ibm cplex global non-convex miqp

Decision OptimizationDecision OptimizationDecision OptimizationDecision Optimization

IBM CPLEXGlobal Non-Convex MIQP

Christian Bliek & Pierre Bonami

© 2013 IBM Corporation2

Standard form

Convex or Positive Semi-Definite

Indefinite

0' Qxx

0

x

bAx

xcQxx ''2

1Min

Qany

Quadratic Program (QP)Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation3

Non-Convex QP

Local optimum

Available since IBM CPLEX 12.3

Interior Point Algorithm

Solution target Parameter FIRSTORDER

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation4

Local Non-Convex QP Benchmark

Performance Cplex versus Ipopt with Wsmp

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

rela

tive

tim

e

time

iterations

© 2013 IBM Corporation5

Non-Convex MIQP

Global optimum

NEW in CPLEX 12.6

Branch and Bound

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation6

Global Non-Convex MIQPGlobal Non-Convex MIQP

1112

yx

xyMin

y

x

Local OptimumLocal Optimum

Global OptimumGlobal Optimum

Example

© 2013 IBM Corporation7

Global Non-Convex QP

Even if Q has only 1 negative eigenvalue, Non-Convex QP is NP-hard

Checking if a feasible solution is not a local minimum is NP-complete

Checking if a Non-Convex QP is unbounded is NP-complete

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation8

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation9

Factorized Eigenvalue Formulation

0

x

bAx

xcQxx ''2

1Min

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation10

Factorized Eigenvalue Formulation

'LBLQ

0

x

bAx

xcQxx ''2

1Min

0'

xyxLbAx

xcByy ''2

1Min

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation11

Factorized Eigenvalue Formulation

0

x

bAx

xcQxx ''2

1Min

0'

xyxLbAx

xcByy ''2

1Min

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation12

Factorized Eigenvalue Formulation

0

x

bAx

xcQxx ''2

1Min

0'

xyxLbAx

xcByy ''2

1Min

0''

xzyyxLbAx

xczz ''2

1Min

'B

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation13

Factorized Eigenvalue Formulation

0

x

bAx

xcQxx ''2

1Min

0'

xyxLbAx

xcByy ''2

1Min

0''

xzyyxLbAx

xczz ''2

1Min

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation14

0

x

bAx

xcQxx ''2

1Min

0'

xyxLbAx

xcByy ''2

1Min

0''

xzyyxLbAx

xczz ''2

1Min

Advantage

– Sparse

– Efficient

– Proper identification of negative eigenvalues

Factorized Eigenvalue FormulationGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation15

1. Original Formulation

2. Factorized Eigenvalue Formulation

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

1112

yx

xyMin

11

1110

0111

112

10110Q

111222

yx

vyxuyx

2

1Min 22 vu

Example

© 2013 IBM Corporation16

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation17

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Automatically select most promising one

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation18

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Automatically select most promising one

Do Term by Term McCormick Relaxation

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation19

0

x

bAx

xcxxqxxq jiN

ijjiP

ij '2

1Min

Relaxation of Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

0

xxxqz

bAxjiijij

xczqxxq ijN

ijjiP

ij '2

1Min

© 2013 IBM Corporation20

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Relaxation of individual Non-Convex quadratic terms using McCormick envelopes

Relaxation of Non-Convex MIQP

© 2013 IBM Corporation21

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Automatically select most promising one

Do Term by Term McCormick Relaxation

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation22

We consider 2 formulations

1. Original

2. Factorized Eigenvalue

Automatically select most promising one

Do Term by Term McCormick Relaxation

Branch and Bound

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Overview

© 2013 IBM Corporation23

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Branch on continuous variables and update envelopes

Branching for Non-Convex MIQP

© 2013 IBM Corporation24

Other Ingredients

QP simplex for convex QP relaxation

Pseudocost branching

Local interior point solver for incumbents

Bound strengthening

Detection of unboundedness

Linearize quadratic terms involving binaries

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

© 2013 IBM Corporation25

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

From miqp testset generated 50% mixed miqp set

Comparison with SCIP and Couenne on 1 thread

internal non-convex miqp testset

globallib GAMS

minlp.org

boxqp

© 2013 IBM Corporation26

CPLEX versus SCIP on individual testsets

at most one timeout

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

[0,10k] [1,10k] [10,10k] [100,10k] 1k,10k]

problem time

rela

tive

tim

e

binary

50% binary

continuous and integer

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

no timeouts

0

0,2

0,4

0,6

0,8

1

1,2

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

rela

tive

tim

e

binary

50% binary

continuous and integer

© 2013 IBM Corporation27

CPLEX versus SCIP and Couenne on combined testset

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

at most one timeout

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

[0,10k] [1,10k] [10,10k] [100,10k] 1k,10k]

problem time

rela

tive

tim

e

scip

couenne

no timeouts

0

0,2

0,4

0,6

0,8

1

1,2

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

rela

tive

tim

e

scip

couenne

© 2013 IBM Corporation28

CPLEX versus SCIP and Couenne on combined testset

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

no timeouts

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

relat

ive n

odes scip

couenne

at most one timeout

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

[0,10k] [1,10k] [10,10k] [100,10k] 1k,10k]

problem time

rela

tive

node

s scip

couenne

© 2013 IBM Corporation29

CPLEX 1 versus 4 threads on combined testset

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

at most one timeout

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

[0,10k] [1,10k] [10,10k] [100,10k] 1k,10k]

problem time

rela

tive

tim

e

4thread

no timeouts

0

0,2

0,4

0,6

0,8

1

1,2

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

rela

tive

tim

e

4thread

© 2013 IBM Corporation30

Available in CPLEX 12.6

By default Non-Convex MIQP are not accepted

Set Solution Target Parameter to OPTIMALGLOBAL

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

How to use it

© 2013 IBM Corporation31

CPLEX versus SCIP and Couenne on combined testset

Global Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQPGlobal Non-Convex MIQP

Global Non-Convex QP Benchmark

at most one timeout

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

[0,10k] [1,10k] [10,10k] [100,10k] 1k,10k]

problem time

rela

tive

tim

e

scip

couenne

no timeouts

0

0,2

0,4

0,6

0,8

1

1,2

[0,1) [1,10) [10,100) [100,1k) [1k,10k)

problem time

rela

tive

tim

e

scip

couenne