a

branch-and-cut framework. Management Science 42, 1229-1246, 1996.

Strong connection with Gomory Mixed Integer (GMI) cuts:

– E. Balas and M. Perregaard, A precise correspondence between lift-and-project cuts,

simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming.

Mathematical Programming 94, 221-245, 2003.

Effective approach to optimize over the Lift and Project closure:

– P. Bonami, On optimizing over lift-and-project closures. Math. Prog. Comp. 4, 151-179,

2012.



Given:

The LP relaxation of a MIP, defined by the underlying polyhedron P:

A fractional solution x* in P (typically, a vertex of P) to cut off

A binary fractional variable xj*: xj in {0,1}, 0 < xj* < 1

Find a violated cut

α x ≥ β

using the simple split disjunction

xj ≤ 0 OR xj ≥ 1

Lift and Project Cuts: The Separation Problem

(P) A x ≥ b

x ≥ 0



The Cut Generating LP (CGLP) of Balas, Ceria and Cornuejols

1. Consider the polyhedra P0 and P1 induced by the disjunction:

2. Separate a cut valid for the disjunctive polyhedron

Q = conv (P0 U P1)

by solving the following CGLP:

3. Strengthen the cut by Monoidal Strengthening (Balas and Jeroslow, Eur. J. Oper. Res. 1980)

P0

(u) A x ≥ b

(w) x ≥ 0

(u0) - xj ≥ 0

P1

(v) A x ≥ b

(z) x ≥ 0

(v0) xj ≥ 1

α = u A + w – u0 ej

β = u b

u, w, u0 ≥ 0

α = v A + z + v0 ej

β = v b + v0

v, z, v0 ≥ 0

min α x* - β

P0

P1

x*

αx ≥ β



The CGLP is unbounded by construction (if a violated cut exists)

Can be normalized in many different ways: see, e.g.,

– Balas, Ceria and Cornuejols (Math. Prog.1993, Man. Science 1996)

– Ceria and Soares (Tech. Report, 1997)

The normalization is the real objective function and makes the difference: see, e.g.,

– Fischetti, Lodi and T. (Math. Prog. 2011)

Two popular normalization constraints are:

– The weak normalization (actually, the one used in CPLEX):

u0 + v0 = 1 (1)

– The strong normalization (Balas, Math. Prog. 1997):

1 u + 1 v + 1 w + 1 z + u0 + v0 = 1 (2)

CGLP: The Role of Normalization



Balas and Perregaard (Math. Prog. 2003) adopt normalization (2) and:

1. Separate high-rank inequalities (x* is a basic solution of P)

2. Provide a precise correspondence between feasible CGLP bases and (possibly

infeasible) bases of the tableau of the original problem

3. Characterize Gomory Mixed Integer (GMI) cuts from the tableau as CGLP bases

4. Develop an efficient approach for solving the CGLP by a sequence of pivoting in the

tableau of the original space.

Different variants of the idea available in commercial and open-source solvers:

FICO XPRESS,

CBC (COIN-OR B&C), see Balas and Bonami (Math. Prog. Comp. 2009)

Lift and Project Cuts à la Balas-Perregaard (BP L&P cuts)

Warmstart CGLP

with GMI basis

Pivots in the space

A x – I s = b

Optimal CGLP

solution



Bonami (Math. Prog. Comp., 2012):

1. Separate only rank-1 L&P cuts (x* is not a vertex of P)

2. Adopts normalization u0 + v0 = 1

3. Applies some algebra to get rid of u0 and v0 in the CGLP

4. Takes the dual of the simplified CGLP and gets the so called Membership LP (MLP):

5. Provides promising computational results on MIPLIB 3.0 and MIPLIB 2003 models

The (strengthened) L&P closure:

– Can be computed in practice

– Provides a good approximation of the split closure

Lift and Project Cuts à la Bonami (this is what we did!)

(MLP) max yj - δ*

(v) b δ* ≤ A y ≤ b δ* + s* (u)

(z) 0 ≤ y ≤ x* (w)

s* := A x* - b

δ* := xj*



L&P cuts à la Balas-Perregaard:

– High-rank inequalities

– Strong normalization (2)

– L&P cuts as improved tableau GMI cuts

– Computationally more expensive

L&P cuts à la Bonami:

– Rank-1 inequalities

– Weak normalization (1)

– Computationally cheaper

L&P cuts à la Balas-Perregaard vs. L&P cuts à la Bonami



Computational investigation in Fischetti, Lodi and T. (Math. Prog. 2011):

Normalization (2) is fundamental when high-rank inequalities are separated, because it:

– Breaks the correlation between optimization and separation

– Tends to produce low-rank inequalities

– Enforces the separation of cuts with sparse dual and primal support

Separating rank-1 cuts:

– Breaks the correlation between optimization and separation

– Enforces the separation of low-rank inequalities (not a big surprise)

– Tends to produce cleaner and sparser cuts

Lift and Project Cuts: Our Idea

Sounds like

normalization (2)

Separate some more rank-1 L&P cuts with u0 + v0 = 1

Keep tableau GMI cuts as they are (if your filter likes them enough), and



Bonami MLP to separate rank-1 cuts only, plus some tricks:

Heuristic reduction of MLP to

– Speed up the separation

– Get rid of constraints you don„t want to use to produce the cut

Clever selection of the disjunctions to use

Lift and Project Cuts in CPLEX 12.5.1



L&P Cuts in CPLEX 12.5.1: Performance Impact

Date: 23 May 2013


Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads


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All Models

Affected Models

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1.10x1.06x 1.16x



Questions and Answers (Q&A)