Progressively Interactive Evolutionary Multi-Objective Optimization ...

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f2 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 B 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 A 0000000000000 1111111111111 0000 1111 0000000000000 1111111111111 0000 1111 0000000000000 1111111111111 0000 1111 0000 1111 0000 1111 f1 Fig. 5. Dominated regions of two points A and B using the modified definition. 4 PI-NSGA-II-PC Procedure Although we do not handle constrained problems in this study, the above modified domination principle can be extended for handling constraints. As defined in [18], when both solutions under consideration for a domination check are feasible, the above domination principle can simply be used to establish dominance of one over the other. However, if one point is feasible and the other is not, the feasible solution can be declared as dominating the other. Finally, if both points are infeasible, the one having smaller overall constraint violation may be declared as dominating the other. We defer consideration of a constrained PI- EMO to a later study. In the PI-NSGA-II-PC procedure, the first τ generations are performed according to the usual NSGA-II algorithm [18]. Thereafter, we modify the NSGA-II algorithm by using the modified domination principle (discussed in Section 3.3) in the elite-preserving operator and also in the tournament selection for creating the offspring population. We also use a different recombination operator in this study. After a child solution xC is created by the SBX (recombination) operator [25], two randomly selected population members x (1) and x (2) are chosen and a small fraction of the difference vector is added to the child solution (similar in principle to a differential evolution operator [26]), as follows: x C = x C + 0.1 x (1) − x (2) . (2) The crowding distance operator of NSGA-II has been replaced with k-mean clustering for maintaining diversity among solutions of the same non-dominated front. An archive A is maintained which contains all the non-dominated members found in the current as well as the previous iterations of the optimization run. The maximum size the archive can have is |A| max . This makes sure that none of the non-dominated solutions generated is lost even if the decision maker makes an error while providing preference information. For termination check (discussed in Section 3.2), we use the SQP code of KNITRO [27] software to solve the single objective optimization problem and the SQP algorithm is terminated (if not terminated due to ds distance check from Abest t discussed earlier) when the KKT error measure is less than or equal to 10−6 . 5 Results In this section we present the results of the PI-NSGA-II procedure on two, three, and five objective test problems. ZDT1 and DTLZ2 test problems are adapted to create maximization problems. In all simulations, we have used the following parameter values: 1. Number of generations between two consecutive DM calls: τ = 5. 64
2. Termination parameter: ds = 0.01 and ds = 0.1. 3. Crossover probability and the distribution index for the SBX operator: pc = 0.9 and ηc = 15. 4. Mutation probability: pm = 0.1. 5. Population size: N = 10M, where M is the number of objectives. 6. Maximum Archive Size: A max = 10N, where N is the population size. In the next section, we perform a parametric study with some of the above parameters. Here, we present the test problems and results obtained with the above setting. 5.1 Two-<strong>Objective</strong> Test Problem Problem 1 is adapted from ZDT1 and has 30 variables. x1 Maximize f(x) = 10− √ x1g(x) , g(x) 30 29 i=2 xi, where g(x) = 1 + 9 0 ≤ xi ≤ 1, for i = 1, 2, . . . , 30, The Pareto-optimal front is given by f2 = 10 − √ f1 and is shown in Figure 6. The solutions are xi = 0 for i = 2, 3, . . . , 30 and x1 ∈ [0, 1]. This maximization problem has a non-convex front. In order to emulate the decision maker, in our simulations, we assume a particular value function which acts as a representative of the DM, but the information is not explicitly used in creating new solutions by the operators of the PI-NSGA-II procedure. In such cases, the most preferred point z ∗ can be determined from the chosen value function beforehand, thereby enabling us to compare our obtained point with z ∗ . In our study, we assume the following nonlinear value function (which acts as a DM) in finding the best point from the archive at every τ generations: 1 V (f1, f2) = (f1 − 0.35) 2 . (4) + (f2 − 9.6) 2 10 9.8 9.6 f2 9.4 9.2 9 0 0.2 Most Preferred Point Pareto Front Value Function Contours (3) 0.4 0.6 0.8 1 f1 Fig. 6. Contours of the chosen value function (acts as a DM) and the most preferred point corresponding to the value function. This value function gives the most preferred solution as z ∗ = (0.25, 9.50). The contours of this value function are shown in Figure 6. Table 1 presents the best, median and worst of 21 different PI-NSGA-II simulations (each starting with a different initial population). The performance (accuracy measure) is computed based on the Euclidean distance of each optimized point with z ∗ . Note that this accuracy measure is different from the termination criterion used in the PI- NSGA-II procedure. Results have been presented for two values of the termination criteria ds. As expected, when the termination criteria is relaxed from 0.01 to 0.1, the 65
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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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Page 24 and 25: Minimize/Maximize F(x) = (f1(x), f2
Page 26 and 27: the search is chosen. A scalarizing
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Page 30: problems. The study discusses some
Page 33 and 34: [9] K. Deb and A. Sinha. An efficie
Page 35 and 36: [31] L. Thiele, K. Miettinen, P. Ko
Page 37 and 38: 1 An interactive evolutionary multi
Page 39 and 40: DM and an MCDM-based EMO algorithm
Page 41 and 42: In Step 2, points in the best non-d
Page 43 and 44: esulting constraints then become ki
Page 45 and 46: mechanisms) and their emphasis of n
Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
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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
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Page 66 and 67: An Interactive Evolutionary Multi-O
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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
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Page 86 and 87: 3.2 AlgorithmicDevelopments Onesimp
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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
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Page 114 and 115: Table 1: Total function evaluations
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Page 118 and 119: Table 4: Comparison of function eva
Page 120 and 121: Deb, K. and Sinha, A. (2009a). Cons
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Sun, D., Benekohal, R. F., and Wall
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Bilevel Multi-Objective Optimizatio
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The constraint functions g(x) and h
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value function which are required t
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to get close to the most preferred
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F2 0.4 0.2 0 −0.2 Lower level P
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F2 2 1.5 1 0.5 Most Preferred Point
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provement in terms of function eval
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9HSTFMG*aeafcd+ ISBN: 978-952-60-40
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