Progressively Interactive Evolutionary Multi-Objective Optimization ...

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Difference in HV, DH(T) 350000 300000 250000 200000 150000 100000 50000 0 −50000 −100000 A2 x (terminated) −150000 0 20 40 60 80 100 120 140 Generation Counter (upper level), T Figure 32: Difference in hypervolume from ideal DH(T ) with upper level generation counter T for problem DS1 using three algorithms. Only algorithm A1 (H- BLEMO)reachesthe Pareto-optimalfront by making DH(T ) = 0. A3 A1 x x Difference in HV, DH(T) 60000 50000 40000 30000 20000 10000 0 A2 x x x (terminated) −10000 0 20 40 60 80 100 120 140 160 Generation Counter (upper level), T Figure 33: Difference in hypervolume from ideal DH(T ) with upper level generation counter T for problem DS2 using three algorithm. Only algorithm A1 (H- BLEMO)reachesthe Pareto-optimalfront by making DH(T ) = 0. Table 3: Comparison of function evaluations for τ = −1 and τ = +1 cases with the H-BLEMOalgorithm. Prob. Best Median Worst No. Total LL Total UL Total LL Total UL Total LL TotalUL FE FE FE FE FE FE DS1(τ = +1) 2,819,770 87,582 3,423,544 91,852 3,829,812 107,659 DS1(τ = −1) 3,139,381 92,624 3,597,090 98,934 4,087,557 113,430 DS2(τ = +1) 4,484,580 105,439 4,695,352 116,605 5,467,633 138,107 DS2(τ = −1) 4,796,131 112,563 4,958,593 122,413 5,731,016 144,428 8 Scalability Study In this section, we consider DS1 and DS2 (with τ = 1) and show the scalability of our proposed procedure up to 40 variables. For this purpose, we consider four different variable sizes: n = 10, 20, 30 and 40. Based on parametric studies performed on these problemsinsection 6,we set Nu = 20n. Allother parametersareautomaticallyset ina self-adaptivemannerduring the course of asimulation, asbefore. Figure34showsthevariationoffunctionevaluationsforobtainingafixedtermination criterionon normalizedhypervolume measure(Hu < 0.0001)calculatedusing the upperlevelobjective valuesfor problemDS1. Sincethe verticalaxisis plotted inalogarithmic scale and the relationship is found to be sub-linear, the hybrid methodology performs better than an exponential algorithm. The break-upof computations needed inthelocalsearch,lowerlevelNSGA-IIandupperlevelNSGA-IIindicatethatmajority of the computations is spent in the lower level optimization task. This is an important insight to the working of the proposed H-BLEMOalgorithm and suggests that further efforts must be put in making the lower level optimization more computationally efficient. Figure 35 shows the similar outcome for problem DS2, but a comparison with that for problem DS1 indicates that DS2 is more difficult to be solved with an increase 108 A1 A3
Number of Function Evaluations 1e+09 1e+08 1e+07 1e+06 100000 10000 Nested Overall H−BLEMO Lower level Local search Upper level 10 20 30 40 Variable Size (n) Figure 34: Variation of function evaluations withproblemsize n forDS1. inproblemsizethanDS1. 9 Comparisonwith aNested Algorithm Number of Function Evaluations 1e+10 1e+09 1e+08 1e+07 1e+06 100000 10000 Lower level Nested Overall H−BLEMO Local search Upper level 10 20 30 40 Variable Size (n) Figure 35: Variation of function evaluations withproblemsize n forDS2. We have argued before that by allowing lower level and upper level NSGA-IIs to proceed partially in tandem, we have created a computationally efficient and accurate algorithm which progresses towards the true Pareto-optimal front on a number of difficult problems (Section 6). The algorithm is even found to converge in problems in which there is a conflict between upper and lower level problems (Section 7). The proposed algorithm is also found to solve scaled-up problems up to 40 real-parameter variables (Section 8). In this section, we compare the proposed H-BLEMO algorithm with an efficient yet nested bilevel optimization algorithm using the NSGA-II-cumlocal-search procedure. This algorithm uses a fixed population structure, but for every xu, the lower level optimization is terminated by performing a local search to all non-dominatedsolutions ofthefinallowerlevelNSGA-IIpopulation. Thetermination criterion for lower and upper level NSGA-IIs and that for the local search procedure are identical to that in H-BLEMO algorithm. Since for every xu, we find a set of wellconverged and well-distributed lower level Pareto-optimal solutions, this approach is truly a nestedbilevel optimizationprocedure. For the simulation with this nested algorithm on DS1 and DS2 problems (τ = 1), weuse Nu = 400. Tomakeafaircomparison,weusethesamesubpopulationsize Nl as thatwasusedintheveryfirstiterationofourH-BLEMOalgorithmusingequation(15). Thenumber ofgenerations forthelower levelNSGA-IIiskept fixedforallupperlevel generations tothat computed by equation (14)inthe initial generation. Similar archiving strategy and other NSGA-II parameter values are used as before. Table 4 shows the comparison of overall function evaluations needed by the nested algorithm and by the hybrid BLEMO algorithm. The table shows that for both problems, the nested algorithm takes at least one order of magnitude of more function evaluations to find a set of solutions having an identical hypervolume measure. The difference between our proposed algorithm and the nested procedurewidens with an increase in number of decision variables. The median number of function evaluations are also plotted in Figures 34 and 35. The computational efficacy of our proposed hybrid approach and 109
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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
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TABLE III DISTANCE OF OBTAINED SOLU
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The PI-EMO-VF algorithm has been te
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Page 74 and 75: Table 1. Final solutions obtained b
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Page 86 and 87: 3.2 AlgorithmicDevelopments Onesimp
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Page 108 and 109: LL Function Evals. UL Function Eval
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Page 120 and 121: Deb, K. and Sinha, A. (2009a). Cons
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Page 134 and 135: F2 2 1.5 1 0.5 Most Preferred Point
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