Progressively Interactive Evolutionary Multi-Objective Optimization ...

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• Theupperlevelproblemhas multi-modalities, therebycausing analgorithmdifficulty infinding the upperlevel Pareto-optimalfront. • With τ = −1, the conflict between upper and lower level problems can be introduced,asinDS1. • The dimension of both upper and lower level problems can be increased by increasing K. • Theparameter γ canbeadjustedtocauseasmallproportionoflowerlevelParetooptimal points tobe responsiblefor the upperlevelPareto-optimalfront. 4.7 ProblemDS3 In this problem, the ΦU2 and the f ∗ 1-f ∗ 2-frontiers lie on constraints and thus are not defined parametrically as in DS1 and DS2 here, but are defined directly as a function of problem variables. Since a constraint function determines the lower level Paretooptimal front, ΦU and ΦL-frontiers are defined with M variables. The linked terms (lj = (xi − yi) 2 for i = 3, . . . , K) are included. We use Lj = τlj. Like before, we do not use any ej term, but use a Ej term in the upper level problem. The variable y1 is considered to be discrete, thereby causing only a few y1 values to represent the upper level Pareto-optimalfront. Theoverall problemis givenbelow: ⎛Minimize F(x, y) = ⎝ y1 + K j=3 (yj − j/2)2 + τ K i=3 (xi − yi)2 − R(y1) cos(4 tan −1 y2 + K j=3 (yj − j/2)2 + τ K i=3 (xi − yi)2 − R(y1) sin(4 tan −1 subjectto (x) ∈ argmin (x) f(x) = K x1 + i=3 (xi − yi)2 x2 + K i=3 (xi − yi)2 y2−x2 ⎞ y1−x1 y2−x2 ⎠ , y1−x1 g1(x) = (x1 − y1) 2 + (x2 − y2) 2 ≤ r 2 , G(y) = y2 − (1 − y 2 1) ≥ 0, −K ≤ xi ≤ K, for i = 1, . . . , K, 0 ≤ yj ≤ K, for j = 1, . . . , K, y1 isamultipleof0.1. Here we suggest a periodically changing radius: R(y1) = 0.1 + 0.15| sin(2π(y1 − 0.1)| and use r = 0.2. For the upper level Pareto-optimal points, yi = j/2 for j ≤ 3. The variables y1 and y2 take values satisfying constraint G(y) = 0. For each such combination, variables x1 and x2 lie on the third quadrant of a circle of radius r and center at (y1, y2) in the F-space. Notice in Figure 6, how lower level Pareto-optimal solutions for y1 = 0.1 and 0.2 (shown in dashed lines in the figure) mapped to corresponding circles in the upper level problemget dominated by that for y1 = 0 and 0.3. Following propertiesareobservedfor this problem: • The Pareto-optimalfronts for both lower and upper level lie on constraint boundaries,therebyrequiringgoodconstrainthandlingstrategiestosolvebothproblems optimally. • Not all lower level Pareto-optimal solutions qualify as upper level Pareto-optimal solutions. • Every lower level front has an unequal contribution to the upper level Paretooptimal front. • Bychoosing τ = −1,conflictbetweentwolevelsofoptimizationcanbeintroduced. 88 (10)
4.8 ProblemDS4 In this problem, the v1-v2 relationship is linear (v1 = 2 − y1, v2 = 2(y1 − 1)), spanning in the first quadrant of F-space. The mapping U1-U2 is not considered here. For every (v1, v2) point, the following relationship is chosen for the lower level Pareto-optimal front: f ∗ 1 + f ∗ 2 = y1. Additional terms having a minimum value of one are multiplied toformthelowerandupperlevelsearchspaces. Thisproblemhas K + L + 1variables, which areall real-valued: Minimize F(x, y) = (1 − x1)(1 + K j=2 x2j)y1 x1(1 + K j=2 x2j)y1 G1(x) = (1 − x1)y1 + 1 x1y1 − 1 ≥ 0, 2 −1 ≤ x1 ≤ 1, 1 ≤ y1 ≤ 2, −(K + L) ≤ xi ≤ (K + L), i = 2, . . . , (K + L). , subject to (x) ∈ argmin f(x) = (x) (1 − x1)(1 + K+L j=K+1 x2j)y1 x1(1 + K+L j=K+1 x2j)y1 The upper level Pareto-optimal front is formed with xi = 0 for all i = 2, . . . , (K + L) and x1 = 2(1 − 1/y1) and y1 ∈ [1, 2]. This problemhasfollowing properties: • By increasing K and L, the problem complexity in converging to the appropriate lower andupper levelfronts canbe increased. • Only one Pareto-optimalpoint fromeachparticipating lower levelproblemqualifies tobe onthe upperlevel front. For our study here,wechoose K = 5and L = 4(anoverall 10-variableproblem). F2 2 1.5 1 0.5 Lower Level Front Upper Level Front 0 0 0.5 1 F1 1.5 2 Figure 7: Pareto-optimal front for problemDS4. 4.9 ProblemDS5 F2 2 1.5 1 0.5 , Lower Level Front (11) Upper Level Front 0 0 0.5 1 F1 1.5 2 Figure 8: Pareto-optimal front for problemDS5. ThisproblemissimilartoproblemDS4exceptthattheupperlevelPareto-optimalfront isconstructedfrommultiplepointsfromafewlowerlevelPareto-optimalfronts. There 89
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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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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
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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 98 and 99: are K + L + 1real-valuedvariablesin
Page 100 and 101: thereafterinaself-adaptivemannerbyd
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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