Harmony Search for Multi-objective Optimization - SBRN 2012

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 , 2012

Pavelski, Almeida, Gon¸alves
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Summary

Introduction

Background
Multi-objective Optimization and MOEAs
Harmony Search and Variants

Proposed Algorithms

Experimental Results

Conclusions

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Introduction

Introduction

Background

Proposed Algorithms


Conclusions

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Introduction

Introduction
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 tests
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Background

Introduction

Background

Proposed Algorithms


Conclusions

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Background
MO and MOEAs

Multi-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 solutions

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Background
MO and MOEAs

Pareto 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]
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Background
MO and MOEAs

Multi-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-II

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Background
MO and MOEAs

Non-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 algorithms

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Background
MO and MOEAs

Non-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 algorithms

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Background
MO and MOEAs

Non-Dominated Sorting Genetic algorithm
(NSGA-II) [Deb et al. 2000]

Figure: Non-dominated Selection
Figure: Non-dominated [Deb et al. 2000]
Sorting [Zitzler 1999]
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Background
MO and MOEAs

Non-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]

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Background
HS and Variants

Harmony 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].

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Background
HS and Variants

Harmony 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].

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Background
HS and Variants

Harmony 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 Practice
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Background
HS and Variants

Harmony 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 function
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Background
HS and Variants

Harmony 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 function
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Background
HS and Variants

Harmony Search Variants

HS: regular Harmony Search algorithm
IHS: Improved Harmony Search
GHS: Global-best Harmony Search
SGHS: Self-adaptive Global-best Harmony Search
...

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Background
HS and Variants

Improved 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)
NI

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Background
HS and Variants

Global-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 function

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Background
HS and Variants

Self-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 otherwise
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Proposed Algorithms

Introduction

Background

Proposed Algorithms


Conclusions

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Proposed Algorithms

Non-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 function

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Proposed Algorithms


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 > 0

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Proposed Algorithms


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 > 0

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Results

Introduction

Background

Proposed Algorithms


Conclusions

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Results

Problems

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 objetives

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Results

Problems

10 unconstrained (bound constrained) problems: UF1,
UF2, . . . , UF10
n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives. UF8,
. . . , UF10: 3 objetives

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Results

Problems

10 unconstrained (bound constrained) problems: UF1,
UF2, . . . , UF10
n = 30 variables. UF1, UF2, . . . , UF7: 2 objetives.
UF8, . . . , UF10: 3 objetives

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Results

Parameters
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.

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Results

Quality 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 test

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Results


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 test

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Results

Kurskal-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 0.04
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Results


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.

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Results


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.

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Results


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.

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Results

Comparison 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-II

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Conclusions

Introduction

Background

Proposed Algorithms


Conclusions

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Conclusions

Conclusions

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-II

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Conclusions

Conclusions

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-II

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Conclusions

Conclusions

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-II

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Conclusions

Future 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)

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Conclusions

Future 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

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Conclusions

Future 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

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Bibliographic References
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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.

Acknowledgments

Fundac˜o Arauc´ria
¸a a
UNICENTRO
Friends and colleagues

Thank you for your attention!
Questions?

Harmony Search for Multi-objective Optimization - SBRN 2012

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Harmony Search for Multi-objective Optimization - SBRN 2012