A Modified Indicator-based Evolutionary Algorithm

Transcription

1 A Modified Indicator-based Evolutionary Algorithm () Wenwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 Automated Scheduling, Optimization and Planning (ASAP) University of Nottingham, UK 2 Operational Research Group, Queen Mary University of London, UK 3 School of Computer Science, The University of Adelaide, Australia {wenwen.li, ender.ozcan, robert.john}@nottingham.ac.uk j.drake@qmul.ac.uk {aneta.neumann,markus.wagner}@adelaide.edu.au June 3, 27 Wenwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 /

2 Content Motivation 2 IBEA VS 3 Results Future Work enwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 2 /

3 Motivation DTLZ NSGA-II SMS-EMOA bf:.75 ndi: 979 bf: 5 ndi: 987 bf:.786 ndi: 892 bf: 26 ndi: 97 bf:. ndi: 82 bf:.22 ndi: 73 bf:.2229 ndi: 555 DTLZ2 bf:.9935 ndi: 66 DTLZ3 IBEA bf:.352 ndi: 988 Figure: Non-dominated solutions from NSGA-II, SMS-EMOA and IBEA on DTLZ-3 over runs, except SMS-EMOA runs due to high computation time. 3, Markus Wagner3 (ASAP) Wenwen Li, Ender O zcan, Robert John John H. Drake2, Aneta Neumann June 3, 27 3 /

4 Observations ndi: # of non-dominated individuals. border fraction (bf ): bf = ndi in A border total ndi, A border := {A f θ f 2 θ 2 f 3 θ 3 }. θ = θ 2 = θ 3 =.3 for DTLZ; θ = θ 2 = θ 3 =. for DTLZ2,3 Observations: NSGA-II can cover the whole fronts of all DTLZ-3; 2 SMS-EMOA reaches almost full coverage of the PF for DTLZ, but leaves empty spaces on DTLZ2, 3; 3 IBEA heavily concentrates in corners on DTLZ and DTLZ3, while show clear gaps on DTLZ2; bf IBEA > bf SMS EMOA > bf NSGA II on DTLZ-3; 5 ndi NSGA II ndi IBEA on DTLZ and 3; enwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 /

5 Observations ndi: # of non-dominated individuals. border fraction (bf ): bf = ndi in A border total ndi, A border := {A f θ f 2 θ 2 f 3 θ 3 }. θ = θ 2 = θ 3 =.3 for DTLZ; θ = θ 2 = θ 3 =. for DTLZ2,3 Observations: NSGA-II can cover the whole fronts of all DTLZ-3; 2 SMS-EMOA reaches almost full coverage of the PF for DTLZ, but leaves empty spaces on DTLZ2, 3; 3 IBEA heavily concentrates in corners on DTLZ and DTLZ3, while show clear gaps on DTLZ2; bf IBEA > bf SMS EMOA > bf NSGA II on DTLZ-3; 5 ndi NSGA II ndi IBEA on DTLZ and 3; Question: why does IBEA deteriorate on DTLZ and 3 so severely? Guess: Similar as the hypervolume drops of SMS-EMOA during the search []? [] Judt, L., Mersmann, O., Naujoks, B. (23) Non-monotonicity of Observed Hypervolume in -Greedy S-Metric Selection. Wiley Online Library, enwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 /

6 IBEA VS and IBEA difference: removes dominated solutions before scaling objective values. No new parameters is introduced in. Initialisation; Randomly generate the initial population with population size µ; 2 while (!Stopping Criteria) do 3 Remove dominated solutions using fast non-dominated sorting (NSGA-II); ) rank the solutions in P: Ranking rankedp = new Ranking(P); 2) get the non-dominated solutions: P = rankedp.getsubfront(); Objective values scaling ; ) find the lower (b i = min x Pf i(x)) and upper (b i = max x Pf i(x)) bound of each objective i.; 2) increase the upper bound b i = bi + ρ (b i b i); 3) scale each objective to the interval [ ]; f i = (f i(x) b i)/(b i bi); 5 Fitness assignment (hypervolume difference) using scaled objectives; 6 Environmental Selection (Survival); ) remove the solutions with least hypervolume loss iteratively; 2) update fitness values of all the remaining individuals; 7 Mating Selection; 8 Variation; end Algorithm : Pseudo Code Wenwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 5 /

7 Overall Results With the same default settings as IBEA, DTLZ DTLZ2 DTLZ3 IBEA bf:.9935 ndi: 66 bf:.786 ndi: 892 bf:. ndi: 82 bf: 297 ndi: 989 bf:.78 ndi: 92 bf:.7897 ndi: 5292 Figure: Non-dominated solutions from IBEA and on DTLZ-3. clearly improves the front coverage of IBEA on DTLZ and 3, but not on DTLZ2. (objective values evolving for DTLZ: [, 5] [,], DTLZ2: [2, 5] [,], DTLZ3: [5, 5] [,]) Wenwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 6 /

8 Performance Analysis w.r.t. different scaling factor ρ with population size µ = bf:.9669 ndi: 5 DTLZ3 ρ =.5 IBEA DTLZ ρ =. bf:.987 ndi: 69 DTLZ DTLZ3 bf:.9998 ndi: 5 bf: 297 ndi: 989 bf: 87 ndi: 9597 bf:.7897 ndi: 5292 bf:.972 ndi: 5356 bf:. ndi: 28 bf:.9729 ndi: 9262 bf:. ndi: 82 bf:.876 ndi: 9 bf:.99 ndi: 66 bf:. ndi: 263 ρ = 6. bf:.9935 ndi: 66 bf:. ndi: 25 ρ = 2. bf:.7252 ndi: 5 Figure: Non-dominated solutions of DTLZ and 3 from IBEA and (µ = ). 3, Markus Wagner3 (ASAP) Wenwen Li, Ender O zcan, Robert John John H. Drake2, Aneta Neumann June 3, 27 7 /

9 Performance analysis w.r.t. different scaling factor ρ with population size µ = ρ =.5 bf:.996 ndi: 956 IBEA DTLZ ρ =. bf:.99 ndi: 837 DTLZ3 2 2 bf:.999 ndi: DTLZ bf:.39 ndi: 2 bf: 259 ndi: 976 bf: 367 ndi: 973 bf:.6226 ndi: bf:.685 ndi: 3797 bf:.985 ndi: bf:.872 ndi: 95 6 bf:.9953 ndi: 86 bf:. ndi: bf:.322 ndi: bf:. ndi: 8 8 bf:.999 ndi: ρ = 6. 6 DTLZ3 ρ = 2. bf:.689 ndi: 3956 Figure: Non-dominated solutions of DTLZ and 3 from IBEA and (µ = ). 3, Markus Wagner3 (ASAP) Wenwen Li, Ender O zcan, Robert John John H. Drake2, Aneta Neumann June 3, 27 8 /

10 Performance Statistics and Running Time Comparison Table: Performance Comparison ρ = 2. hypervolume ɛ+ µ = IBEA IBEA DTLZ DTLZ DTLZ DTLZ DTLZ DTLZ DTLZ Table: Running Time Comparison Average Running Time per Run (in minutes) µ = µ = ρ = 2. IBEA IBEA DTLZ DTLZ DTLZ DTLZ DTLZ DTLZ DTLZ Summary: ρ affects significantly, but little impact on IBEA since all the solutions already collapse to the extreme points; µ has little influence on IBEA while solving DTLZ (as all the solutions are located in the border area), but deteriorates on DTLZ3. improves with larger µ. outperforms IBEA on both convergence (hypervolume, ɛ+) and front coverage perspectives. However, doesn t beat NSGA-II on most DTLZ problems. Over 8-fold speed-ups are obtained from comparing with IBEA enwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 9 /

11 Future Work DTLZ DTLZ2 3(.33).36().29(.6).79(.9).3(.7).(.6) DTLZ3.7(.9).(.7).8(.6) Figure: Solution distribution of on DTLZ-3 still have uneven distribution while solving DTLZ and gaps on DTLZ2 and 3 (see above figure). Systematically examine the courses of the gaps of IBEA s solution fronts on DTLZ2 and 3; Combining the investigation of hypervolume drops in SMS-EMOA to see if the gaps are caused by the computation of hypervolume; Improve and SMS-EMOA by redirecting its attention away from extreme points. 3, Markus Wagner3 (ASAP) Wenwen Li, Ender O zcan, Robert John John H. Drake2, Aneta Neumann June 3, 27 /

12 Thank you! Q & A Wenwen Li, Ender Özcan, Robert John John H. Drake 2, Aneta Neumann 3, Markus Wagner 3 (ASAP) June 3, 27 /