adaptation of the musical composition method for solving constrained optimization problems

Soft Comput (2014) 18:1931–1948 DOI 10.1007/s00500-013-1177-5 METHODOLOGIES AND APPLICATION Adaptation of the musical composition method for solving constrained optimization problems Roman Anselmo Mora-Gutiérrez · Javier Ramírez-Rodríguez · Eric Alfredo Rincón-García · Antonin Ponsich · Oscar Herrera · Pedro Lara-Velázquez Published online: 29 November 2013 © Springer-Verlag Berlin Heidelberg 2013 Abstract Many real-world problems may be expressed as nonlinear constrained optimization problems (CNOP). For this kind of problems, the set of constraints specifies the feasible solution space. In the last decades, several algorithms have been proposed and developed for tackling CNOP. In this paper, we present an extension of the “Musical Composition Method” (MMC) for solving constrained optimization problems. MMC was proposed by Mora et al. (Artif Intell Rev 1–15, doi:10.1007/s10462-011-9309-8, 2012a). The MMC is based on a social creativity system used to compose music. We evaluated and analyzed the performance of MMC on 12 CNOP benchmark cases. The experimental results demon- strate that MMC significantly improves the global perfor- mances of the other tested metaheuristics on some benchmark functions. Communicated by D. Liu. R. A. Mora-Gutiérrez (B ) · J. Ramírez-Rodríguez · E. A. Rincón-García · A. Ponsich · O. Herrera, P · Lara-Velázquez Departamento de Sistemas, Universidad Autónoma Metropolitana, C. P. 02200 D.F. México, México e-mail: [email protected] E. A. Rincón-García e-mail: [email protected] A. Ponsich e-mail: [email protected] O. Herrera e-mail: [email protected] P. Lara-Velázquez e-mail: [email protected] J. Ramírez-Rodríguez LIA Université d’Avignon et des Pays de Vaucluse, Avignon, France e-mail: [email protected] 1 Introduction Many real-world problems can be formulated as a nonlinear optimization problem (NLP). Every NLP including constraints is referred to as “Constrained Optimization Problem” (CNOP). A CNOP is made up of three basic components: a set of decision variables (x ), a fitness function to be opti- mized (min or max f (x )) and a set of constraints that spec- ify the feasible space for decision variables (g j (x ) ≤ 0 and h j (x ) = 0) (Michalewicz 1995; Hu and Eberhart 2002). Without loss of generality, the general constrained optimization problem can be formulated as: min x ∈F ⊆S f (x ) g j (x ) ≤ 0 ∀ j = 1, 2,..., p h j (x ) = 0 ∀ j = p + 1,..., m x = (x 1 ,..., x n ) ∈ R n (1) where n is the number of decision variables, the objective function f : R n → R is defined over search space S ⊆ R n and F ⊆ S is the feasible region. Usually, S is defined as a n-dimensional rectangle included in R n , defined by the decision variables lower and upper bounds (S ={x ∈ R n : x l ∈[x L l , x U l ]}). Besides, the feasible set F is described by a set of inequality (g : R n → R p ) and/or equality (h : R n → R m ) constraints (F ={x ∈ R n : g j (x ) ≤ 0, ∀ j = 1, 2,..., p; h j (x ) = 0, ∀ j = p + 1,..., m; m ≤ n})(Koziel and Michalewicz 1999; Michalewicz 1995; Cai and Wang 2006; Michalewicz et al. 2000). A point x ∗ ∈ S is a feasible solution if x ∗ ∈ F . If f (x ∗ )< f (x ), ∀x ∈ F then x ∗ is the optimal solution of the problem. In practice, determining the optimal solution(s) of a CNOP is generally a very difficult task. 123

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Soft Comput (2014) 18:19311948DOI 10.1007/s00500-013-1177-5

METHODOLOGIES AND APPLICATION

Adaptation of the musical composition method for solvingconstrained optimization problems

Roman Anselmo Mora-Gutirrez Javier Ramrez-Rodrguez Eric Alfredo Rincn-Garca Antonin Ponsich Oscar Herrera Pedro Lara-Velzquez

Published online: 29 November 2013 Springer-Verlag Berlin Heidelberg 2013

Abstract Many real-world problems may be expressed asnonlinear constrained optimization problems (CNOP). Forthis kind of problems, the set of constraints specifies the fea-sible solution space. In the last decades, several algorithmshave been proposed and developed for tackling CNOP. In thispaper, we present an extension of the Musical CompositionMethod (MMC) for solving constrained optimization prob-lems. MMC was proposed by Mora et al. (Artif Intell Rev115, doi:10.1007/s10462-011-9309-8, 2012a). The MMCis based on a social creativity system used to compose music.We evaluated and analyzed the performance of MMC on 12CNOP benchmark cases. The experimental results demon-strate that MMC significantly improves the global perfor-mances of the other tested metaheuristics on some bench-mark functions.

Communicated by D. Liu.

R. A. Mora-Gutirrez (B) J. Ramrez-Rodrguez E. A. Rincn-Garca A. Ponsich O. Herrera, P Lara-VelzquezDepartamento de Sistemas, Universidad Autnoma Metropolitana,C. P. 02200 D.F. Mxico, Mxicoe-mail: [email protected]

E. A. Rincn-Garcae-mail: [email protected]

A. Ponsiche-mail: [email protected]

O. Herrerae-mail: [email protected]

P. Lara-Velzqueze-mail: [email protected]

J. Ramrez-RodrguezLIA Universit dAvignon et des Pays de Vaucluse, Avignon, Francee-mail: [email protected]

1 Introduction

Many real-world problems can be formulated as a nonlin-ear optimization problem (NLP). Every NLP including con-straints is referred to as Constrained Optimization Problem(CNOP). A CNOP is made up of three basic components: aset of decision variables (x), a fitness function to be opti-mized (min or max f (x)) and a set of constraints that spec-ify the feasible space for decision variables (g j (x) 0 andh j (x) = 0) (Michalewicz 1995; Hu and Eberhart 2002).Without loss of generality, the general constrained optimiza-tion problem can be formulated as:

minxFS

f (x)g j (x) 0 j = 1, 2, . . . , ph j (x) = 0 j = p + 1, . . . , mx = (x1, . . . , xn) Rn

(1)

where n is the number of decision variables, the objectivefunction f : Rn R is defined over search space S Rnand F S is the feasible region. Usually, S is defined asa n-dimensional rectangle included in Rn , defined by thedecision variables lower and upper bounds (S = {x Rn :xl [x Ll , xUl ]}). Besides, the feasible set F is describedby a set of inequality (g : Rn Rp) and/or equality(h : Rn Rm) constraints (F = {x Rn : g j (x) 0, j = 1, 2, . . . , p; h j (x) = 0, j = p + 1, . . . , m; m n}) (Koziel and Michalewicz 1999; Michalewicz 1995; Caiand Wang 2006; Michalewicz et al. 2000).

A point x S is a feasible solution if x F . If f (x)