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 <strong>Multi</strong>disciplinary Analyis and <strong>Optimization</strong>, pages810–816. Bard,J.F. (1998). PracticalBilevel <strong>Optimization</strong>: 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). <strong>Multi</strong>-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
- Page 1 and 2:
Department of Business Technology P
- Page 3 and 4:
Aalto University publication series
- Page 5 and 6:
Progressively Interactive Evolution
- Page 7:
Acknowledgements The dissertation h
- Page 10 and 11:
II List of Papers 27 1 An interacti
- Page 12 and 13:
1 Introduction Many real-world appl
- Page 14 and 15:
• f(x (1) ) ≥ f(x (2) ) fi(x (1
- Page 16 and 17:
for a maximization problem. Mathema
- Page 18 and 19:
• x (1) ∼ x (2) ⇔ x (1) and x
- Page 20 and 21:
Start Initialise Population Assign
- Page 22 and 23:
ithm can approximate the Pareto-opt
- Page 24 and 25:
Minimize/Maximize F(x) = (f1(x), f2
- Page 26 and 27:
the search is chosen. A scalarizing
- Page 28 and 29:
of the papers is provided in this s
- 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
- 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
- Page 55 and 56:
2 Progressively interactive evoluti
- 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
- Page 72 and 73: f2 0000000000000 1111111111111 0000
- 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
- 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 110 and 111: LL Function Evals. UL Function Eval
- Page 112 and 113: 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 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