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

Summary

Introduction

BackgroundMulti-objective Optimization and MOEAsHarmony Search and Variants

Proposed Algorithms

Experimental Results

Conclusions

Summary

Introduction

Proposed Algorithms

Conclusions

Summary

Introduction

Proposed Algorithms

Conclusions

Summary

Introduction

Proposed Algorithms

Conclusions

Summary

Introduction

Proposed Algorithms

Conclusions

Harmony Search for Multi-objective OptimizationIntroduction

Introduction

Proposed Algorithms

Conclusions

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

Harmony Search for Multi-objective OptimizationBackground

Introduction

Proposed Algorithms

Conclusions

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

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

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]

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

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

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

MO and MOEAs

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

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]

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]

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].

HS and Variants

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].

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

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

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

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 . . .

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

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

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)

Harmony Search for Multi-objective OptimizationProposed Algorithms

Introduction

Proposed Algorithms

Conclusions

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

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

I A different selection scheme: memory is doubled andnon-dominated sorting + crowding distance are applied

Harmony Search for Multi-objective OptimizationResults

Introduction

Proposed Algorithms

Conclusions

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

Problems

I 10 unconstrained (bound constrained) problems: UF1,UF2, . . . , UF10

Problems

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

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.

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

I Mann-Whitney and dominance rankingI Quality indicators: hypervolume, additive unary-ε

and R2

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

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.

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.

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

Harmony Search for Multi-objective OptimizationConclusions

Introduction

Proposed Algorithms

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

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

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

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)

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

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

Bibliographic ReferencesCOELLO, C. A. C.; LAMONT, G. B.; VELDHUIZEN, D. A. V. Evolutionary Algorithms for SolvingMulti-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 ofthe American Statistical Association, v. 75, n. 372, p. 947–951, 1980.

MAHDAVI, M.; FESANGHARY, M.; DAMANGIR, E. An improved harmony search algorithm for solvingoptimization 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.

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?

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