ibm cplex global non-convex miqp
DESCRIPTION
IBM CPLEX Global Non-Convex MIQP. Christian Bliek & Pierre Bonami. Global Non-Convex MIQP. Quadratic Program (QP). Standard form Convex or Positive Semi-Definite Indefinite. Non-Convex QP. Local optimum Available since IBM CPLEX 12.3 Interior Point Algorithm - PowerPoint PPT PresentationTRANSCRIPT
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