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Harmony Search for Multi-objective Optimization - SBRN 2012

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Slides used in the presentation of the article "Harmony Search for Multi-objective
Optimization" in the 2012 Brazilian Symposium on Neural Networks (SBRN). Link to the article: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=6374852

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  • 1. Harmony Search for Multi-objective Optimization Harmony Search for Multi-objective Optimization Lucas M. Pavelski Carolina P. Almeida Richard A. Goncalves ¸ 2012 Brazilian Symposium on Neural Networks — SBRN October 25th , 2012Pavelski, Almeida, Gon¸alves c SBRN 2012 1 of 34
  • 2. Harmony Search for Multi-objective OptimizationSummary Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 2 of 34
  • 3. Harmony Search for Multi-objective OptimizationSummary Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 2 of 34
  • 4. Harmony Search for Multi-objective OptimizationSummary Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 2 of 34
  • 5. Harmony Search for Multi-objective OptimizationSummary Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 2 of 34
  • 6. Harmony Search for Multi-objective OptimizationSummary Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 2 of 34
  • 7. Harmony Search for Multi-objective Optimization Introduction Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 3 of 34
  • 8. Harmony Search for Multi-objective Optimization IntroductionIntroduction Multi-objective Optimization Extends Mono-objective Optimization Lack of extensive studying and comparison between existing techniques Computationally expensive methods Harmony Search A recent, emergent metaheuristic Little exploration of its operands Simple implementation Objectives: Explore the Harmony Search in MO, using the well-known NSGA-II framework Test on 10 MO problems from CEC 2009 Evaluate results with statistical testsPavelski, Almeida, Gon¸alves c SBRN 2012 4 of 34
  • 9. Harmony Search for Multi-objective Optimization IntroductionIntroduction Multi-objective Optimization Extends Mono-objective Optimization Lack of extensive studying and comparison between existing techniques Computationally expensive methods Harmony Search A recent, emergent metaheuristic Little exploration of its operands Simple implementation Objectives: Explore the Harmony Search in MO, using the well-known NSGA-II framework Test on 10 MO problems from CEC 2009 Evaluate results with statistical testsPavelski, Almeida, Gon¸alves c SBRN 2012 4 of 34
  • 10. Harmony Search for Multi-objective Optimization IntroductionIntroduction Multi-objective Optimization Extends Mono-objective Optimization Lack of extensive studying and comparison between existing techniques Computationally expensive methods Harmony Search A recent, emergent metaheuristic Little exploration of its operands Simple implementation Objectives: Explore the Harmony Search in MO, using the well-known NSGA-II framework Test on 10 MO problems from CEC 2009 Evaluate results with statistical testsPavelski, Almeida, Gon¸alves c SBRN 2012 4 of 34
  • 11. Harmony Search for Multi-objective Optimization Background Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 5 of 34
  • 12. Harmony Search for Multi-objective Optimization Background MO and MOEAsMulti-objective Optimization Problem Mathematically [Deb 2011]: Min/Max fm (x), m = 1, . . . , M subject to gj (x) ≥ 0, j = 1, . . . , J hk (x) = 0, k = 1, . . . , K (L) (U) xi ≤ xi ≤ xi i = 1, . . . , n M where f : Ω → Y (⊆ ) Conflicting objectives Multiple optimal solutionsPavelski, Almeida, Gon¸alves c SBRN 2012 6 of 34
  • 13. Harmony Search for Multi-objective Optimization Background MO and MOEAsMulti-objective Optimization Problem Mathematically [Deb 2011]: Min/Max fm (x), m = 1, . . . , M subject to gj (x) ≥ 0, j = 1, . . . , J hk (x) = 0, k = 1, . . . , K (L) (U) xi ≤ xi ≤ xi i = 1, . . . , n M where f : Ω → Y (⊆ ) Conflicting objectives Multiple optimal solutionsPavelski, Almeida, Gon¸alves c SBRN 2012 6 of 34
  • 14. Harmony Search for Multi-objective Optimization Background MO and MOEAsPareto dominance u v: ∀i ∈ {1, . . . , M}, ui ≥ vi and ∃i ∈ {1, . . . , M} : ui < vi [Coello, Lamont e Veldhuizen 2007] Figure: Graphical representation of Pareto dominance [Zitzler 1999]Pavelski, Almeida, Gon¸alves c SBRN 2012 7 of 34
  • 15. Harmony Search for Multi-objective Optimization Background MO and MOEAsMulti-Objective Evolutionary Algorithms (MOEAs) Two main issues in Multi-objective Optimization: [Zitzler 1999]: Approximate Pareto-optimal solutions Maintain diversity Evolutionary Algorithms: Maintain a population of solutions Explore the solution’s similarities Multi-objective Evolutionary Algorithms (MOEAs), like the NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 8 of 34
  • 16. Harmony Search for Multi-objective Optimization Background MO and MOEAsMulti-Objective Evolutionary Algorithms (MOEAs) Two main issues in Multi-objective Optimization: [Zitzler 1999]: Approximate Pareto-optimal solutions Maintain diversity Evolutionary Algorithms: Maintain a population of solutions Explore the solution’s similarities Multi-objective Evolutionary Algorithms (MOEAs), like the NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 8 of 34
  • 17. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 18. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 19. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 20. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 21. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 22. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-dominated Sorting Genetic Algorithm II(NSGA-II) Proposed in [Deb et al. 2000] Successfully applied to many problems Non-dominated sorting to obtain close Pareto-optimal optimal solutions Crowding distance to maintain the diversity Genetic Algorithm operands to create new solutions Basic framework is used in the proposed algorithmsPavelski, Almeida, Gon¸alves c SBRN 2012 9 of 34
  • 23. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-Dominated Sorting Genetic algorithm(NSGA-II) [Deb et al. 2000] Figure: Non-dominated Selection Figure: Non-dominated [Deb et al. 2000] Sorting [Zitzler 1999]Pavelski, Almeida, Gon¸alves c SBRN 2012 10 of 34
  • 24. Harmony Search for Multi-objective Optimization Background MO and MOEAsNon-Dominated Sorting Genetic algorithm(NSGA-II) [Deb et al. 2000] + Figure: Crowding Figure: Non-dominated distance Selection [Deb et al. 2000] [Deb et al. 2000]Pavelski, Almeida, Gon¸alves c SBRN 2012 11 of 34
  • 25. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search (HS) Overview New metaheuristic, proposed in [Geem, Kim e Loganathan 2001] Simplicity of implementation and customization Little exploration on MO Inspired by jazz musicians: just like musical performers seek an aesthetically good melody, by varying the set of sounds played on each practice; the optimization seeks the global optimum of a function, by evolving its components variables on each iteration [Geem, Kim e Loganathan 2001].Pavelski, Almeida, Gon¸alves c SBRN 2012 12 of 34
  • 26. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search (HS) Overview New metaheuristic, proposed in [Geem, Kim e Loganathan 2001] Simplicity of implementation and customization Little exploration on MO Inspired by jazz musicians: just like musical performers seek an aesthetically good melody, by varying the set of sounds played on each practice; the optimization seeks the global optimum of a function, by evolving its components variables on each iteration [Geem, Kim e Loganathan 2001].Pavelski, Almeida, Gon¸alves c SBRN 2012 12 of 34
  • 27. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search (HS) Overview New metaheuristic, proposed in [Geem, Kim e Loganathan 2001] Simplicity of implementation and customization Little exploration on MO Inspired by jazz musicians: just like musical performers seek an aesthetically good melody, by varying the set of sounds played on each practice; the optimization seeks the global optimum of a function, by evolving its components variables on each iteration [Geem, Kim e Loganathan 2001].Pavelski, Almeida, Gon¸alves c SBRN 2012 12 of 34
  • 28. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search (HS) Overview New metaheuristic, proposed in [Geem, Kim e Loganathan 2001] Simplicity of implementation and customization Little exploration on MO Inspired by jazz musicians: just like musical performers seek an aesthetically good melody, by varying the set of sounds played on each practice; the optimization seeks the global optimum of a function, by evolving its components variables on each iteration [Geem, Kim e Loganathan 2001].Pavelski, Almeida, Gon¸alves c SBRN 2012 12 of 34
  • 29. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search (HS) Best state Global Optimum Fantastic Harmony Estimated by Objective Function Aesthetic Standard Estimated with Values of Variables Pitches of Instruments Process unit Each Iteration Each PracticePavelski, Almeida, Gon¸alves c SBRN 2012 13 of 34
  • 30. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search Algorithm 1: function HarmonySearch 2: /* 1. Harmony Memory Initialization */ 3: HM = xi ∈ Ω, i ∈ (1, . . . , HMS) 4: for t = 0, . . . , NI do 5: /* 2. Improvisation */ 6: x new = improvise(HM) 7: /* 3. Memory Update */ 8: x worst = minxi f (xi ), xi ∈ HM 9: if f (x new ) > f (x worst ) then 10: HM = (HM ∪ {x new }) {x worst } 11: end if 12: end for 13: end functionPavelski, Almeida, Gon¸alves c SBRN 2012 14 of 34
  • 31. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search – Improvise Method 1: function Improvise(HM) : x new 2: for i = 0, . . . , n do 3: if r1 < HMCR then 4: /* 1. Memory Consideration */ 5: xinew = xik , k ∈ (1, . . . , HMS) 6: if r2 < PAR then 7: /* 2. Pitch Adjustment */ 8: xinew = xinew ± r3 × BW 9: end if 10: else 11: /* 3. Random Selection */ (L) (U) (L) 12: xinew = xi + r × (xi − xi ) 13: end if 14: end for 15: end functionPavelski, Almeida, Gon¸alves c SBRN 2012 15 of 34
  • 32. Harmony Search for Multi-objective Optimization Background HS and VariantsHarmony Search Variants HS: regular Harmony Search algorithm IHS: Improved Harmony Search GHS: Global-best Harmony Search SGHS: Self-adaptive Global-best Harmony Search ...Pavelski, Almeida, Gon¸alves c SBRN 2012 16 of 34
  • 33. Harmony Search for Multi-objective Optimization Background HS and VariantsImproved Harmony Search (IHS) Fine adjustment of the parameters PAR and BW [Mahdavi, Fesanghary e Damangir 2007]: (PAR max − PAR min ) PAR(t) = PAR min + ×t (1) NI BW min ln BW max BW (t) = BW max exp ×t (2) NIPavelski, Almeida, Gon¸alves c SBRN 2012 17 of 34
  • 34. Harmony Search for Multi-objective Optimization Background HS and VariantsGlobal-best Harmony Search (GHS) Inspired by swarm intelligence approaches, involves the best harmony in the improvisation of new ones [Omran e Mahdavi 2008]: function Improvise(HM) : x new ... if r2 < PAR then /* Pitch Adjustment */ xinew = xk , k ∈ (1, . . . , n) best end if ... end functionPavelski, Almeida, Gon¸alves c SBRN 2012 18 of 34
  • 35. Harmony Search for Multi-objective Optimization Background HS and VariantsSelf-adaptive Global-best Harmony Search (SGHS) Involves the best harmony and provides self-adaptation to the PAR and HMCR parameters [Pan et al. 2010]: function Improvise(HM) : x new ... if r1 < HMCR then xinew = xik ± r × BW , k ∈ (1, . . . , HMS) if r2 < PAR then xinew = xibest end if end if ... end function BW max −BW min BW max − NI if t < NI/2, BW (t) = (3) BW min otherwisePavelski, Almeida, Gon¸alves c SBRN 2012 19 of 34
  • 36. Harmony Search for Multi-objective Optimization Proposed Algorithms Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 20 of 34
  • 37. Harmony Search for Multi-objective Optimization Proposed AlgorithmsNon-dominated Sorting Harmony Search – NSHS 1: function nshs 2: HM = xi ∈ Ω, i ∈ (1, . . . , HMS) 3: for j = 0, . . . , NI/HMS do 4: HM new = ∅ 5: for k = 0, . . . , HMS do 6: x new = improvise(HM) 7: HM new = HM new ∪ {x new } 8: end for 9: HM = HM ∪ HM new 10: HM = NondominatedSorting(HM) 11: end for 12: end functionPavelski, Almeida, Gon¸alves c SBRN 2012 21 of 34
  • 38. Harmony Search for Multi-objective Optimization Proposed AlgorithmsNon-dominated Sorting Harmony Search – NSHS A different selection scheme: memory is doubled and non-dominated sorting + crowding distance are applied NSIHS: t is the amount of harmonies improvised NSGHS: xibest is a random non-dominated solution NSSGHS: lp = HMS (a generation), learning from solutions where cd > 0Pavelski, Almeida, Gon¸alves c SBRN 2012 22 of 34
  • 39. Harmony Search for Multi-objective Optimization Proposed AlgorithmsNon-dominated Sorting Harmony Search – NSHS A different selection scheme: memory is doubled and non-dominated sorting + crowding distance are applied NSIHS: t is the amount of harmonies improvised NSGHS: xibest is a random non-dominated solution NSSGHS: lp = HMS (a generation), learning from solutions where cd > 0Pavelski, Almeida, Gon¸alves c SBRN 2012 22 of 34
  • 40. Harmony Search for Multi-objective Optimization Proposed AlgorithmsNon-dominated Sorting Harmony Search – NSHS A different selection scheme: memory is doubled and non-dominated sorting + crowding distance are applied NSIHS: t is the amount of harmonies improvised NSGHS: xibest is a random non-dominated solution NSSGHS: lp = HMS (a generation), learning from solutions where cd > 0Pavelski, Almeida, Gon¸alves c SBRN 2012 22 of 34
  • 41. Harmony Search for Multi-objective Optimization Proposed AlgorithmsNon-dominated Sorting Harmony Search – NSHS A different selection scheme: memory is doubled and non-dominated sorting + crowding distance are applied NSIHS: t is the amount of harmonies improvised NSGHS: xibest is a random non-dominated solution NSSGHS: lp = HMS (a generation), learning from solutions where cd > 0Pavelski, Almeida, Gon¸alves c SBRN 2012 22 of 34
  • 42. Harmony Search for Multi-objective Optimization Results Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 23 of 34
  • 43. Harmony Search for Multi-objective Optimization ResultsProblems 10 unconstrained (bound constrained) problems: UF1, UF2, . . . , UF10 Taken from CEC 2009 [Zhang et al. 2009] Difficult to solve, with different characteristics n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8, . . . , UF10: 3 objetivesPavelski, Almeida, Gon¸alves c SBRN 2012 24 of 34
  • 44. Harmony Search for Multi-objective Optimization ResultsProblems 10 unconstrained (bound constrained) problems: UF1, UF2, . . . , UF10 Taken from CEC 2009 [Zhang et al. 2009] Difficult to solve, with different characteristics n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8, . . . , UF10: 3 objetivesPavelski, Almeida, Gon¸alves c SBRN 2012 24 of 34
  • 45. Harmony Search for Multi-objective Optimization ResultsProblems 10 unconstrained (bound constrained) problems: UF1, UF2, . . . , UF10 Taken from CEC 2009 [Zhang et al. 2009] Difficult to solve, with different characteristics n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8, . . . , UF10: 3 objetivesPavelski, Almeida, Gon¸alves c SBRN 2012 24 of 34
  • 46. Harmony Search for Multi-objective Optimization ResultsProblems 10 unconstrained (bound constrained) problems: UF1, UF2, . . . , UF10 Taken from CEC 2009 [Zhang et al. 2009] Difficult to solve, with different characteristics n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8, . . . , UF10: 3 objetivesPavelski, Almeida, Gon¸alves c SBRN 2012 24 of 34
  • 47. Harmony Search for Multi-objective Optimization ResultsParameters 30 executions, 150.000 objective functions evaluations, population size or HMS of 200 HMCR PAR BW NSHS 0.95 0.10 0.01 ∗ ∆x NSIHS 0.95 PAR min = 0.01 BW min = 0.0001 PAR max = 0.20 BW max = 0.05 ∗ ∆x NSGHS 0.95 PAR min = 0.01 - PAR max = 0.40 NSSGHS 0.95 0.90 BW min = 0.001 BW max = 0.10 ∗ ∆x NSGA-II: polynomial mutation with probability 1/n and SBX crossover with probability 0.7.Pavelski, Almeida, Gon¸alves c SBRN 2012 25 of 34
  • 48. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests Non-parametric tests, PISA framework [Zitzler, Knowles e Thiele 2008] Mann-Whitney and dominance ranking Quality indicators: hypervolume, additive unary- and R2 Overall performance of each algorithm (macro-evaluation): Mack-Skillings variation of the Friedman test [Mack e Skillings 1980] Each algorithm, each instance (micro-evaluation): Kruskal-Wallis testPavelski, Almeida, Gon¸alves c SBRN 2012 26 of 34
  • 49. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests Non-parametric tests, PISA framework [Zitzler, Knowles e Thiele 2008] Mann-Whitney and dominance ranking Quality indicators: hypervolume, additive unary- and R2 Overall performance of each algorithm (macro-evaluation): Mack-Skillings variation of the Friedman test [Mack e Skillings 1980] Each algorithm, each instance (micro-evaluation): Kruskal-Wallis testPavelski, Almeida, Gon¸alves c SBRN 2012 26 of 34
  • 50. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests Non-parametric tests, PISA framework [Zitzler, Knowles e Thiele 2008] Mann-Whitney and dominance ranking Quality indicators: hypervolume, additive unary- and R2 Overall performance of each algorithm (macro-evaluation): Mack-Skillings variation of the Friedman test [Mack e Skillings 1980] Each algorithm, each instance (micro-evaluation): Kruskal-Wallis testPavelski, Almeida, Gon¸alves c SBRN 2012 26 of 34
  • 51. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests Non-parametric tests, PISA framework [Zitzler, Knowles e Thiele 2008] Mann-Whitney and dominance ranking Quality indicators: hypervolume, additive unary- and R2 Overall performance of each algorithm (macro-evaluation): Mack-Skillings variation of the Friedman test [Mack e Skillings 1980] Each algorithm, each instance (micro-evaluation): Kruskal-Wallis testPavelski, Almeida, Gon¸alves c SBRN 2012 26 of 34
  • 52. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests Non-parametric tests, PISA framework [Zitzler, Knowles e Thiele 2008] Mann-Whitney and dominance ranking Quality indicators: hypervolume, additive unary- and R2 Overall performance of each algorithm (macro-evaluation): Mack-Skillings variation of the Friedman test [Mack e Skillings 1980] Each algorithm, each instance (micro-evaluation): Kruskal-Wallis testPavelski, Almeida, Gon¸alves c SBRN 2012 26 of 34
  • 53. Harmony Search for Multi-objective Optimization ResultsKurskal-Wallis test for hypervolume NSHS NSHS NSHS NSIHS NSIHS NSGHS x x x x x x NSIHS NSGHS NSSGHS NSGHS NSSGHS NSSGHS UF1 0.19 0.42 1.0 0.75 1.0 1.0 UF2 0.65 0.21 1.0 0.12 1.0 1.0 UF3 0.09 0.0 0.0 0.0 0.0 0.0 UF4 0.94 0.0 0.96 0.0 0.59 1.0 UF5 0.07 0.0 0.0 0.0 0.0 0.07 UF6 0.5 0.5 0.5 0.5 0.5 0.5 UF7 0.02 0.02 0.8 0.55 1.0 1.0 UF8 0.25 1.0 0.0 1.0 0.0 0.0 UF9 0.97 0.11 0.07 0.0 0.0 0.41 UF10 0.0 0.0 0.0 0.25 0.01 0.04Pavelski, Almeida, Gon¸alves c SBRN 2012 27 of 34
  • 54. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests NSHS was among the best algorithms for solving UF3, UF5, UF6, UF7, UF9 and UF10. NSIHS, many times incomparable to NSHS, had a good performance on in UF3, UF4, UF5, UF6 and UF9. NSGHS obtained good results on the 3 objective problems, namely UF8, UF9 and UF10. NSSGHS performed well on UF1, UF4 and UF7.Pavelski, Almeida, Gon¸alves c SBRN 2012 28 of 34
  • 55. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests NSHS was among the best algorithms for solving UF3, UF5, UF6, UF7, UF9 and UF10. NSIHS, many times incomparable to NSHS, had a good performance on in UF3, UF4, UF5, UF6 and UF9. NSGHS obtained good results on the 3 objective problems, namely UF8, UF9 and UF10. NSSGHS performed well on UF1, UF4 and UF7.Pavelski, Almeida, Gon¸alves c SBRN 2012 28 of 34
  • 56. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests NSHS was among the best algorithms for solving UF3, UF5, UF6, UF7, UF9 and UF10. NSIHS, many times incomparable to NSHS, had a good performance on in UF3, UF4, UF5, UF6 and UF9. NSGHS obtained good results on the 3 objective problems, namely UF8, UF9 and UF10. NSSGHS performed well on UF1, UF4 and UF7.Pavelski, Almeida, Gon¸alves c SBRN 2012 28 of 34
  • 57. Harmony Search for Multi-objective Optimization ResultsQuality Indicators and Statistical Tests NSHS was among the best algorithms for solving UF3, UF5, UF6, UF7, UF9 and UF10. NSIHS, many times incomparable to NSHS, had a good performance on in UF3, UF4, UF5, UF6 and UF9. NSGHS obtained good results on the 3 objective problems, namely UF8, UF9 and UF10. NSSGHS performed well on UF1, UF4 and UF7.Pavelski, Almeida, Gon¸alves c SBRN 2012 28 of 34
  • 58. Harmony Search for Multi-objective Optimization ResultsComparison against NSGA-II (Mann-Whitney) Hyp. Unary- R2 MS Friedman test: critical UF1 0.11 0.00 0.08 difference of 2.795. UF2 1.00 0.03 1.00 Hypervolume: 28.87 for UF3 0.00 0.00 0.00 NSHS and 32.13 for UF4 1.00 1.00 1.00 NSGA-II UF5 0.00 0.00 0.00 UF6 0.00 0.00 0.00 Unary- : 23.57 for UF7 0.54 0.45 0.57 NSHS and 37.43 for UF8 0.00 0.00 0.00 NSGA-II UF9 1.00 0.02 0.64 R2 : 27.83 for NSHS UF10 0.00 0.00 0.00 and 33.16 for NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 29 of 34
  • 59. Harmony Search for Multi-objective Optimization Conclusions Introduction Background Multi-objective Optimization and MOEAs Harmony Search and Variants Proposed Algorithms Experimental Results ConclusionsPavelski, Almeida, Gon¸alves c SBRN 2012 30 of 34
  • 60. Harmony Search for Multi-objective Optimization ConclusionsConclusions Objectives: propose hybridization of four HS versions with the NSGA-II framework, run benchmark functions used in CEC 2009, evaluate results with quality indicators Tests showed that NSHS, the original HS algorithm using non-dominated sorting, was the best among all proposed multi-objective versions NSHS algorithm was favorably compared with the original NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 31 of 34
  • 61. Harmony Search for Multi-objective Optimization ConclusionsConclusions Objectives: propose hybridization of four HS versions with the NSGA-II framework, run benchmark functions used in CEC 2009, evaluate results with quality indicators Tests showed that NSHS, the original HS algorithm using non-dominated sorting, was the best among all proposed multi-objective versions NSHS algorithm was favorably compared with the original NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 31 of 34
  • 62. Harmony Search for Multi-objective Optimization ConclusionsConclusions Objectives: propose hybridization of four HS versions with the NSGA-II framework, run benchmark functions used in CEC 2009, evaluate results with quality indicators Tests showed that NSHS, the original HS algorithm using non-dominated sorting, was the best among all proposed multi-objective versions NSHS algorithm was favorably compared with the original NSGA-IIPavelski, Almeida, Gon¸alves c SBRN 2012 31 of 34
  • 63. Harmony Search for Multi-objective Optimization ConclusionsFuture works Effects of other HS variants and parameter values in problems with different characteristics Analysis of different aspects: computational effort, and comparisons against other MOEAs, etc Adaptation of HS operators on other state-of-art frameworks (in progress)Pavelski, Almeida, Gon¸alves c SBRN 2012 32 of 34
  • 64. Harmony Search for Multi-objective Optimization ConclusionsFuture works Effects of other HS variants and parameter values in problems with different characteristics Analysis of different aspects: computational effort, and comparisons against other MOEAs, etc Adaptation of HS operators on other state-of-art frameworks (in progress)Pavelski, Almeida, Gon¸alves c SBRN 2012 32 of 34
  • 65. Harmony Search for Multi-objective Optimization ConclusionsFuture works Effects of other HS variants and parameter values in problems with different characteristics Analysis of different aspects: computational effort, and comparisons against other MOEAs, etc Adaptation of HS operators on other state-of-art frameworks (in progress)Pavelski, Almeida, Gon¸alves c SBRN 2012 32 of 34
  • 66. Bibliographic References COELLO, C. A. C.; LAMONT, G. B.; VELDHUIZEN, D. A. V. Evolutionary Algorithms for Solving Multi-Objective Problems. 2. ed. USA: Springer, 2007. DEB, K. Multi-Objective Optimization Using Evolutionary Algorithms: An Introduction. [S.l.], 2011. DEB, K. et al. A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II. In: Proceedings of the 6th International Conference on Parallel Problem Solving from Nature. London, UK, UK: Springer-Verlag, 2000. (PPSN VI), p. 849–858. ISBN 3-540-41056-2. GEEM, Z. W.; KIM, J. H.; LOGANATHAN, G. A new heuristic optimization algorithm: Harmony search. SIMULATION, v. 76, n. 2, p. 60–68, 2001. MACK, G. A.; SKILLINGS, J. H. A friedman-type rank test for main effects in a two-factor anova. Journal of the American Statistical Association, v. 75, n. 372, p. 947–951, 1980. MAHDAVI, M.; FESANGHARY, M.; DAMANGIR, E. An improved harmony search algorithm for solving optimization problems. Applied Mathematics and Computation, v. 188, n. 2, p. 1567–1579, maio 2007. OMRAN, M.; MAHDAVI, M. Global-best harmony search. Applied Mathematics and Computation, v. 198, n. 2, p. 643–656, 2008. PAN, Q.-K. et al. A self-adaptive global best harmonysearch algorithm for continuous optimization problems. Applied Mathematics and Computation, v. 216, n. 3, p. 830 – 848, 2010. ZHANG, Q. et al. Multiobjective optimization test instances for the cec 2009 special session and competition. Mechanical Engineering, CEC2009, n. CES-487, p. 1–30, 2009. ZITZLER, E. Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. Tese (Doutorado) — ETH Zurich, Switzerland, 1999. ZITZLER, E.; KNOWLES, J.; THIELE, L. Quality assessment of pareto set approximations. Springer-Verlag, Berlin, Heidelberg, p. 373–404, 2008.
  • 67. Acknowledgments Fundac˜o Arauc´ria ¸a a UNICENTRO Friends and colleagues Thank you for your attention! Questions?
  • 68. Acknowledgments Fundac˜o Arauc´ria ¸a a UNICENTRO Friends and colleagues Thank you for your attention! Questions?

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