Extended'Formula/ons'for'Quadra/c'...

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'


•  NLP'(QCP)'Based'B&B''   ''''''''''''''''''''''''

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•  NLP'(QCP)'Based'B&B''•  (Dynamic)'LP'Based'B&B''

–  Few'cuts'='high'speed.'–  Possible'slow'convergence.'

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•  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'


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


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]'


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


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��


Significant'Improvement'For'Many'Problems'

•  Tawarmalani'and'Sahinidis'’05:'

  ''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''

Li_ed'Relaxa/on'of''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':''


Significant'Improvement'For'Many'Problems'

•  Tawarmalani'and'Sahinidis'’05:'

•  Polynomial'(degree'n)'increment'in'#'of'constraints'through'projec/on'

Li_ed'Relaxa/on'of''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':''


Separable'Approach'for'Conic'Quadra/c?'

•  “Separable'Sets”'include'many'quadra/cs:'–  Euclidean'Ball'

–  Paraboloids'

•  Does'not'include'Lorentz/SOCP'cone:'


•  From'Euclid'to'Lorentz''

Separable'to'Conic:'Homogenize'


Li_ed'Relaxa/on'for'Separable'Conic'Sets'

Li_ed'Relaxa/on'of'''''''''''''''''''''''''''''''''''':''


Li_ed'Reformula/on'for'Conic'Quadra/c'Sets'

•  Li_ed'Reformula/on'of'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''':'


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''


Computa/onal'Experiments'1:'Solvers'

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  ''''''''''''''''''''''''''''''''   ''''''''''''''''''''''''''''''''''''''''''' Extended'Formula/ons'for'MIQCP' 10'/'25'


•  NLPSbased'BranchSandSBound:'–  CPLEXCP':'MIQCPSTRAT'='1'–  GurobiCP':'MIQCPMethod'='0'

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•  Tradi/onal'LPSbased'BranchSandSBound:'–  CPLEXLP':'MIQCPSTRAT'='2'–  GurobiLP':'MIQCPMethod'='1'

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•  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'





•  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:'


Results'for'CPLEX:'100'instances'per'n'

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Results'for'Gurobi:'100'instances'per'n'Time'[s]'

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�� 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'

  '''''''''''''''''''''''''''''''''''''''''''''''''   !!!!!!!!!!!!'''!!!!!!!!!!!!!'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

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•  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'



•  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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Computa/onal'Experiments'2':'More'Instances'


Classical:'




Shorfall:'




Robust:'


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


Summary''

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Summary''

•  Extended'Formula/ons'can'help'in'–  LPSbased'B&B:'Both'in'theory'and'prac/ce.'

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Summary''


•  'Most'ideas'can'be'extended'beyond'quadra/c'–  pSorder'cones'almost'directly'–  General'nonlinear'through'perspec/ve'func/ons'

  '''''''''''''''''''''''''''


Summary''


•  '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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