Progressively Interactive Evolutionary Multi-Objective Optimization ...

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ulation of Nl(x (1) u ) lower level variable vectors are created by genetic operations from PT and AT. However, if the lower level population size (Nl(x (1) u )) is less than N (0) l (indicating that the variable set xu is close to the archive members), a different strategy is used. First, a specific archive member (say, x (a) u ) closest to x (1) u is identified. Insteadofcreatingnewlowerlevelvariablevectors, Nl(x (1) u )vectorsarechosenfrom the subpopulation to which x (a) u belongs. Complete child solutions are createdby concatenating upper and lower level variables vectors together. If however the previous subpopulation does not have Nl(x (1) u ) members, the remaining slots are filled by creatingnew child solutions by the procedureof the previousparagraph. Step2: LowerlevelNSGA-II: For each subpopulation, we now perform a NSGA-II procedure using lower level objectives (f) and constraints (g) for tl generations (equation (18)). It is important to reiterate that in each lower level NSGA-II run, theupperlevelvariablevector xu isnotchanged. Theselectionprocessisdifferent from that in the usual NSGA-II procedure. If the subpopulation has no member in the current archive AT, the parent solutions are chosen as usual by the binary tournament selection using NDl and CDl lexicographically. If, however, the subpopulation has a member or members which already exist in the archive, only these solutions are used in the binary tournament selection. This is done to emphasize already-foundgood solutions. The mutation operator is appliedas usual. After the lower level NSGA-II simulation is performed on a subpopulation, the members aresortedaccordingtotheconstrained non-domination level(Debet al., 2002) and are assigned their non-dominated rank (NDl) and crowding distance value(CDl)basedonlower level objectives (f)andlower levelconstraints (g). Step3: Localsearch: The local search operator is employed next to provide us with a solution which is guaranteed to be on a locally Pareto-optimal front. Since the local search operator can be expensive, we use this operator sparingly. We apply the local search operator to good solutions having the following properties: (i) it is a non-dominated solution in the lower level having NDl = 1, (ii) it is a nondominated solution in the upper level having NDu = 1, and (iii) it does not get dominated by any current archive member, or it is located at a distance less than δU Nl/N (0) l fromanyofthecurrentarchivemembers. Inthelocalsearchprocedure, theachievementscalarizingfunctionproblem(Wierzbicki,1980)formulatedatthe currentNSGA-IIsolution (xl)with zj = fj(xl) is solved: Minimizep m max j=1 subject to p ∈ Sl, fj (p)−zj f max j −f min j + ρ m j=1 fj(p)−zj f max j −f min j where Sl is the feasible search space for the lower level problem. The minimum and maximum function values are taken from the NSGA-II minimum and maximum function values of the current generation. The optimal solution p ∗ to the above problem is guaranteed to be a Pareto-optimal solution to the lower level problem(Miettinen,1999). Here,weuse ρ = 10 −6 ,whichprohibitsthelocalsearch to converge to a weak Pareto-optimal solution. We use a popular software KNI- TRO(Byrdetal.,2006)(whichemploysasequentialquadraticprogramming(SQP) algorithm) to solve the above single objective optimization problem. The KNI- TROsoftwareterminateswhenasolutionsatisfiestheKarush-Kuhn-Tucker(KKT) 94 , (19)
conditions (Reklaitis et al., 1983) with a pre-specified error limit. We fix this error limit to 10 −2 in all problems of this study here. The solutions which meet this KKT satisfaction criterion are assigned an ‘optimal tag’ for further processing. For handling non-differentiable problems, a non-gradient, adaptive step-size basedhill-climbing procedure(Nolle,2006)canbe used. Step4: Updatingthearchive: The optimally tagged members, if feasible with respect to the upper level constraints (G), are then compared with the current archive members. If these members are non-dominated when compared to the members of the archive, they become eligible to be added into the archive. The dominated members in the archive are also eliminated, thus the archive always keeps nondominated solutions. We limit the size of archive to 10Nu. If and when more members are to be entered in the archive, the archive size is maintained to the abovelimitbyeliminatingextramembersusingthecrowdingdistance(CDa)measure. Step5: Formationofthecombinedpopulation: Steps 1 to 4 are repeated until the population QT is filled with newly created solutions. Each member of QT is now evaluated with F and G. Populations PT and QT are combined together to form RT. The combined population RT is then ranked according to constrained nondomination (Deb et al., 2002) based on upper level objectives (F) and upper level constraints (G). Solutions are thus, assigned a non-dominated rank (NDu) and members within an identical non-dominated rank are assigned a crowding distance(CDu)computed inthe F-space. Step7: Choosing halfthepopulation: Fromthecombinedpopulation RT ofsize 2Nu, half of its members are retained in this step. First, the members of rank NDu = 1 are considered. From them, solutions having NDl = 1 are noted one by one in the order of reducing crowding distance CDu. For each such solution, the entire Nl subpopulation from its source population (either PT or QT) are copied in an intermediate population ST. If a subpopulation is already copied in ST and a future solution from the same subpopulation is found to have NDu = NDl = 1, the subpopulation is not copied again. When all members of NDu = 1 are considered, a similar consideration is continued with NDu = 2 and so on till ST has Nu populationmembers. Step6: Upgradingoldlowerlevelsubpopulations: Each subpopulation of ST which arenotcreatedinthecurrentgenerationaremodifiedusingthelowerlevelNSGA- II procedure (Step 2) applied with f and g. This step helps progress each lower level populationtowardstheir individual Pareto-optimalfrontiers. The final population is renamed as PT +1. This marks the end of one generation of the overallH-BLEMOalgorithm. 5.4 AlgorithmicComplexity With self-adaptive operations to update population sizes and number of generations, it becomes difficult to compute an exact number of function evaluations (FE) needed inthe proposedH-BLEMOalgorithm. However,using the maximumallowablevalues of these parameters, we estimate that the worst case function evaluations is Nu(2Tu + 1)(tmax l + 1) + FELS. Here Tu is the number of upper level generations and FELS is the total function evaluations used by the local search (LS) algorithm. Lower level 95
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Department of Business Technology P
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Aalto University publication series
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Progressively Interactive Evolution
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Acknowledgements The dissertation h
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II List of Papers 27 1 An interacti
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1 Introduction Many real-world appl
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• f(x (1) ) ≥ f(x (2) ) fi(x (1
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for a maximization problem. Mathema
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• x (1) ∼ x (2) ⇔ x (1) and x
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Start Initialise Population Assign
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ithm can approximate the Pareto-opt
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Minimize/Maximize F(x) = (f1(x), f2
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the search is chosen. A scalarizing
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of the papers is provided in this s
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problems. The study discusses some
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[9] K. Deb and A. Sinha. An efficie
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[31] L. Thiele, K. Miettinen, P. Ko
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1 An interactive evolutionary multi
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DM and an MCDM-based EMO algorithm
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In Step 2, points in the best non-d
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esulting constraints then become ki
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mechanisms) and their emphasis of n
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f2 10 8 6 4 2 P1 P2 Most preferred
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DM calls. As mentioned earlier, the
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Page 53 and 54: on developed value function. For ex
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Page 57 and 58: DM one or more pairs of alternative
Page 59 and 60: f 2 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 P
Page 61 and 62: TABLE III DISTANCE OF OBTAINED SOLU
Page 63: The PI-EMO-VF algorithm has been te
Page 66 and 67: An Interactive Evolutionary Multi-O
Page 68 and 69: direction has been done by Phelps a
Page 70 and 71: 0 ∀ i ∈ {1, . . . , M}, then th
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Page 74 and 75: Table 1. Final solutions obtained b
Page 76 and 77: Table 4. Distance of obtained solut
Page 78 and 79: DM Calls and Func. Evals. (in thous
Page 80 and 81: 13. P. Korhonen and J. Laakso, “A
Page 82 and 83: An EfficientandAccurateSolution Met
Page 84 and 85: eight differentproblems areshown. A
Page 86 and 87: 3.2 AlgorithmicDevelopments Onesimp
Page 88 and 89: more studies are performed, the alg
Page 90 and 91: 4.4 TestProblemTP4 The next problem
Page 92 and 93: F2 Step 3: Map (U1,U2) to (f1*,f2*)
Page 94 and 95: F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A y1
Page 96 and 97: • Theupperlevelproblemhas multi-m
Page 98 and 99: are K + L + 1real-valuedvariablesin
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Page 104 and 105: NSGA-II is able to bring the member
Page 106 and 107: F2 0 −0.2 −0.4 −0.6 −0.8
Page 108 and 109: LL Function Evals. UL Function Eval
Page 114 and 115: Table 1: Total function evaluations
Page 116 and 117: Difference in HV, DH(T) 350000 3000
Page 118 and 119: Table 4: Comparison of function eva
Page 120 and 121: Deb, K. and Sinha, A. (2009a). Cons
Page 122 and 123: Sun, D., Benekohal, R. F., and Wall
Page 124 and 125: Bilevel Multi-Objective Optimizatio
Page 126 and 127: The constraint functions g(x) and h
Page 128 and 129: value function which are required t
Page 130 and 131: to get close to the most preferred
Page 132 and 133: F2 0.4 0.2 0 −0.2 Lower level P
Page 134 and 135: F2 2 1.5 1 0.5 Most Preferred Point
Page 136 and 137: provement in terms of function eval
Page 138: 9HSTFMG*aeafcd+ ISBN: 978-952-60-40
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