Multi-level approach in multi-objective optimisationutopiae.eu/wp-content/uploads/2017/12/20171121_4...§ Actually we have transferred it to single-objective problem §Lexicographic

Multi-level approach in multi-objective optimisationPeter Korošec

utopiae_network http://utopiae.eu

OVERVIEW§ Overviewofapproachestoreal-worldproblemsinoptimisation

§ Simplifyingapproaches§ Generalmulti-levelapproach

§ Searchspace§ Evolution

§ Quicklookatsurrogatemodelling

Multi-levelapproachinmulti-objectiveoptimisation

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PROBLEMS IN REAL-WORLD OPTIMISATION§ Real-worldoptimisation problemsareoftencomplexandtime-consumingtoevaluate

§ Optimisationalgorithms needtofindhighqualitysolutions“quickly”

§ Somepossibleapproaches§ Usageofspeciallydesignedoptimisationalgorithms

§ Providequickconvergence§ Simplifyingmulti-objectiveoptimisation§ Usageofmulti-levelapproach§ Usageofsurrogatemodelling


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APPROACHES TO “SIMPLIFYING”MO§ Weightedsummethod

§ 𝑓 𝒙 = ∑ 𝑤&𝑓&(𝒙))&*+

§ Actuallywehavetransferredittosingle-objectiveproblem

§ Lexicographicmethod§ Letusassumethati definesthelevelofobjectiveimportance§ Goalmin

𝒙∈𝑿𝑓& 𝒙 ,𝑖 = 1, 2, … ,𝑀

§ Subjectto§ 𝑓7 𝒙 ≤ 𝑓7 𝒙7∗ , 𝑗 = 1, 2, … , 𝑖 − 1, 𝑖 > 1

§ SequentiallysolvingMsingle-objectiveproblems

§ Manyothers


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BILEVEL MO§ Letusdefinevector𝒙 = (𝑥+, … , 𝑥?) = (𝒙u, 𝒙l)§ Goal min

(𝒙u,𝒙l)𝒇 𝒙 = (𝑓1 𝒙 ,… , 𝑓𝑀 𝒙 )

§ Subjectto§ 𝒙l ∈ min(𝒙l)

𝒇′ 𝒙 = (𝑓′1 𝒙 , … , 𝑓′𝑚 𝒙 )

§ 𝒙l ⊂ 𝒙, 𝒙uvariablesarefixed


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MULTI-LEVEL APPROACH§ Problemsinoptimisation

§ Big/complexsearchspace§ E.g.,dimensionality,multimodality,islandsoffeasiblesolutions…

§ Expensivesolutionevaluation§ Time/spacecomplexity

§ Canresultinpoor/infeasibleoptimisation§ Onepossiblesolution

§ Multi-levelapproach§ Applicabletosingle- andmulti-objectiveproblems


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MULTI-LEVEL APPROACH§ Generalidea

§ Firstwedecreasethecomplexityoftheproblem§ Thiscanbedonenotonlyonce,butmanytimes§ Whenwearesatisfiedwiththecomplexity,weapplyoptimisationonsuch(lesscomplex)problem

§ Thenwe“slowly”increasethecomplexityduringtheoptimisation

§ Reductionofcomplexitycomesatcost§ Withit,accuracyoftheoptimisationisalsodecreased§ Thisiswhyweneedtoincreasethecomplexity/accuracy

§ Result§ Atthebeginningofoptimisationwearequicklyidentifyingpromisingsearchspaces– exploration

§ Lateronwearemoreandmoreaccuratelyresearchingonlypromising(muchsmaller)searchspaces- exploitation


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MULTI-LEVEL APPROACH§ LetusdefinelevelsL§ Wherewitheachlevelwedecreasecomplexity

§ 𝐿G ⊐ 𝐿+ ⊐ ⋯ ⊐ 𝐿J§ Mostcomplex⊐ ⋯ ⊐ leastcomplexproblem

§ 𝐿G → 𝐿J letuscallcoarsening§ 𝐿J → 𝐿G letuscallrefining


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MULTI-LEVEL APPROACH§ Withregardtoaccuracy

§ Mostaccurate⊐ ⋯ ⊐ leastaccurate

§ Managingofcomplexitycanbeachievedthrough§ Searchspace

§ 𝑋 ⊐ 𝑋+ ⊐ ⋯ ⊐ 𝑋J§ Neighbourhood property

§ Evaluations§ 𝑓 𝒙 ⊐ 𝑓+ 𝒙 ⊐ ⋯ ⊐ 𝑓J(𝒙)


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MANAGING SEARCH SPACE COMPLEXITY§ Especiallyusefuloncombinatorialoptimisationproblems

§ Everycombinatorialproblemhascorrespondinggraphrepresentation§ Graphshasanicepropertyofneighbourhood

§ Usefulforqualitycoarsening

§ Butcanbealsoappliedtonumericaloptimisationproblems§ E.g.,usingdiscretization

§ Discretizationisoftenusedinreal-worldproblems,sinceaccuracyfound/requiredisnotinfinite


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MANAGING SEARCH SPACE COMPLEXITY§ Graphpartitioningproblem

§ LetusdefinegraphG=(V,E)§ k-waypartitioningdividesgraphG intok smallersubgraphsaccordingtosomeproperties

§ E.g.,minimizethenumberofedgesbetweensubgraphswhilehavingthesamenumberofverticesinsubgraphs


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MANAGING SEARCH SPACE COMPLEXITY§ Graphpartitioningproblemexample


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MANAGING SEARCH SPACE COMPLEXITY§ Numericalproblem

§ Letassume§ 𝑥& ∈ 𝒙§ 𝑥& ∈ ℝ§ 𝑥& ∈ (𝑥&N, 𝑥&O)

§ Definethemaximalaccuracyforeach𝑥& anddiscretizevariableaccordingly

§ Oneachcoarseningleveldecreasethenumberofdiscretevaluesforeachvariable𝑥& (accordingtoneighbourhood)

§ Startoptimisationonlowestlevelandslowlyrefinethroughlevels

§ IMPORTANT:duringrefiningonemustalwayscarrygainedinformation/knowledgefrompreviouslevel!


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MANAGING EVALUATION COMPLEXITY§ Evaluations

§ Mostaccurateevaluation⊐ ⋯ ⊐ leastaccurateevaluation§ Hightimecomplexity⊐ ⋯ ⊐ lowtimecomplexity

§ Usuallynoneedforexplicitcoarsening§ Startoptimisationwithleastaccurateevaluation(smallesttime/spacecomplexity)andgraduallymovetowardsmostaccurateevaluation(largesttime/spacecomplexity)


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MANAGING EVALUATION COMPLEXITY§ Especiallyusefulontimecomplexevaluations

§ Exactmathematicalevaluations§ Simulations

§ Equallyapplicabletocombinatorialornumericaloptimisationproblems


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MANAGING EVALUATION COMPLEXITY§ Severaloptionstoachievethis

§ Predefinedifferentapproximationfunctionstoexactevaluation§ Createapproximationfunctions(cheapersimulation)withregardtoalreadyknowninformation/knowledge

§ Decreasetheaccuracyoftheevaluation(whichalsoreducestimecomplexity)

§ E.g.,decreasethenumberoftrianglesinmeshes(finiteelementmethods)

§ Startwith“poor”evaluation(model)andimproveitthroughouttheoptimisationwithgainedknowledge

§ Surrogatemodelling


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SURROGATE MODELLING§ Basicsteps

1. Sampletheexactevaluationfunctiontogenerateasetofsolutions2. Selectasuitablesurrogatemodel3. Constructthesurrogateusingevaluatedsolutions4. Usesurrogatetopredictnewpromisingsolutions5. Evaluateone(ormore)promisingsolutions6. Updatethesurrogatebygoingto3untilendingconditionmet


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SURROGATE MODELLING§ Typesofmodels

§ Linearmodels§ Randomforests§ Artificialneuralnetworks§ Symbolicregression§ Kriging§ …


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SURROGATE MODELLING§ Linearmodels

§ 𝑦 = 𝛼0 + 𝛼1𝑥1 + ⋯+ 𝛼𝑛𝑥𝑛§ 𝑥𝑖 …datatobemodelled§ 𝛼𝑖 …corespondingcoefficients§ Easytoanalyzeandinterpret

§ Extensioninpolynomialregression§ 𝑦 = 𝛽0 + 𝛽1𝑥 + 𝛽2𝑥2 + ⋯+ 𝛽𝑛𝑥𝑛§ Possibletomodelcomplexfunctions


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SURROGATE MODELLING§ Randomforests

§ Largenumberofdecisiontreescombinedintoanensemblepredictor§ Decisiontree

§ Ateachnodeofthetreeasplitismadeonthebasisofadecisionvariablevalue

§ Thepredictionofanewpointisgivenbythemeanvalueofassociatedpoints

§ Eachtreeisfittedusingdifferentsubsetofevaluatedsolutionstoavoidoverfitting

§ Finalpredictioniscumulatedmeanofallpredictorsintheensemble

§ Relativelyfastcomparedtoothers§ Oneneedstoavoidbigtreestopreventoverfitting


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SURROGATE MODELLING§ Artificialneuralnetworks

§ Inspiredbybrain,whichutilisesconnectedneuronstolearnandapproximatethebehaviourofafunction

§ Neuronsareweightedtransformfunctions§ Neuronsareorderedinlayersandneuronswithinlayersareconnectedforwardand/orbackwardtootherlayers

§ Deep….manylayerswithnon-lineartransformation

§ Goodespeciallyinclassificationtasks§ Computationallyexpensive(learning)anddifficulttointerpret


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SURROGATE MODELLING§ Symbolicregression

§ Geneticprogrammingthatevolvesmodelusingevolutionarypopulation-basedapproach

§ Individualsaretreesforbuildinghigh-levelexpressionsbasedonmathematicaloperators(+,−,sin,cos,exp...)andtheinputdata

§ Computationaldemandingbuildingamodel§ Fastprediction§ Modeliseasytoanalyseandinterpret


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§ KrigingorGaussianprocessregression

§ Krigingmodelprovidesuncertaintymeasurefortheprediction

takenfromWikipedia

§ Enablescalculationofexpectedimprovement

§ Promisingsolutionisdeterminedbybestexpectedimprovement

§ Verysuitableforcomplexproblems

§ Notefficientforhigh-dimensionaldata(computationalcomplexity)

SURROGATE MODELLINGMulti-levelapproachinmulti-objectiveoptimisation

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SEARCH SPACE +EVALUATION§ Naturally,wecancombinebothapproaches§ Option1(solutionspaceandevaluation)

§ Duringcoarseningwereducesearchspace§ Duringrefinement

§ Westartwithcoarsestsearchspace,whileapplyingleastaccurateevaluation

§ Oneachrefinementweincreasethesearchspaceandimprovetheaccuracyofevaluation


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SEARCH SPACE +EVALUATION§ Option2(searchspacewithmulti-levelevaluation)

§ Duringcoarseningwereducesearchspace§ Duringrefinement

§ Westartwithcoarsestsearchspace§ Oneachrefinementweincreasethesearchspace§ Oneachlevelweapplyevaluationsinmulti-levelfashion

§ Option3(evaluationwithmulti-levelsearchspace)


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SUMMARY§ Optimisationalgorithmsneedtoreturnresultsinfeasibletime§ Mostreal-worldoptimisationproblemshavecomplexsearchspacesandrequiretime-consumingevaluations

§ Lownumberofevaluationsareavailabletofindhighqualitysolutions

§ Approachestospeed-upthesearchprocess§ Multi-levelapproach§ Surrogatemodelling

§ Withmulti-levelapproach,wereducetheaccuracyofthealgorithmatthebeginningofthesearchandgraduallyincreaseitthroughoutthesearchprocess

§ Theaccuracyisreducedbycoarseningsearchspaceorsimplifyingtheproblemevaluation

§ Duringthesearch,theaccuracyisimprovedbyrefiningsearchspaceorimprovingtheaccuracyofproblemevaluation


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ACKNOWLEDGEMENT§ PartofthisworkwasfundedfromtheEuropeanUnion’sHorizon2020researchandinnovationprogrammeundergrantagreementNo692286– project„SYNERGYforsmartmulti-objectiveoptimisation“.


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Multi-level approach in multi-objective optimisationutopiae.eu/wp-content/uploads/2017/12/20171121_4...§ Actually we have transferred it to single-objective problem §Lexicographic