What's new in CPLEX 12.6?

Decision Optimization

IFORS 2014 – CPLEX 12.6

IFORS 2014, Barcelona

What’s new in CPLEX 12.6

Xavier Nodet

Software Development Manager

IBM CPLEX Optimizer

[email protected]

© 2014 IBM Corporation


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Disclaimer

IBM’s statements regarding its plans, directions, and intent are subject to change or withdrawal without notice at IBM’s sole discretion. Information regarding potential future products is intended to outline our general product direction and it should not be relied on in making a purchasing decision. The information mentioned regarding potential future products is not a commitment, promise, or legal obligation to deliver any material, code or functionality. Information about potential future products may not be incorporated into any contract. The development, release, and timing of any future features or functionality described for our products remains at our sole discretion.

Performance is based on measurements and projections using standard IBM®benchmarks in a controlled environment. The actual throughput orperformance that any user will experience will vary depending upon manyfactors, including considerations such as the amount of multiprogramming inthe user's job stream, the I/O configuration, the storage configuration, and theworkload processed. Therefore, no assurance can be given that an individualuser will achieve results similar to those stated here.




Agenda

MIP performance

Distributed MIP

Nonconvex (MI)QP



MIP performance

Continuous improvements in MIP performance with each release

Main ingredients since 12.5:– Concurrent root cut loop– Lift and Project cuts

With 12.6:– 5% on nontrivial models (more than 1 sec)– 15% on hard models (more than 100 sec)




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tota

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x217Same hardware

3147 models

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Date: 28 September 2013Testset: 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

v6.0

v6.5.3

v7.0v8.0

v9.0

v10.0

v11.0v12.1

v12.2v12.4

v12.5v12.6




Agenda

MIP performance

Distributed MIP

Nonconvex (MI)QP



One master, several workers

Distributed MIP





Racing rampup: – Each worker solves a copy of the problem using different settings

Distributed MIP





Racing rampup: – Each worker solves a copy of the problem using different settings– The master gathers primal and dual bounds from the workers and redistributes them

Distributed MIP

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Racing rampup: – Each worker solves a copy of the problem using different settings– The master gathers primal and dual bounds from the workers and redistributes them– A winner is selected

Distributed MIP

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Distributed B&B

Distributed MIP

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Distributed B&B– The master distributes subtrees coming from the winner to all workers

Distributed MIP

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Distributed B&B– The master distributes subtrees coming from the winner to all workers– And coordinates the exploration of the whole tree

Distributed MIP

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Distributed MIP

Use infinite rampup (default) to exploit performance variability– 1.5x to 2x speed-up with 4 workers on non-trivial problems

Use Distributed B&B (set CPXPARAM_DistMIP_Rampup_Duration to 1) – for very hard problems, many nodes to enumerate

Hardware doesn’t need to be the same

But workers should be as similar as possible to facilitate load balancing

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Agenda

MIP performance

Distributed MIP

Nonconvex (MI)QP



Evolution of CPLEX capabilities

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LP-Simplex

MIP-B&B




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LP-Simplex-Barrier

Convex QP

-QP Barrier

Nonconvex QP-Local: QP Barrier

MIP-B&B




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LP-Simplex-Barrier

Convex QP-QP Simplex-QP Barrier


(convex) MI(Q)P-B&B




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LP-Simplex-Barrier



Convex QCP-SOCP Barrier

(convex) MI(QC)P-B&B




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LP-Simplex-Barrier


Nonconvex QP-Local: QP Barrier-Global: Spatial B&B

Convex QCP-SOCP Barrier

(convex) MI(QC)P-B&B

Nonconvex MIQP-Global: Spatial B&B



Solving a nonconvex (MI)QP

QP Simplex doesn’t handle nonconvex problems

QP Barrier only gives locally optimal solutions

Relax the problem into a convex QP

Loop:– Solve the relaxed model to optimality using QP simplex– Do we need to branch?– Choose a variable to branch on– Create the nodes, update the relaxations

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Relaxing to convex QP

If we had only positive squares, the problem would be convex

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Relax negative squares using Secant approximations

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......min 2 iiixq





Relax negative squares using Secant approximations– Replace with new variable

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2

ix iy2

......min

ii

iii

xy

yq





Relax negative squares using Secant approximations– Replace with new variable– Coef. is negative, no need for equality

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2

ix iy2

......min

ii

iii

xy

yq





Relax negative squares using Secant approximations– Replace with new variable– Coef. is negative, no need for equality– Use secant instead of

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iiiiii

iii

ulxuly

yq

......min2

ix iy

2

ix





Relax negative squares using Secant approximations– Replace with new variable– Coef. is negative, no need for equality– Use secant instead of – Tight bounds are important!

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iiiiii

iii

ulxuly

yq

......min2

ix iy

2

ix






Use McCormick formulas for products

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2

ix iy

2

ix

ji xx






Use McCormick formulas for products

Convex quadratic parts remain

Relaxed problem depends on the bounds!

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ix iy

2

ix

ji xx



Branching

If not all integer variables have an integer value, we have to branch

Compute a solution for nonconvex QP. If its value is equal to optimal solution of convex relaxation, done.

Else there must be a pair of variables not equal to the artificial and we have to branch

Choose a variable to branch on

Choose a value to divide this variable’s domain

Update Secant / McCormick approximations

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ji xx ijy



EigenValue factorization

CPLEX may decide to transform the nonconvex (MI)QP given by the user

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0

2

1:min

x

bAx

xcQxx





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0

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1:min

x

yxL

bAx

xcByy

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1:min

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bAx

xcQxx

'LBLQ





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1:min

x

zyM

yxL

bAx

xcDzz

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1:min

x

yxL

bAx

xcByy

MDM'=B





D being diagonal, there are only squares in the objective

Maybe only few of them are negative, and trigger branching

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0

2

1:min

x

zyM

yxL

bAx

xcDzz





D being diagonal, there are only squares in the objective

Maybe only few of them are negative, and trigger branching

But…

Problem is larger

Bounds on z are larger than bounds on x

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2

1:min

x

zyM

yxL

bAx

xcDzz



Other ingredients…

Multi-threading in the tree

Branching decisions based on pseudocosts

Unboundedness of the problem may be diagnosed

Quadratic terms involving binaries are linearized

KKT system used to strengthen the bounds of the variables

…

Remember to set CPXPARAM_SolutionTarget to OPTIMALGLOBAL!

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No other solver aimed specifically at nonconvex MIQP

Comparison with SCIP 3.0.1 and Couenne 0.4 using 1 thread

390 models (internal, or publicly available), time limit 3 hours

Computational testing

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no timeouts

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couenne

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CPLEX Optimizer 12.6

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What's new in CPLEX 12.6?

Software

Transcript of What's new in CPLEX 12.6?