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Extended'Formula/ons'for'Quadra/c'Mixed'Integer'Programming'
Juan'Pablo'Vielma''
Sloan'School'of'Business,''Massachuse@s'Ins/tute'of'Technology'''IP'for'Lunch,''IBM'T.'J.'Watson'Research'Center,'Yorktown'Heights,'NY.'December,'2014.''Supported'by'NSF'grants'CMMIS1233441'and'CMMIS1351619'
Nonlinear'MIP'B&B'Algorithms'
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Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B'' ''''''''''''''''''''''''
''''''''''''''''''''''' '''''''''''''''''''''''''''
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Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B'' ''''''''''''''''''''''''
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Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B'' ''''''''''''''''''''''''
''''''''''''''''''''''' '''''''''''''''''''''''''''
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Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B'' ''''''''''''''''''''''''
''''''''''''''''''''''' '''''''''''''''''''''''''''
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'''''''''''''''' '''''''''''''''''''
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Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
''''''''''''''' ''''''''''''''''''''''''''''''' ''''''''''''''''''
'''''''''''''''' '''''''''''''''''''
''''''''''''''''
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
• Li_ed'LP'B&B''– Extended'or'Li_ed'relaxa/on.'– Sta/c'relaxa/on'
• Mimic'NLP'B&B.''– Dynamic'relaxa/on'
• Standard'LP'B&B'
Extended'Formula/ons'for'MIQCP' 1'/'25'
Nonlinear'MIP'B&B'Algorithms'
• NLP'(QCP)'Based'B&B''• (Dynamic)'LP'Based'B&B''
– Few'cuts'='high'speed.'– Possible'slow'convergence.'
• Li_ed'LP'B&B''– Extended'or'Li_ed'relaxa/on.'– Sta/c'relaxa/on'
• Mimic'NLP'B&B.''– Dynamic'relaxa/on'
• Standard'LP'B&B'
Extended'Formula/ons'for'MIQCP' 1'/'25'
Sta/c'Li_ed'LP'for'Conic'Quadra/c'MIP'
• Approxima/on'of'Second'Order'Cone'of'dimension'n by'BenSTal'and'Nemirovski'(Glineur).'
• '''''''''''''''''''''''''''variables'and'constraints'for'quality''''.'– Exponen/al'increment'in'#'of'constraints'through'projec/on.'
• Problem:'– Fixed'aSpriori'quality:'no'dynamic'improvement'(e.g.'''''''''''''''''''best'for''some'porfolio'op/miza/on'problems)'
r
Extended'Formula/ons'for'MIQCP' 2'/'25'
Correct'Quality'='Significant'Improvement'
• Results'from'V.,'Ahmed'and''Nemhauser'’08.'Introduction Lifted LP Algorithm Computational Results Final Remarks
Avg. of Solve Times [s] for n 2 {20, 30} (CPLEX v11)
0.1
1
10
100
LP(!)-BB CPLEX QCP CPLEX LP 0.1
1
10
100
LP(!)-BB CPLEX QCP CPLEX LP
Average'Solve'Times'[s]'
Extended'Formula/ons'for'MIQCP' 3'/'25'
Dynamic'Li_ed'LP'for'Separable'Problems''
• Mo/va/ng'example'from'Hijazi'et'al.'‘14'
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Extended'Formula/ons'for'MIQCP' 4'/'25'
Dynamic'Li_ed'LP'for'Separable'Problems''
• Mo/va/ng'example'from'Hijazi'et'al.'‘14'
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Extended'Formula/ons'for'MIQCP' 4'/'25'
Dynamic'Li_ed'LP'for'Separable'Problems''
• Mo/va/ng'example'from'Hijazi'et'al.'‘14'
0 0.5 1��0
0.25
0.5��
Extended'Formula/ons'for'MIQCP' 4'/'25'
Dynamic'Li_ed'LP'for'Separable'Problems''
• Mo/va/ng'example'from'Hijazi'et'al.'‘14'
0 0.5 1��0
0.25
0.5��
Extended'Formula/ons'for'MIQCP' 4'/'25'
Significant'Improvement'For'Many'Problems'
• Tawarmalani'and'Sahinidis'’05:'
''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''
Li_ed'Relaxa/on'of''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':''
Extended'Formula/ons'for'MIQCP' 5'/'25'
Significant'Improvement'For'Many'Problems'
• Tawarmalani'and'Sahinidis'’05:'
''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''
Li_ed'Relaxa/on'of''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':''
Extended'Formula/ons'for'MIQCP' 5'/'25'
Significant'Improvement'For'Many'Problems'
• Tawarmalani'and'Sahinidis'’05:'
• Polynomial'(degree'n)'increment'in'#'of'constraints'through'projec/on'
Li_ed'Relaxa/on'of''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':''
Extended'Formula/ons'for'MIQCP' 5'/'25'
Separable'Approach'for'Conic'Quadra/c?'
• “Separable'Sets”'include'many'quadra/cs:'– Euclidean'Ball'
– Paraboloids'
• Does'not'include'Lorentz/SOCP'cone:'
Extended'Formula/ons'for'MIQCP' 6'/'25'
• From'Euclid'to'Lorentz''
Separable'to'Conic:'Homogenize'
Extended'Formula/ons'for'MIQCP' 7'/'25'
• From'Euclid'to'Lorentz''
Separable'to'Conic:'Homogenize'
Extended'Formula/ons'for'MIQCP' 7'/'25'
• From'Euclid'to'Lorentz''
Separable'to'Conic:'Homogenize'
Extended'Formula/ons'for'MIQCP' 7'/'25'
Li_ed'Relaxa/on'for'Separable'Conic'Sets'
Li_ed'Relaxa/on'of'''''''''''''''''''''''''''''''''''':''
Extended'Formula/ons'for'MIQCP' 8'/'25'
Li_ed'Reformula/on'for'Conic'Quadra/c'Sets'
• Li_ed'Reformula/on'of'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':'
Extended'Formula/ons'for'MIQCP' 9'/'25'
Li_ed'Reformula/on'for'Conic'Quadra/c'Sets'
• Li_ed'Reformula/on'of'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':'
Extended'Formula/ons'for'MIQCP' 9'/'25'
Li_ed'Reformula/on'for'Conic'Quadra/c'Sets'
• Li_ed'Reformula/on'of'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':'
• Li_ed'polyhedral'relaxa/on'automa/c'from'standard'polyhedral'approxima/on'of'(rotated)'SOCP'cone:'– 'Dynamic'Li_ed'LPSbased'algorithm:'
1. Replace'every'SOCP'cone'with'li_ed'reformula/on'2. Solve'with'standard'LPSbased'algorithm''
Extended'Formula/ons'for'MIQCP' 9'/'25'
Computa/onal'Experiments'1:'Solvers'
'''''''''''''''''''''''''''' !!!!!!!'''''''''''''''''' !!!!!!!!'''''''''''''''''''
''''''''''''''''''''''''''''''''''''''' !!!!!!!'''''''''''''''''' !!!!!!!!'''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''' !!!!!!!!!!'''''''''''''''''''''''''''''''''''' !!!!!!!!!!!''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Solvers'
• NLPSbased'BranchSandSBound:'– CPLEXCP':'MIQCPSTRAT'='1'– GurobiCP':'MIQCPMethod'='0'
''''''''''''''''''''''''''''''''''''''' !!!!!!!'''''''''''''''''' !!!!!!!!'''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''' !!!!!!!!!!'''''''''''''''''''''''''''''''''''' !!!!!!!!!!!''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Solvers'
• NLPSbased'BranchSandSBound:'– CPLEXCP':'MIQCPSTRAT'='1'– GurobiCP':'MIQCPMethod'='0'
• Tradi/onal'LPSbased'BranchSandSBound:'– CPLEXLP':'MIQCPSTRAT'='2'– GurobiLP':'MIQCPMethod'='1'
'''''''''''''''''''''''''''''''''''''''''' !!!!!!!!!!'''''''''''''''''''''''''''''''''''' !!!!!!!!!!!''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Solvers'
• NLPSbased'BranchSandSBound:'– CPLEXCP':'MIQCPSTRAT'='1'– GurobiCP':'MIQCPMethod'='0'
• Tradi/onal'LPSbased'BranchSandSBound:'– CPLEXLP':'MIQCPSTRAT'='2'– GurobiLP':'MIQCPMethod'='1'
• Dynamic'Li_ed'LPSbased'BranchSandSBound:'– CPLEXSepLP':'CPLEXLP'on''li_ed'reformula/on'– GurobiSepLP:'GurobiLP'on''li_ed'reformula/on'
'''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Solvers'
• NLPSbased'BranchSandSBound:'– CPLEXCP':'MIQCPSTRAT'='1'– GurobiCP':'MIQCPMethod'='0'
• Tradi/onal'LPSbased'BranchSandSBound:'– CPLEXLP':'MIQCPSTRAT'='2'– GurobiLP':'MIQCPMethod'='1'
• Dynamic'Li_ed'LPSbased'BranchSandSBound:'– CPLEXSepLP':'CPLEXLP'on''li_ed'reformula/on'– GurobiSepLP:'GurobiLP'on''li_ed'reformula/on'
• CPLEX'v'12.6'and'Gurobi'v'5.6.3' ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Solvers'
• NLPSbased'BranchSandSBound:'– CPLEXCP':'MIQCPSTRAT'='1'– GurobiCP':'MIQCPMethod'='0'
• Tradi/onal'LPSbased'BranchSandSBound:'– CPLEXLP':'MIQCPSTRAT'='2'– GurobiLP':'MIQCPMethod'='1'
• Dynamic'Li_ed'LPSbased'BranchSandSBound:'– CPLEXSepLP':'CPLEXLP'on''li_ed'reformula/on'– GurobiSepLP:'GurobiLP'on''li_ed'reformula/on'
• CPLEX'v'12.6'and'Gurobi'v'5.6.3'• Time'limit'of'3,600's'on'i7S3770'3.40GHz''' Extended'Formula/ons'for'MIQCP' 10'/'25'
Computa/onal'Experiments'1:'Instances'
• Porfolio'op/miza/on'problems:'
Classical:'
Extended'Formula/ons'for'MIQCP' 11'/'25'
Results'for'CPLEX:'100'instances'per'n'
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1000
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Time'[s]'
n'
Extended'Formula/ons'for'MIQCP' 12'/'25'
Results'for'Gurobi:'100'instances'per'n'Time'[s]'
n'
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1
10
100
1000
�������� ������� ����������Extended'Formula/ons'for'MIQCP' 13'/'25'
Computa/onal'Experiments'2:'More'Solvers'
• Sta/c'Li_ed'LPSbased'BranchSandSBound:'– Li0edLP:'from'V.,'Ahmed'and''Nemhauser'’08'– Fixed'approxima/on'by'BenSTal'and'Nemirovski'(Glineur)'– No'refinement:'integer'nodes'='solve'NLP'and'process'– Heuris/c:'Correct'integral'solu/ons'(fix'integers,'solve'NLP)''– CPLEX'Branch,'heuris/c'and'incumbent'callbacks'in'JuMP'
''''''''''''''''''''''''''''''''''''''''''''''''' !!!!!!!!!!!!'''!!!!!!!!!!!!!'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''
Extended'Formula/ons'for'MIQCP' 14'/'25'
Computa/onal'Experiments'2:'More'Solvers'
• Sta/c'Li_ed'LPSbased'BranchSandSBound:'– Li0edLP:'from'V.,'Ahmed'and''Nemhauser'’08'– Fixed'approxima/on'by'BenSTal'and'Nemirovski'(Glineur)'– No'refinement:'integer'nodes'='solve'NLP'and'process'– Heuris/c:'Correct'integral'solu/ons'(fix'integers,'solve'NLP)''– CPLEX'Branch,'heuris/c'and'incumbent'callbacks'in'JuMP'
• Sta/c/Dynamic'Li_ed'LPSbased'BranchSandSBound:'– CPLEXSepLazy'/'GurobiSepLazy':'Li_edLP'+'Refinement'through'Separable'extended'formula/on'
– Solver'independent'implementa/on'in'JuMP'
'''''''''''''''''''''''''''''''''''''''''Extended'Formula/ons'for'MIQCP' 14'/'25'
Computa/onal'Experiments'2:'More'Solvers'
• Sta/c'Li_ed'LPSbased'BranchSandSBound:'– Li0edLP:'from'V.,'Ahmed'and''Nemhauser'’08'– Fixed'approxima/on'by'BenSTal'and'Nemirovski'(Glineur)'– No'refinement:'integer'nodes'='solve'NLP'and'process'– Heuris/c:'Correct'integral'solu/ons'(fix'integers,'solve'NLP)''– CPLEX'Branch,'heuris/c'and'incumbent'callbacks'in'JuMP'
• Sta/c/Dynamic'Li_ed'LPSbased'BranchSandSBound:'– CPLEXSepLazy'/'GurobiSepLazy':'Li_edLP'+'Refinement'through'Separable'extended'formula/on'
– Solver'independent'implementa/on'in'JuMP'
• ~'400'lines'of'JuMP'code'in'about'a'week'Extended'Formula/ons'for'MIQCP' 14'/'25'
JuMP':'Julia'for'Mathema/cal'Programming'
• Developed'by'ORC'students:'Iain'Dunning,'Joey'Huche@e'and'Miles'Lubin.'
• “As'easy'as'AMPL'and'faster'than'C++”'(JPV'2014).'• Linear/Quadra/c'MIP'and'general'nonlinear'
– 'Cbc/Clp,'CPLEX,'ECOS,'GLPK,'Gurobi,'Ipopt,'KNITRO,'MOSEK,'and'Nlopt.'
• Automa/c'differen/a/on,'solver'independent'MIP'callbacks,'etc.'
• h@ps://github.com/JuliaOpt/JuMP.jl'
11/29/14, 7:36 PM
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Extended'Formula/ons'for'MIQCP' 15'/'25'
Computa/onal'Experiments'2':'More'Instances'
• Porfolio'op/miza/on'problems:'
Classical:'
Extended'Formula/ons'for'MIQCP' 16'/'25'
Computa/onal'Experiments'2':'More'Instances'
• Porfolio'op/miza/on'problems:'
Shorfall:'
Extended'Formula/ons'for'MIQCP' 16'/'25'
Computa/onal'Experiments'2':'More'Instances'
• Porfolio'op/miza/on'problems:'
Robust:'
Extended'Formula/ons'for'MIQCP' 16'/'25'
Performance'Profile'for'n'“=“'20S60,'100'and'200''
100 101 102 103 104 1050
0.2
0.4
0.6
0.8
1
Performance Ratio
Problem
ssolved
CPLEXCP
CPLEXLP
CPLEXSepLP
CPLEXSepLazy
GurobiCP
GurobiLP
GurobiSepLP
GurobiSepLazy
LiftedLP
Extended'Formula/ons'for'MIQCP' 17'/'25'
Performance'Profile'for'n'“=“'20S60,'100'and'200''
100 101 102 103 104 1050
0.2
0.4
0.6
0.8
1
Performance Ratio
Problem
ssolved
CPLEXCP
CPLEXLP
CPLEXSepLP
CPLEXSepLazy
GurobiCP
GurobiLP
GurobiSepLP
GurobiSepLazy
LiftedLP
Extended'Formula/ons'for'MIQCP' 17'/'25'
Performance'Profile'for'n'“=“'20S60,'100'and'200''
100 101 102 103 104 1050
0.2
0.4
0.6
0.8
1
Performance Ratio
Problem
ssolved
CPLEXCP
CPLEXLP
CPLEXSepLP
CPLEXSepLazy
GurobiCP
GurobiLP
GurobiSepLP
GurobiSepLazy
LiftedLP
Extended'Formula/ons'for'MIQCP' 17'/'25'
Performance'Profile'for'n'“=“'20S60,'100'and'200''
100 101 102 103 104 1050
0.2
0.4
0.6
0.8
1
Performance Ratio
Problem
ssolved
CPLEXCP
CPLEXLP
CPLEXSepLP
CPLEXSepLazy
GurobiCP
GurobiLP
GurobiSepLP
GurobiSepLazy
LiftedLP
Extended'Formula/ons'for'MIQCP' 17'/'25'
Performance'Profile'for'n'“=“'20S60,'100'and'200''
100 101 102 103 104 1050
0.2
0.4
0.6
0.8
1
Performance Ratio
Problem
ssolved
CPLEXCP
CPLEXLP
CPLEXSepLP
CPLEXSepLazy
GurobiCP
GurobiLP
GurobiSepLP
GurobiSepLazy
LiftedLP
Extended'Formula/ons'for'MIQCP' 17'/'25'
Summary''
'''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''
''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''
Extended'Formula/ons'for'MIQCP' 18'/'25'
Summary''
• Extended'Formula/ons'can'help'in'– LPSbased'B&B:'Both'in'theory'and'prac/ce.'
''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''
'''''''''''''''''''''''''''
Extended'Formula/ons'for'MIQCP' 18'/'25'
Summary''
• Extended'Formula/ons'can'help'in'– LPSbased'B&B:'Both'in'theory'and'prac/ce.'
• 'Most'ideas'can'be'extended'beyond'quadra/c'– pSorder'cones'almost'directly'– General'nonlinear'through'perspec/ve'func/ons'
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Extended'Formula/ons'for'MIQCP' 18'/'25'
Summary''
• Extended'Formula/ons'can'help'in'– LPSbased'B&B:'Both'in'theory'and'prac/ce.'
• 'Most'ideas'can'be'extended'beyond'quadra/c'– pSorder'cones'almost'directly'– General'nonlinear'through'perspec/ve'func/ons'
• You'should'definitely'try:'
11/29/14, 7:36 PM
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Extended'Formula/ons'for'MIQCP' 18'/'25'
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