Progressively Interactive Evolutionary Multi-Objective Optimization ...

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LL Function Evals. UL Function Evals. 6.0e+06 5.6e+06 5.2e+06 4.8e+06 4.4e+06 4.0e+06 150000 140000 130000 120000 110000 100000 200 200 400 Population Size 400 Population Size Figure 25: Average lower and upper level function evaluations with different populationsizesforproblemDS2indicates Nu = 400isthebestchoice. 21runsareperformed ineachcase. F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0 0.2 0.4 0.6 F1 0.8 1 1.2 1.4 Figure 26: Final archive solutions for problemDS3. 6.8 ProblemDS5 F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 600 600 800 800 0 0.2 0.4 0.6 F1 0.8 1 1.2 1.4 Figure 27: Attainment surfaces(0%, 50% and100%)forproblemDS3from21runs. This problem is also considered with 10 variables. We have used Nu = 200. Figure 30 showsthefinalarchivepopulationforatypicalrun. Figure31showsthecorresponding attainmentsurfaceplots,whichareveryclosetoeachotherindicatingtheefficacyofthe procedure. The hypervolumes for the obtained attainment surfaces are 0.5216, 0.5281 and0.5308,respectively. The differencein hypervolume is only 1.7%forthis problem. 6.9 ComputationalEfforts Next, we investigate two aspects related to the computational issues. First, we record the total function evaluations needed by the overall algorithm to achieve the specified 104
F2 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 F1 Figure 28: Final archive solutions for problemDS4. F2 2 1.5 1 0.5 0 0 0.5 1 F1 1.5 2 Figure 30: Final archive solutions for problemDS5. F2 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 F1 Figure 29: Attainment surfaces(0%, 50% and100%)forproblemDS4from21runs. F2 2 1.5 1 0.5 0 0 0.5 1 F1 1.5 2 Figure 31: Attainment surfaces(0%, 50% and100%)forproblemDS5from21runs. terminationcriterion(ǫu = 0.0001)andthesameneededexclusivelyforthelowerlevel optimization task, which includes the local search. Table 1 shows these values for the best, median and worst of 21 simulation runs for all eight problems. It is clear from thetablethatthemostofthecomputationaleffortsarespentinthelowerlevelsolution evaluations. Despite our efforts being differentfromanested algorithm in not solving a lower level problem all the way for every upper level solution, the nature of bilevel programming problem demands that the lower level optimization task must be emphasized. The use of archive in sizing lower level subpopulations in a self-adaptive mannerandtheuseofacoarseterminatingconditionforlowerleveloptimizationtask enabled our algorithm to use comparatively smaller number of function evaluations than that would be needed in a nested algorithm. We compare the function evalua- 105
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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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A linear value function similar to
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on developed value function. For ex
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2 Progressively interactive evoluti
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DM one or more pairs of alternative
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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 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 110 and 111: 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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