Progressively Interactive Evolutionary Multi-Objective Optimization ...

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are K + L + 1real-valuedvariablesin this problemaswell: Minimize F(x, y) = (1 − x1)(1 + K j=2 x2 j )y1 x1(1 + K j=2 x2 j )y1 subject to (x) ∈ argmin (x) f(x) = (1 − x1)(1 + K+L j=K+1 x2 j)y1 x1(1 + K+L j=K+1 x2 j)y1 , , G1(x) = (1 − x1)y1 + 1 (12) 1 x1y1 − 2 + [5(1 − x1)y1 + 0.2] ≥ 0, [·] denotesgreatestint. function, 2 5 −1 ≤ x1 ≤ 1, 1 ≤ y1 ≤ 2, −(K + L) ≤ xi ≤ (K + L), i = 2, . . . , (K + L). For the upper level Pareto-optimal front, xi = 0 for i = 2, . . . , (K + L), x1 ∈ [2(1 − 1/y1), 2(1 − 0.9/y1)], y1 ∈ {1, 1.2, 1.4, 1.6, 1.8}(Figure8). For this test problemwe have chosen K = 5 and L = 4 (an overall 10-variable problem). This problem has similar difficulties as in DS4, except that only a finite number of y1 qualifies at the upper level Pareto-optimal front and that a consecutive set of lower level Pareto-optimal solutions now qualify tobeon theupper levelPareto-optimalfront. 5 HybridBilevel EvolutionaryMulti-Objective Optimization (H-BLEMO) Algorithm The proposed hybrid BLEMO procedure is motivated from our previously suggested algorithms (Deb and Sinha, 2009a,b),but differs in many different fundamental ways. Beforewe describethe differences,we first outline the proposed hybridprocedure. A sketch of the population structure is shown in Figure 9. The initial population Lower level NSGA−II Local search x_u t=0 x_l t=1 x_l t=t* ND x_l Archive T=0 ND ND Upper level NSGA−II Archive Figure9: A sketchof theproposed bileveloptimization algorithm. (markedwith upperlevel generationcounter T = 0 ofsize Nu) has asubpopulation of lower level variable set xl for each upper level variable set xu. Initially the subpopulation size (N (0) l ) is kept identical for each xu variable set, but it is allowed to change adaptivelywithgeneration T. Initially,anemptyarchive A0 iscreated. Foreach xu,we perform a lower level NSGA-II operation on the corresponding subpopulation having variables xl alone, not till the true lower level Pareto-optimal front is found, but only till a small number of generations at which the specified lower level termination criterion (discussed in subsection 5.2) is satisfied. Thereafter, a local search is performed 90 T=1
on a few rank-one lower level solutions until the local search termination criterion is met (discussed in Step 3 in subsection 5.3). The archive is maintained at the upper level containing solution vectors (xua, xla), which are optimal at the lower level and non-dominated at the upper level. The solutions in the archive are updated after every lower level NSGA-II call. The members of the lower level population undergoing a local search are lower level optimal solutions and hence are assigned an ‘optimality tag’. Theselocalsearchedsolutions(xl)arethencombinedwithcorresponding xu variables and become eligible to enter the archive if it is non-dominated when compared to the existing members of the archive. The dominated members in the archive are then eliminated. The solutions obtained from the lower level (xl) are combined with corresponding xu variables and are processed by the upper level NSGA-II operators to create a new upper level population. This process is continued till an upper level termination criterion(describedinsubsection 5.2)is satisfied. To make the proposed algorithm computationally faster, we have used two different strategies: (i) for every upper level variable vector xu, we do not completely solvethelowerlevelmulti-objectiveoptimizationproblem,therebynotmakingourapproachanested procedure,and (ii) the subpopulation size and number of generations foralowerlevelNSGA-IIsimulationarecomputedadaptivelybasedontherelativelocationof xu comparedtoarchivesolutions, therebymakingtheoverallproceduremore less parametric and more computationally efficient in terms of overall function evaluations. However, before we present a detailed step-by-step procedure, we discuss the automatic update procedure of population size and termination criteria of the lower level NSGA-II. 5.1 UpdateofPopulationSizes The upper level population size Nu is kept fixed and is chosen to be proportional to the number of variables. However, the subpopulation size (Nl) for each lower level NSGA-IIis sizedinaself-adaptivemanner. Herewe describethe procedure. In alower level problem, xl is updatedby amodified NSGA-IIprocedureand the corresponding xu is kept fixedthroughout. Initially, The populationsize of eachlower level NSGA-II (N (0) l ) is set depending upon the dimension of lower and upper level variables(|xl| and |xu|, respectively). The number of lower level subpopulations (n (0) s ) signifies the number of independent population members for xu in a population. Our intention is to set the population sizes (n (0) s and N (0) l ) for xu and xl proportionately to their dimensions, yielding n (0) s N (0) l = |xu| . (13) |xl| Noting alsothat n (0) s N (0) l = Nu, weobtainthe following sizing equations: n (0) s N (0) l = = |xu| |xl| Nu, (14) |xl| |xu| Nu. (15) For an equal number of lower and upper level variables, n (0) s = N (0) l = √ Nu. The above values are set for the initial population only, but are allowed to get modified 91
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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
Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
Page 49 and 50: DM calls. As mentioned earlier, the
Page 51 and 52: A linear value function similar to
Page 53 and 54: on developed value function. For ex
Page 55 and 56: 2 Progressively interactive evoluti
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
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Page 102 and 103: ulation of Nl(x (1) u ) lower level
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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