an optimization algorithm inspired by musical...

IntroductionMMC algorithmExperimentation

ConclusionsReferencias

An Optimization Algorithm Inspired by MusicalComposition

R. A. Mora-Gutiérrez; J. Ramírez-Rodríguez; E. A.Rincón-García;

Universidad Autónoma Metropolitana

May 7, 2012

Mora, Ramírez, Rincón An Optimization Algorithm Inspired by Musical Composition



Outline

1 IntroductionCultural algorithmsMusical composition

2 MMC algorithm3 Experimentation

MethodologyTest problemsPrevious WorkExperimental desingNumerical resul

4 Conclusions5 Referencias




Cultural algorithmsMusical composition

Cultural algorithmsCulture is the shared patterns of behaviors and interactions,cognitive constructs, and affective understanding that are learnedthrough a process of socialization.In the human society, the cultural changes are faster than genetic[?].




Cultural algorithmsMusical composition

Musical composition

Musical composition is the artistic process of creating andinnovating an artwork through a recursive process in a creativesystem socio-cultural. Composer’s creativity results from makingconnections between disjoint ideas [de Bono (1993)]:

Moment of geniusRecursive reasoning process about a thought, called “hardwork” [Jacob (1996)]




Optimization and musical compositionIn this work, we present a cultural algorithm based on a socialcreativity system used for music composition, whereby we called it“Musical Composition Method” or MCM [Mora et al 2012].




MMC algorithm

The MMC algorithm is based on ideas following:Musical composition can be considered as an algorithm, sincethis process use rules, principles and a finite number of stepsto create original music of a particular style [Cope (2000)].Musical composition is a system of creativity, so there areinteractions among agents.Usually agents learn form their experience and use what theylearn in future decisions.




Table: Analogy between optimization and musical composition.

Comparative ques-tion

Process of optimiza-tion

Process of musicalcomposition

Why is performed? Solving problems ofmathematical pro-gramming

Generate musicalworks that satisfydesires of the goalaudience

Which seeks? Global optimum Musical work withthe best acceptanceof the goal public




Table: Analogy between optimization and musical composition(continuation).

How is the quality ofthe solutions deter-mined?

Objective function Degree acceptancefor part of target au-dience

Which are the deci-sion elements?

Decision variables Motives

What properties de-termine the qualityof the solutions?

Values of variables Sound characteris-tics of the motives

Is a finite process? Yes YesWhich is the processunit?

Each iteration Each of the arrange-ment to the track




Figure: Model of the creative system for musical composition




Figure: Model of the creative system for musical compositionMora, Ramírez, Rincón An Optimization Algorithm Inspired by Musical Composition



Figure: Dynamic social network





Experimentation

We probed our proposed MMC algorithm on 13 benchmarkcontinuous optimization functions [Bersini et al (1996),Ali et al (2005), Molga and Smutnicki (2005), Pohlheim (2006),Liang et al (2006), Riley (2010), Yang (2010), Brest et al (2006),de los Cobos Silva et al (2010)]. Those results were comparedwith the results obtained by metaheuristics following:

Harmony search (HS) .Improved harmony search(IHS)Global-best harmony search (GHS) .Self-adaptative harmony search (SHS).





Test problems

The general structure of optimization problems is:

min(x∈X)

f (x) (1)

Such that:

xLi ≤ xi ≤ xU

i for all i = 1, 2, 3, . . . , n xi ∈ R (2)

where:f : Rn → R

X ⊆ Rn

xLi ≤ xi ≤ xU

i feasible range of decision variables.





Test problems

We divided the functions into three groups: unimodal problems,unrotated multimodal problems and rotated multimodal problems.Some of benchmark functions used in this work, are following:

UnimodalRosenbrock function.Step function.

Unrotated multimodal problemsRastrigin function.Ackley’s function.Griewank function.

Rotated multimodal problemsRotate hyper-ellipsoid function.Shifted sphere function.





Some of the 13 benchmark functions have been solved by exactmethods like: Newton, Davidon-Fletcher-Powell and quasi-Newtonmethods [Luenberger (1984), Moudgalya and Joshi (2004)], whilesome others have been solves by metaheuristics like: Tabu Search(TS)[Chelouaha and Siarry (2000)], Genetic Algorithmos (GA)[Salomon (1996), Ali et al (2005)] , HS[Wang and Huang (2010), Pan et al (2010)],PSO[Liang et al (2006)] etc.





Experimental desin

We made 30 independent replications were performed for eachinstance (An experiment consisted in 50000 evaluations ofobjective function)For each replication, we were registered both thecomputational execution time and the best target functionvalue.We made statistical analysis of data, next the results obtainedwere compared versus the results of other heuristics.





Values for parameters

max _arrangement = 12500ifg = 0.01cfg = 0.09fcla = 0.045Nc = 4Ns = 5.





Table: Mean and standard deviation of the benchmark functionoptimization.

Problem A BGlobal optimum 0 0

HS 350.297± 266.691 4.233± 3.030HIS 624.323± 559.847 3.333± 2.196GHS 49.669± 59.161 0± 0.000SHS 150.93± 131.055 0± 0.000

MMC 245.43± 297.927 1.0667± 1.143where:

L is Rosenbrock functionM is step function





Table: Mean and standard deviation of the benchmark functionoptimization.

Problem L MGlobal optimum −180 −330

HS 547.869± 0.501 −274.687± 12.863HIS 494.756± 6.717 −270.695± 16.223GHS 546.62± 8686070.61 −192.096± 18.646SHS −47.189± 13.313 −232.14± 30.033

MMC −178.509± 0.160 −283.677± 10.416

where:L is shifted rotated Griewank functionM is shifted rotated Rastrigin function





The MMC algorithm has obtained good results for the set ofrotated multimodal problems, as the solutions generated by thisalgorithm are for all instances of test, the best or the second-bestoption.





Table: Results of Wilcoxon rank test.

Parameter HS IHS GHS SHSp 0.1119 0.1008 0.3051 0.9183h 0 0 0 0




Conclusions

In this paper, we have presented a novel cultural algorithm forCNOP, which mimics creativity within music composition process.The creativity in MCM is made up two different levels: personaland social-cultural. The results also illustrate that the MCM has ahigher exploration capability of the solution.




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Referencias I

Mora-GutiÃĹrrez, R., RamÃŇrez-RodrÃŇguez,J.,RincÃŻn-GarcÃŇa, E.:An optimization algorithm inspired bymusical composition. Artificial Intelligence Review. SpringerNetherlands. Url:http://dx.doi.org/10.1007/s10462–011–9309–8. Doi:10.1007/s10462–011–9309–8

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Referencias II

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Referencias III

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de los Cobos Silva, S.G., Close, J.G., Andrade, M.A.G.,Licona, A.E.M.: Búsqueda y exploración estocástica.Universidad Autónoma Metropolitana, México (2010)

Cope, D.: The algorithmic composer.A-R Editions Inc, Wisconsin USA (2000)




Referencias IV

Cope, D.: Computer model of musical creativity.MIT Press, London England (2005)

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Referencias V

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Referencias VI

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Referencias VIII

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Molga, M., Smutnicki, C.: Test functions for optimizationneeds.http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf.(2005)




Referencias IX

Pan, Q.K., Suganthan, P.N., Tasgetiren, M.F., Liang, J.J.: Aself-adaptive global best harmony search algorithm forcontinuous optimization problems. Applied Mathematics andComputation 216, 830–848 (2010)

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Referencias X

Riley, M.J.W., Jenkins, K.W., Thompson, C.P.: A study ofearly stopping, ensembling, and patchworking for cascadecorrelation neural networks. IAENG International Journal ofApplied Mathematics 40:4, 307–316 (2010)

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Referencias XI

Yang, X.S.: Test problems in optimization. EngineeringOptimization: An Introduction with MetaheuristicApplications. John Wiley & Sons (2010)


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