IBM CPLEX Global Non-Convex MIQP

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Transcript IBM CPLEX Global Non-Convex MIQP 

Decision Optimization
IBM CPLEX
Global Non-Convex MIQP
Christian Bliek & Pierre Bonami
Global Non-Convex MIQP
Quadratic Program (QP)
 Standard form
1
Min
x
'Qx
c
'x
2
Ax  b
x0
 Convex or Positive Semi-Definite
x' Qx  0
 Indefinite
2
any Q
© 2013 IBM Corporation
Global Non-Convex MIQP
Non-Convex QP
 Local optimum
 Available since IBM CPLEX 12.3
 Interior Point Algorithm
 Solution target Parameter FIRSTORDER
3
© 2013 IBM Corporation
Global Non-Convex MIQP
Local Non-Convex QP Benchmark
Performance Cplex versus Ipopt with Wsmp
1,6
1,4
relative time
1,2
1
time
0,8
iterations
0,6
0,4
0,2
0
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
4
© 2013 IBM Corporation
Global Non-Convex MIQP
Non-Convex MIQP
 Global optimum
 NEW in CPLEX 12.6
 Branch and Bound
5
© 2013 IBM Corporation
Global Non-Convex MIQP
Example
Min xy
 2  x 1
1  y  1
y
Local Optimum
Global Optimum
6
x
© 2013 IBM Corporation
Global Non-Convex MIQP
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 NPcomplete
7
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Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
8
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
9
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
1
Min y ' By  c' x
2
Ax  b
L' x  y
x0
Q  LBL'
10
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
11
1
Min y ' By  c' x
2
Ax  b
L' x  y
x0
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
1
Min y ' By  c' x
2
Ax  b
L' x  y
x0
1
Min z ' z  c' x
2
Ax  b
L' x  y
' y  z
x0
B   '
12
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
13
1
Min y ' By  c' x
2
Ax  b
L' x  y
x0
1
Min z ' z  c' x
2
Ax  b
L' x  y
' y  z
x0
© 2013 IBM Corporation
Global Non-Convex MIQP
Factorized Eigenvalue Formulation
1
Min x' Qx  c' x
2
Ax  b
x0
1
Min y ' By  c' x
2
Ax  b
L' x  y
x0
1
Min z ' z  c' x
2
Ax  b
L' x  y
' y  z
x0
Advantage
– Sparse
– Efficient
– Proper identification of negative eigenvalues
14
© 2013 IBM Corporation
Global Non-Convex MIQP
Example
1. Original Formulation
Min xy
 2  x 1
1  y  1
2. Factorized Eigenvalue Formulation
Q  0

1

1  1
0

2
1 2
Min u  v 2
2
15
1

1
1  1
 1

0
0  1
 1

1
1 
 1


x  y  2u
x  y  2v
 2  x 1
1  y  1
© 2013 IBM Corporation
Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
16
© 2013 IBM Corporation
Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
 Automatically select most promising one
17
© 2013 IBM Corporation
Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
 Automatically select most promising one
 Do Term by Term McCormick Relaxation
18
© 2013 IBM Corporation
Global Non-Convex MIQP
Relaxation of Non-Convex MIQP
Min
1

q
x
x

q
x
x
  ij i j  ij i j   c' x
2 P
N

Ax  b
x0
1

Min   qij xi x j   qij zij   c' x
2 P
N

Ax  b
zij  qij xi x j
x0
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© 2013 IBM Corporation
Global Non-Convex MIQP
Relaxation of Non-Convex MIQP
 Relaxation of individual Non-Convex quadratic
terms using McCormick envelopes
20
© 2013 IBM Corporation
Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
 Automatically select most promising one
 Do Term by Term McCormick Relaxation
21
© 2013 IBM Corporation
Global Non-Convex MIQP
Overview
 We consider 2 formulations
1. Original
2. Factorized Eigenvalue
 Automatically select most promising one
 Do Term by Term McCormick Relaxation
 Branch and Bound
22
© 2013 IBM Corporation
Global Non-Convex MIQP
Branching for Non-Convex MIQP
 Branch on continuous variables and update
envelopes
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Global Non-Convex MIQP
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
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© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
 internal non-convex miqp testset
 globallib GAMS
 minlp.org
 boxqp
 From miqp testset generated 50% mixed miqp set
 Comparison with SCIP and Couenne on 1 thread
25
© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
CPLEX versus SCIP on individual testsets
at most one timeout
no timeouts
0,35
1,2
0,3
1
binary
0,2
50% binary
0,15
continuous and integer
0,1
0,8
binary
0,6
50% binary
continuous and integer
0,4
0,2
0,05
0
0
[0,10k]
[1,10k]
[10,10k] [100,10k] 1k,10k]
problem time
26
relative time
relative time
0,25
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
CPLEX versus SCIP and Couenne on combined testset
at most one timeout
no timeouts
0,35
1,2
0,3
1
0,2
scip
couenne
0,15
0,1
0,8
scip
0,6
couenne
0,4
0,2
0,05
0
0
[0,10k]
[1,10k]
[10,10k]
problem time
27
relative time
relative time
0,25
[100,10k]
1k,10k]
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
CPLEX versus SCIP and Couenne on combined testset
no timeouts
at most one timeout
1,6
0,7
1,4
0,6
1,2
0,4
scip
couenne
0,3
relative nodes
relative nodes
0,5
1
scip
0,8
couenne
0,6
0,2
0,4
0,1
0,2
0
0
[0,10k]
[1,10k]
[10,10k]
problem time
28
[100,10k]
1k,10k]
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
CPLEX 1 versus 4 threads on combined testset
no timeouts
at most one timeout
1,2
0,9
0,8
1
0,6
0,5
4thread
0,4
0,3
relative time
relative time
0,7
0,8
0,6
4thread
0,4
0,2
0,2
0,1
0
0
[0,10k]
[1,10k]
[10,10k]
problem time
29
[100,10k]
1k,10k]
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
© 2013 IBM Corporation
Global Non-Convex MIQP
How to use it
 Available in CPLEX 12.6
 By default Non-Convex MIQP are not accepted
 Set Solution Target Parameter to OPTIMALGLOBAL
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© 2013 IBM Corporation
Global Non-Convex MIQP
Global Non-Convex QP Benchmark
CPLEX versus SCIP and Couenne on combined testset
at most one timeout
no timeouts
0,35
1,2
0,3
1
0,2
scip
couenne
0,15
0,1
0,8
scip
0,6
couenne
0,4
0,2
0,05
0
0
[0,10k]
[1,10k]
[10,10k]
problem time
31
relative time
relative time
0,25
[100,10k]
1k,10k]
[0,1)
[1,10)
[10,100)
[100,1k)
[1k,10k)
problem time
© 2013 IBM Corporation