Progressively Interactive Evolutionary Multi-Objective Optimization ...

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Table 4: Comparison of function evaluations needed by a nested algorithm and by H- BLEMOonproblems DS1andDS2. Results from21runs aresummarized. ProblemDS1 n Algo. Median Min. overall Max. overall LowerFE UpperFE OverallFE FE FE 10 Nested 12,124,083 354,114 12,478,197 11,733,871 14,547,725 10 Hybrid 1,454,194 36,315 1,490,509 1,437,038 1,535,329 20 Nested 51,142,994 1,349,335 52,492,329 42,291,810 62,525,401 20 Hybrid 3,612,711 94,409 3,707,120 2,907,352 3,937,471 30 Nested 182,881,535 4,727,534 187,609,069 184,128,609 218,164,646 30 Hybrid 7,527,677 194,324 7,722,001 6,458,856 8,726,543 40 Nested 538,064,283 13,397,967 551,462,250 445,897,063 587,385,335 40 Hybrid 12,744,092 313,861 13,057,953 10,666,017 15,146,652 ProblemDS2 n Algo. Median Min. overall Max. overall LowerFE UpperFE OverallFE FE FE 10 Nested 13,408,837 473,208 13,882,045 11,952,650 15,550,144 10 Hybrid 1,386,258 50,122 1,436,380 1,152,015 1,655,821 20 Nested 74,016,721 1,780,882 75,797,603 71,988,726 90,575,216 20 Hybrid 4,716,205 117,632 4,833,837 4,590,019 5,605,740 30 Nested 349,242,956 5,973,849 355,216,805 316,279,784 391,648,693 30 Hybrid 13,770,098 241,474 14,011,572 14,000,057 15,385,316 40 Nested 1,248,848,767 17,046,212 1,265,894,979 1,102,945,724 1,366,734,137 40 Hybrid 28,870,856 399,316 29,270,172 24,725,683 30,135,983 differenceofourapproachfromanestedapproachareclearlyevidentfromtheseplots. 10 Conclusions Bilevel programming problems appearcommonly inpractice,however dueto complications associated in solving them, often they are treated as single-level optimization problemsbyadoptingapproximatesolutionprinciplesforthelowerlevelproblem. Although single-objective bilevel programming problems are studied extensively, there does not seem to be enough emphasis for multi-objective bilevel optimization studies. This paperhas madeasignificant step inpresenting pastkeyresearchefforts,identifying insights for solving such problems, suggesting scalable test problems, and implementing aviable hybridevolutionary-cum-local-searchalgorithm. The proposed algorithm is also self-adaptive, allowing an automatic update of the key parameters from generation to generation. Simulation results on eight different multi-objective bilevel programming problems and their variants, and a systematic overall analysis amply demonstrate the usefulness of the proposed approach. Importantly, due to the multisolution natureoftheproblemandintertwinedinteractions betweenbothlevelsofoptimization, this study helps to showcase the importance of evolutionary algorithms in solving suchcomplex problems. Thestudyofmulti-objectivebilevelproblemsolvingmethodologieselevatesevery aspect of anoptimization effortat a higher level, therebymaking them interesting and challenging to pursue. Although formulation of theoretical optimality conditions is possible and has been suggested, viable methodologies to implement them in practice are challenging. Although a nested implementation of lower level optimization from the upper level is an easy fix-up and has been attempted by many researchers, suitablepracticalalgorithms couplingthetwolevelsofoptimizationsinacomputationally 110
efficientmannerisnoteasyanddefinitelychallenging. Developinghybridbileveloptimization algorithms involving various optimization techniques, such as evolutionary, classical, mathematical, simulated annealing methods etc., are possible in both levels independently or synergistically, allowing a plethora of implementational opportunities. This paper has demonstrated one such implementation involving evolutionary algorithms, a classical local search method, and a mathematical optimality condition for termination, but certainly many other ideas are possible and must be pursued urgently. Every upper level Pareto-optimal solution comes from a lower level Paretooptimalsolutionset. Thus,adecisionmakingtechniqueforchoosingasinglepreferred solution in such scenarios must involve both upper and lower level objective space considerations, which may require and give birth to new and interactive multiple criterion decision making (MCDM) methodologies. Finally, a successful implementation andunderstandingofmulti-objectivebilevelprogrammingtasksshouldmotivateusto understandanddevelophigherlevel(say,threeorfour-level)optimizationalgorithms, which should also be of great interest to computational science due to the hierarchical natureof systems approachoftenfollowed incomplex computational problemsolving tasks today. Acknowledgments Authors wishtothank Academyof FinlandandFoundationofHelsinki School ofEconomics (undergrant 118319)for theirsupport of this study. References Abass,S.A.(2005).Bilevelprogrammingapproachappliedtotheflowshopscheduling problemunder fuzziness. ComputationalManagement Science, 4(4),279–293. Alexandrov, N. and Dennis, J. E. (1994). Algorithms for bilevel optimization. In AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analyis and Optimization, pages810–816. Bard,J.F. (1998). PracticalBilevel Optimization: AlgorithmsandApplications. The Netherlands: Kluwer. Bianco, L., Caramia, M., and Giordani, S. (2009). A bilevel flow model for hazmat transportation network design. Transportation Research. Part C: Emergingtechnologies, 17(2),175–196. Byrd, R. H., Nocedal, J., and Waltz, R. A. (2006). KNITRO: An integrated package for nonlinear optimization,pages 35–59. Springer-Verlag. Calamai, P. H. and Vicente, L. N. (1994). Generating quadratic bilevel programming test problems. ACM Trans.Math.Software, 20(1),103–119. Colson, B., Marcotte, P., and Savard, G. (2007). An overview of bilevel optimization. Annalsof OperationalResearch, 153,235–256. Deb, K. (2001). Multi-objective optimization using evolutionary algorithms. Wiley, Chichester,UK. Deb, K. and Agrawal, R. B. (1995). Simulated binary crossover for continuous search space. ComplexSystems,9(2),115–148. 111
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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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An Interactive Evolutionary Multi-O
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Page 114 and 115: Table 1: Total function evaluations
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