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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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Harmony Search for Multi-objective Optimization
Harmony Search for Multi-objectiveOptimization
Lucas M. PavelskiCarolina P. Almeida
Richard A. Goncalves
2012 Brazilian Symposium on Neural Networks — SBRN
October 25th, 2012
Pavelski, Almeida, Goncalves SBRN 2012 1 of 34
Harmony Search for Multi-objective Optimization
Summary
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 2 of 34
Harmony Search for Multi-objective Optimization
Summary
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 2 of 34
Harmony Search for Multi-objective Optimization
Summary
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 2 of 34
Harmony Search for Multi-objective Optimization
Summary
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 2 of 34
Harmony Search for Multi-objective Optimization
Summary
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 2 of 34
Harmony Search for Multi-objective OptimizationIntroduction
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 3 of 34
Harmony Search for Multi-objective OptimizationIntroduction
IntroductionI Multi-objective Optimization
I Extends Mono-objective OptimizationI Lack of extensive studying and comparison between
existing techniquesI Computationally expensive methods
I Harmony SearchI A recent, emergent metaheuristicI Little exploration of its operandsI Simple implementation
I Objectives:I Explore the Harmony Search in MO, using the
well-known NSGA-II frameworkI Test on 10 MO problems from CEC 2009I Evaluate results with statistical tests
Pavelski, Almeida, Goncalves SBRN 2012 4 of 34
Harmony Search for Multi-objective OptimizationIntroduction
IntroductionI Multi-objective Optimization
I Extends Mono-objective OptimizationI Lack of extensive studying and comparison between
existing techniquesI Computationally expensive methods
I Harmony SearchI A recent, emergent metaheuristicI Little exploration of its operandsI Simple implementation
I Objectives:I Explore the Harmony Search in MO, using the
well-known NSGA-II frameworkI Test on 10 MO problems from CEC 2009I Evaluate results with statistical tests
Pavelski, Almeida, Goncalves SBRN 2012 4 of 34
Harmony Search for Multi-objective OptimizationIntroduction
IntroductionI Multi-objective Optimization
I Extends Mono-objective OptimizationI Lack of extensive studying and comparison between
existing techniquesI Computationally expensive methods
I Harmony SearchI A recent, emergent metaheuristicI Little exploration of its operandsI Simple implementation
I Objectives:I Explore the Harmony Search in MO, using the
well-known NSGA-II frameworkI Test on 10 MO problems from CEC 2009I Evaluate results with statistical tests
Pavelski, Almeida, Goncalves SBRN 2012 4 of 34
Harmony Search for Multi-objective OptimizationBackground
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 5 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Multi-objective Optimization Problem
Mathematically [Deb 2011]:
Min/Max fm(x), m = 1, . . . ,Msubject to gj(x) ≥ 0, j = 1, . . . , J
hk(x) = 0, k = 1, . . . ,Kx (L)
i ≤ xi ≤ x (U)i i = 1, . . . , n
where f : Ω→ Y (⊆ <M)
Conflicting objectives Multiple optimal solutions
Pavelski, Almeida, Goncalves SBRN 2012 6 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Multi-objective Optimization Problem
Mathematically [Deb 2011]:
Min/Max fm(x), m = 1, . . . ,Msubject to gj(x) ≥ 0, j = 1, . . . , J
hk(x) = 0, k = 1, . . . ,Kx (L)
i ≤ xi ≤ x (U)i i = 1, . . . , n
where f : Ω→ Y (⊆ <M)
Conflicting objectives Multiple optimal solutions
Pavelski, Almeida, Goncalves SBRN 2012 6 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Pareto dominanceu ≺ 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, Goncalves SBRN 2012 7 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Multi-Objective Evolutionary Algorithms (MOEAs)
Two main issues in Multi-objective Optimization:[Zitzler 1999]:I Approximate Pareto-optimal solutionsI Maintain diversity
Evolutionary Algorithms:I Maintain a population of solutionsI Explore the solution’s similarities Multi-objective Evolutionary Algorithms (MOEAs), like theNSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 8 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Multi-Objective Evolutionary Algorithms (MOEAs)
Two main issues in Multi-objective Optimization:[Zitzler 1999]:I Approximate Pareto-optimal solutionsI Maintain diversity
Evolutionary Algorithms:I Maintain a population of solutionsI Explore the solution’s similarities Multi-objective Evolutionary Algorithms (MOEAs), like theNSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 8 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close Pareto-optimal
optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close Pareto-optimal
optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close
Pareto-optimal optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close Pareto-optimal
optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close Pareto-optimal
optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-dominated Sorting Genetic Algorithm II(NSGA-II)
I Proposed in [Deb et al. 2000]I Successfully applied to many problemsI Non-dominated sorting to obtain close Pareto-optimal
optimal solutionsI Crowding distance to maintain the diversityI Genetic Algorithm operands to create new solutionsI Basic framework is used in the proposed algorithms
Pavelski, Almeida, Goncalves SBRN 2012 9 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-Dominated Sorting Genetic algorithm(NSGA-II) [Deb et al. 2000]
Figure: Non-dominatedSorting [Zitzler 1999]
Figure: Non-dominated Selection[Deb et al. 2000]
Pavelski, Almeida, Goncalves SBRN 2012 10 of 34
Harmony Search for Multi-objective OptimizationBackground
MO and MOEAs
Non-Dominated Sorting Genetic algorithm(NSGA-II) [Deb et al. 2000]
Figure: Non-dominatedSelection [Deb et al. 2000]
+Figure: Crowdingdistance[Deb et al. 2000]
Pavelski, Almeida, Goncalves SBRN 2012 11 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search (HS) Overview
I New metaheuristic, proposed in[Geem, Kim e Loganathan 2001]
I Simplicity of implementation and customizationI Little exploration on MOI Inspired by jazz musicians: just like musical performers
seek an aesthetically good melody, by varying the set ofsounds played on each practice; the optimization seeksthe global optimum of a function, by evolving itscomponents variables on each iteration[Geem, Kim e Loganathan 2001].
Pavelski, Almeida, Goncalves SBRN 2012 12 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search (HS) Overview
I New metaheuristic, proposed in[Geem, Kim e Loganathan 2001]
I Simplicity of implementation and customizationI Little exploration on MOI Inspired by jazz musicians: just like musical performers
seek an aesthetically good melody, by varying the set ofsounds played on each practice; the optimization seeksthe global optimum of a function, by evolving itscomponents variables on each iteration[Geem, Kim e Loganathan 2001].
Pavelski, Almeida, Goncalves SBRN 2012 12 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search (HS) Overview
I New metaheuristic, proposed in[Geem, Kim e Loganathan 2001]
I Simplicity of implementation and customizationI Little exploration on MOI Inspired by jazz musicians: just like musical performers
seek an aesthetically good melody, by varying the set ofsounds played on each practice; the optimization seeksthe global optimum of a function, by evolving itscomponents variables on each iteration[Geem, Kim e Loganathan 2001].
Pavelski, Almeida, Goncalves SBRN 2012 12 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search (HS) Overview
I New metaheuristic, proposed in[Geem, Kim e Loganathan 2001]
I Simplicity of implementation and customizationI Little exploration on MOI Inspired by jazz musicians: just like musical
performers seek an aesthetically good melody, byvarying the set of sounds played on each practice;the optimization seeks the global optimum of afunction, by evolving its components variables oneach iteration [Geem, Kim e Loganathan 2001].
Pavelski, Almeida, Goncalves SBRN 2012 12 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search (HS)
Best state Global Optimum Fantastic HarmonyEstimated by Objective Function Aesthetic StandardEstimated with Values of Variables Pitches of InstrumentsProcess unit Each Iteration Each Practice
Pavelski, Almeida, Goncalves SBRN 2012 13 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search Algorithm1: function HarmonySearch2: /* 1. Harmony Memory Initialization */3: HM = xi ∈ Ω, i ∈ (1, . . . ,HMS)4: for t = 0, . . . ,NI do5: /* 2. Improvisation */6: xnew = improvise(HM)7: /* 3. Memory Update */8: xworst = minxi f (xi ), xi ∈ HM9: if f (xnew) > f (xworst) then
10: HM = (HM ∪ xnew) \ xworst11: end if12: end for13: end function
Pavelski, Almeida, Goncalves SBRN 2012 14 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search – Improvise Method1: function Improvise(HM) : xnew
2: for i = 0, . . . , n do3: if r1 < HMCR then4: /* 1. Memory Consideration */5: xnew
i = xki , k ∈ (1, . . . , HMS)
6: if r2 < PAR then7: /* 2. Pitch Adjustment */8: xnew
i = xnewi ± r3 × BW
9: end if10: else11: /* 3. Random Selection */12: xnew
i = x (L)i + r × (x (U)
i − x (L)i )
13: end if14: end for15: end function
Pavelski, Almeida, Goncalves SBRN 2012 15 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Harmony Search Variants
I HS: regular Harmony Search algorithmI IHS: Improved Harmony SearchI GHS: Global-best Harmony SearchI SGHS: Self-adaptive Global-best Harmony SearchI . . .
Pavelski, Almeida, Goncalves SBRN 2012 16 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Improved Harmony Search (IHS)
Fine adjustment of the parameters PAR and BW[Mahdavi, Fesanghary e Damangir 2007]:
PAR(t) = PARmin +(PARmax − PARmin)
NI × t (1)
BW (t) = BW max exp( ln
(BW minBW max
)NI × t
)(2)
Pavelski, Almeida, Goncalves SBRN 2012 17 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Global-best Harmony Search (GHS)
Inspired by swarm intelligence approaches, involves the bestharmony in the improvisation of new ones[Omran e Mahdavi 2008]:
function Improvise(HM) : xnew
. . .if r2 < PAR then
/* Pitch Adjustment */xnew
i = xbestk , k ∈ (1, . . . , n)
end if. . .
end function
Pavelski, Almeida, Goncalves SBRN 2012 18 of 34
Harmony Search for Multi-objective OptimizationBackground
HS and Variants
Self-adaptive Global-best Harmony Search (SGHS)Involves the best harmony and provides self-adaptation to thePAR and HMCR parameters [Pan et al. 2010]:
function Improvise(HM) : xnew
. . .if r1 < HMCR then
xnewi = xk
i ± r × BW , k ∈ (1, . . . , HMS)if r2 < PAR then
xnewi = xbest
iend if
end if. . .
end function
BW (t) =
BW max − BW max −BW min
NI if t < NI/2,BW min otherwise (3)
Pavelski, Almeida, Goncalves SBRN 2012 19 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 20 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Non-dominated Sorting Harmony Search – NSHS1: function nshs2: HM = xi ∈ Ω, i ∈ (1, . . . ,HMS)3: for j = 0, . . . ,NI/HMS do4: HMnew = ∅5: for k = 0, . . . ,HMS do6: xnew = improvise(HM)7: HMnew = HMnew ∪ xnew8: end for9: HM = HM ∪ HMnew
10: HM = NondominatedSorting(HM)11: end for12: end function
Pavelski, Almeida, Goncalves SBRN 2012 21 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Non-dominated Sorting Harmony Search – NSHS
I A different selection scheme: memory is doubledand non-dominated sorting + crowding distanceare applied
I NSIHS: t is the amount of harmonies improvisedI NSGHS: xbest
i is a random non-dominated solutionI NSSGHS: lp = HMS (a generation), learning from
solutions where cd > 0
Pavelski, Almeida, Goncalves SBRN 2012 22 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Non-dominated Sorting Harmony Search – NSHS
I A different selection scheme: memory is doubled andnon-dominated sorting + crowding distance are applied
I NSIHS: t is the amount of harmonies improvisedI NSGHS: xbest
i is a random non-dominated solutionI NSSGHS: lp = HMS (a generation), learning from
solutions where cd > 0
Pavelski, Almeida, Goncalves SBRN 2012 22 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Non-dominated Sorting Harmony Search – NSHS
I A different selection scheme: memory is doubled andnon-dominated sorting + crowding distance are applied
I NSIHS: t is the amount of harmonies improvisedI NSGHS: xbest
i is a random non-dominated solutionI NSSGHS: lp = HMS (a generation), learning from
solutions where cd > 0
Pavelski, Almeida, Goncalves SBRN 2012 22 of 34
Harmony Search for Multi-objective OptimizationProposed Algorithms
Non-dominated Sorting Harmony Search – NSHS
I A different selection scheme: memory is doubled andnon-dominated sorting + crowding distance are applied
I NSIHS: t is the amount of harmonies improvisedI NSGHS: xbest
i is a random non-dominated solutionI NSSGHS: lp = HMS (a generation), learning from
solutions where cd > 0
Pavelski, Almeida, Goncalves SBRN 2012 22 of 34
Harmony Search for Multi-objective OptimizationResults
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 23 of 34
Harmony Search for Multi-objective OptimizationResults
Problems
I 10 unconstrained (bound constrained) problems:UF1, UF2, . . . , UF10
I Taken from CEC 2009 [Zhang et al. 2009]I Difficult to solve, with different characteristicsI n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8,
. . . , UF10: 3 objetives
Pavelski, Almeida, Goncalves SBRN 2012 24 of 34
Harmony Search for Multi-objective OptimizationResults
Problems
I 10 unconstrained (bound constrained) problems: UF1,UF2, . . . , UF10
I Taken from CEC 2009 [Zhang et al. 2009]I Difficult to solve, with different characteristicsI n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8,
. . . , UF10: 3 objetives
Pavelski, Almeida, Goncalves SBRN 2012 24 of 34
Harmony Search for Multi-objective OptimizationResults
Problems
I 10 unconstrained (bound constrained) problems: UF1,UF2, . . . , UF10
I Taken from CEC 2009 [Zhang et al. 2009]I Difficult to solve, with different characteristicsI n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8,
. . . , UF10: 3 objetives
Pavelski, Almeida, Goncalves SBRN 2012 24 of 34
Harmony Search for Multi-objective OptimizationResults
Problems
I 10 unconstrained (bound constrained) problems: UF1,UF2, . . . , UF10
I Taken from CEC 2009 [Zhang et al. 2009]I Difficult to solve, with different characteristicsI n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives.
UF8, . . . , UF10: 3 objetives
Pavelski, Almeida, Goncalves SBRN 2012 24 of 34
Harmony Search for Multi-objective OptimizationResults
Parameters30 executions, 150.000 objective functions evaluations,population size or HMS of 200
HMCR PAR BWNSHS 0.95 0.10 0.01 ∗∆xNSIHS 0.95 PARmin = 0.01 BW min = 0.0001
PARmax = 0.20 BW max = 0.05 ∗∆xNSGHS 0.95 PARmin = 0.01 -
PARmax = 0.40NSSGHS 0.95 0.90 BW min = 0.001
BW max = 0.10 ∗∆xNSGA-II: polynomial mutation with probability 1/n and SBXcrossover with probability 0.7.
Pavelski, Almeida, Goncalves SBRN 2012 25 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I Non-parametric tests, PISA framework[Zitzler, Knowles e Thiele 2008]
I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε and R2
I Overall performance of each algorithm(macro-evaluation): Mack-Skillings variation of theFriedman test [Mack e Skillings 1980]
I Each algorithm, each instance (micro-evaluation):Kruskal-Wallis test
Pavelski, Almeida, Goncalves SBRN 2012 26 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I Non-parametric tests, PISA framework[Zitzler, Knowles e Thiele 2008]
I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε and R2
I Overall performance of each algorithm(macro-evaluation): Mack-Skillings variation of theFriedman test [Mack e Skillings 1980]
I Each algorithm, each instance (micro-evaluation):Kruskal-Wallis test
Pavelski, Almeida, Goncalves SBRN 2012 26 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I Non-parametric tests, PISA framework[Zitzler, Knowles e Thiele 2008]
I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε
and R2
I Overall performance of each algorithm(macro-evaluation): Mack-Skillings variation of theFriedman test [Mack e Skillings 1980]
I Each algorithm, each instance (micro-evaluation):Kruskal-Wallis test
Pavelski, Almeida, Goncalves SBRN 2012 26 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I Non-parametric tests, PISA framework[Zitzler, Knowles e Thiele 2008]
I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε and R2
I Overall performance of each algorithm(macro-evaluation): Mack-Skillings variation of theFriedman test [Mack e Skillings 1980]
I Each algorithm, each instance (micro-evaluation):Kruskal-Wallis test
Pavelski, Almeida, Goncalves SBRN 2012 26 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I Non-parametric tests, PISA framework[Zitzler, Knowles e Thiele 2008]
I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε and R2
I Overall performance of each algorithm(macro-evaluation): Mack-Skillings variation of theFriedman test [Mack e Skillings 1980]
I Each algorithm, each instance (micro-evaluation):Kruskal-Wallis test
Pavelski, Almeida, Goncalves SBRN 2012 26 of 34
Harmony Search for Multi-objective OptimizationResults
Kurskal-Wallis test for hypervolumeNSHS 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 1.0UF2 0.65 0.21 1.0 0.12 1.0 1.0UF3 0.09 0.0 0.0 0.0 0.0 0.0UF4 0.94 0.0 0.96 0.0 0.59 1.0UF5 0.07 0.0 0.0 0.0 0.0 0.07UF6 0.5 0.5 0.5 0.5 0.5 0.5UF7 0.02 0.02 0.8 0.55 1.0 1.0UF8 0.25 1.0 0.0 1.0 0.0 0.0UF9 0.97 0.11 0.07 0.0 0.0 0.41UF10 0.0 0.0 0.0 0.25 0.01 0.04
Pavelski, Almeida, Goncalves SBRN 2012 27 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I NSHS was among the best algorithms for solvingUF3, UF5, UF6, UF7, UF9 and UF10.
I NSIHS, many times incomparable to NSHS, had a goodperformance on in UF3, UF4, UF5, UF6 and UF9.
I NSGHS obtained good results on the 3 objectiveproblems, namely UF8, UF9 and UF10.
I NSSGHS performed well on UF1, UF4 and UF7.
Pavelski, Almeida, Goncalves SBRN 2012 28 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I NSHS was among the best algorithms for solving UF3,UF5, UF6, UF7, UF9 and UF10.
I NSIHS, many times incomparable to NSHS, had agood performance on in UF3, UF4, UF5, UF6 andUF9.
I NSGHS obtained good results on the 3 objectiveproblems, namely UF8, UF9 and UF10.
I NSSGHS performed well on UF1, UF4 and UF7.
Pavelski, Almeida, Goncalves SBRN 2012 28 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I NSHS was among the best algorithms for solving UF3,UF5, UF6, UF7, UF9 and UF10.
I NSIHS, many times incomparable to NSHS, had a goodperformance on in UF3, UF4, UF5, UF6 and UF9.
I NSGHS obtained good results on the 3 objectiveproblems, namely UF8, UF9 and UF10.
I NSSGHS performed well on UF1, UF4 and UF7.
Pavelski, Almeida, Goncalves SBRN 2012 28 of 34
Harmony Search for Multi-objective OptimizationResults
Quality Indicators and Statistical Tests
I NSHS was among the best algorithms for solving UF3,UF5, UF6, UF7, UF9 and UF10.
I NSIHS, many times incomparable to NSHS, had a goodperformance on in UF3, UF4, UF5, UF6 and UF9.
I NSGHS obtained good results on the 3 objectiveproblems, namely UF8, UF9 and UF10.
I NSSGHS performed well on UF1, UF4 and UF7.
Pavelski, Almeida, Goncalves SBRN 2012 28 of 34
Harmony Search for Multi-objective OptimizationResults
Comparison against NSGA-II (Mann-Whitney)Hyp. Unary-ε R2
UF1 0.11 0.00 0.08UF2 1.00 0.03 1.00UF3 0.00 0.00 0.00UF4 1.00 1.00 1.00UF5 0.00 0.00 0.00UF6 0.00 0.00 0.00UF7 0.54 0.45 0.57UF8 0.00 0.00 0.00UF9 1.00 0.02 0.64
UF10 0.00 0.00 0.00
MS Friedman test: criticaldifference of 2.795.I Hypervolume: 28.87 for
NSHS and 32.13 forNSGA-II
I Unary-ε: 23.57 forNSHS and 37.43 forNSGA-II
I R2: 27.83 for NSHSand 33.16 for NSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 29 of 34
Harmony Search for Multi-objective OptimizationConclusions
Introduction
BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants
Proposed Algorithms
Experimental Results
Conclusions
Pavelski, Almeida, Goncalves SBRN 2012 30 of 34
Harmony Search for Multi-objective OptimizationConclusions
Conclusions
I Objectives: propose hybridization of four HSversions with the NSGA-II framework, runbenchmark functions used in CEC 2009, evaluateresults with quality indicators
I Tests showed that NSHS, the original HS algorithm usingnon-dominated sorting, was the best among all proposedmulti-objective versions
I NSHS algorithm was favorably compared with the originalNSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 31 of 34
Harmony Search for Multi-objective OptimizationConclusions
Conclusions
I Objectives: propose hybridization of four HS versions withthe NSGA-II framework, run benchmark functions used inCEC 2009, evaluate results with quality indicators
I Tests showed that NSHS, the original HS algorithmusing non-dominated sorting, was the best amongall proposed multi-objective versions
I NSHS algorithm was favorably compared with the originalNSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 31 of 34
Harmony Search for Multi-objective OptimizationConclusions
Conclusions
I Objectives: propose hybridization of four HS versions withthe NSGA-II framework, run benchmark functions used inCEC 2009, evaluate results with quality indicators
I Tests showed that NSHS, the original HS algorithm usingnon-dominated sorting, was the best among all proposedmulti-objective versions
I NSHS algorithm was favorably compared with theoriginal NSGA-II
Pavelski, Almeida, Goncalves SBRN 2012 31 of 34
Harmony Search for Multi-objective OptimizationConclusions
Future works
I Effects of other HS variants and parameter valuesin problems with different characteristics
I Analysis of different aspects: computational effort, andcomparisons against other MOEAs, etc
I Adaptation of HS operators on other state-of-artframeworks (in progress)
Pavelski, Almeida, Goncalves SBRN 2012 32 of 34
Harmony Search for Multi-objective OptimizationConclusions
Future works
I Effects of other HS variants and parameter values inproblems with different characteristics
I Analysis of different aspects: computational effort,and comparisons against other MOEAs, etc
I Adaptation of HS operators on other state-of-artframeworks (in progress)
Pavelski, Almeida, Goncalves SBRN 2012 32 of 34
Harmony Search for Multi-objective OptimizationConclusions
Future works
I Effects of other HS variants and parameter values inproblems with different characteristics
I Analysis of different aspects: computational effort, andcomparisons against other MOEAs, etc
I Adaptation of HS operators on other state-of-artframeworks (in progress)
Pavelski, Almeida, Goncalves SBRN 2012 32 of 34
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Acknowledgments
I Fundacao AraucariaI UNICENTROI Friends and colleagues
Thank you for your attention!Questions?
Acknowledgments
I Fundacao AraucariaI UNICENTROI Friends and colleagues
Thank you for your attention!Questions?