# Generacion de funciones benchmark complejas no lineales para optimizacion usando conjuntos difusos y funciones de prueba clasicas.

GENERATION OF COMPLEX NONLINEAR BENCHMARK FUNCTIONS FOR OPTIMIZATION USING FUZZY SETS AND CLASSICAL TEST FUNCTIONS

INTRODUCTION

Optimization (maximization or minimization) of continuous nonlinear functions is an important problem in sciences, mathematics, engineering and economics, due to many real-world problems would be solved in these terms [1]. However, nonlinear optimization is usually a hard problem; their difficulties are related to the complexity of goal function, constrains, topology of the search region and limitations of the optimization methodology.

Optimization field has several types of problems, e.g. combinatorial optimization, integer optimization, quadratic optimization, and other; and a universal optimizer does not exist. Talking about optimizers, Wolpert and Macready emphasize two points: First, when an optimizer is designed for a specific problem, it works better for the problem than other generic optimizer. Second, there are not harder problems, but there are different types of problems. It means that each type of problem has specific characteristics and it agrees to them an optimizer could have better performance that another one [2]. I.e. For developing and comparing new methods, and for solving real world problems efficiently, we will want benchmark functions with the highest number of characteristics available or the most closed characteristic to the real problem. Moreover, scientific community provides special test to compare methods [3].

There are some well-known test functions [4-7]. They are characterized by their behavior, which is manifested by the features of the generated surface, but regularly these functions are limited to specific characteristics. For example, the Rosenbrock function has its global optimum inside of a narrow valley, while in the Bukin's F4 function the minimum is located inside a canon. The aim of this paper is to present a new methodology for generating or specifying complex test functions to prove and tune heuristic optimization algorithms. To do that, we use fuzzy sets theory to combining well-known to generate new functions.

The paper is organized as follows. In Section 2, we review some aspects of classical test functions. Following, fuzzy set theory is introduced in Section 3. In Section 4, we state the description of the proposed methodology. Next, some examples are presented in Section 5. Finally, we conclude in Section 6.

1. CLASSICAL TEST FUNCTIONS

The use of well-known test functions is a common procedure for testing global optimization algorithms, such that, these functions are standard literature benchmarks [4-7]. Thus, researchers are able to compare their results against other methodologies whose results are validated and accepted by the scientific community. Usually, a set of functions is used to evaluate the behavior of a specific algorithm in different conditions and to determine their robustness.

A classical benchmark is the Rosenbrock's function (Figure 1):

z = f (x, y) = 100 [([y.sup.2] - x).sup.2] + [(1 - x).sup.2] (1)

[FIGURE 1 OMITTED]

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It has a global minimum in f (1,1) = 0 . A standard initial point is [x.sub.0] = 1 and [y.sub.0] = -1.2 . DeJong bounded the function to the interval defined as -2.048 [menor que o igual a] x [menor que o igual a] 2.048 to use it in heuristic optimization [8]. And Yao et al. [9] report and extend this function to n dimension using the borders in [[-30,30].sup.n], which increase the complexity because it is a bigger search region. Other classical benchmark is the Bukin's F4 function (Figure 2):

z = f (x, y) = 100 [y.sup.2] + 0.01 [valor absolute de x + y] (2)

Proposed by Bukin [10] with a global minimum in f (0,0) = 0 . The standard initial point [x.sub.0] = 1 and [y.sub.0] = 1, and there are not borders available.

Test functions are classified in terms of surface's features and the primitive functions used for its construction (or setting-up). Several features are described as following:

* The dimensionality of the functions definition: for a fixed number of dimensions or in a general way.

* The modality or ruggedness: quantity of local optimal (maximum and minimum) points. They are classified in unimodal functions with a global minimum and multimodal with several minimums.

* The homogeneity: the characteristic of the relationship between variables (separable or no separable).

The above features would have a great impact in the behavior of a specific optimization algorithm. For example, it is well known that gradient based methods have poor behavior optimizing functions with a huge number of local optimal points, because they are trapped in a local optimum and they are not able to escape out.

2. FUZZY SETS AND FUZZY OPERATIONS

In classical sets theory, the membership of an element to a set is a binary function returning zero when the element not belongs to the set and one in otherwise. In fuzzy set theory, the discrete set {0, 1} for the range of membership function is changed by the interval [0, 1]. Zero and one still represents the absolute certainty of that element belongs or not belongs to the set. The intermediate values between zero and one represent uncertainty, such that, partial membership is allowed [11].

The generalized bell function is a typical method for specifying the membership function of a fuzzy set [11]:

[mu] (x; a, b, c) = 1 / 1 + [[absolute value of x - c / b].sup.2a] (3)

Examples of the generalized bell function with different values for the parameters a, b and c are shown in Figure 3.

One-dimensional fuzzy sets, as in equation (3), can be extended to the X x Y plane, using a fuzzy set for each axis, and then applying any fuzzy operator. This process is the so-called composition of fuzzy sets e.g. several bi-dimensional fuzzy sets would be obtained using the one-dimensional fuzzy membership functions [mu](x; 1, 3, 0) and [mu](y; 1, 3, 0) for representing a fuzzy set for each axis, Figure 3. Membership functions for several parameters. and composite it by means of any fuzzy operator; In Figure 4, we present the fuzzy set obtained using the product operator defined as:

[mu][(x, y) = [mu](x; 1, 3, 0) x [mu](y; 1, 3, 0) (4)

while the two-dimensional fuzzy set presented in Figure 5 is obtained using the minimum operator:

[mu](x, y) = [mu]{(x; 1, 3, 0), [mu](y; 1, 3, 0)} (5)

An extensive recompilation of fuzzy operators is presented in [11] and [12].

[FIGURE 3 OMITTED]

3. PROPOSED METHODOLOGY

We propose the generation of new test functions by means of a weighted sum of two functions defined by the modeler. Two-dimensional fuzzy membership is restricted to the interval [0, 1], then the operator are used as the weights in the sum of functions. Thus, if [F.sub.1] (x, y) and [F.sub.2] (x, y) are classical benchmark test functions, a new test function would be generated as:

z = [F.sub.*] (x, y) = [mu](x, y) x H ([F.sub.1] (x, y)) + [1 - [mu] (x, y)] x H ([F.sub.2] (x, y)) (6)

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In the previous equation, H(*) is the function F(x, y) scaled to the interval [0, 1]. The main idea of this methodology is to obtain new functions with similar characteristics to a certain problem, with the advantage, over real problem, of knowing function's characteristics like minimums, topology, etc. Also, this methodology provides a large number of functions with different characteristics. So, we can obtain a nearer simulation of the method's performance for a specific sort of problems.

4. EXAMPLES

To demonstrate the proposed methodology, we create three new test function combining Rosenbrock's (see Equation (1)) and Bukin's F4 (see Equation (2)) test functions using the operators described. In the sake of a clear vision of the functions' details, we limited the plots in a range in [[-3,3].sup.2].

In Figure 6, we present the surface obtained when [mu](x, y) is defined as in equation (4); the obtained function shows that the function BukinF4's canon is extended with the features of the Rosenbrock's function, while the edges retain the regularity of the function Bukin F4, but with the softness of the Rosenbrock's function. It would be the first function.

The generated second function is shown in Figure 7. For this, [mu] (x, y) is defined as in equation (5), as in the previous figure. It still has a combination of both functions that produce a different surface, but reminds the same minimum value: min f (1,1) = f (0,0) = 0.

[FIGURE 6 OMITTED]

Moreover, complex surfaces may be generated using only one function too. For example, we use a rotated version of the original Rosenbrock's function as [F.sub.2] (x, y) in Eq. (6). Figure 8 presents the obtained surface using only the Rosenbrock function and the product operator. The gradient [derivada parcial]/[derivada parcial] x [derivada parcial] y z of the generated function is plot in Figure 9. One of the criteria of complexity for optimization problems is the existence of planar regions on the gradient plot, which is evident in the figure.

5. CONCLUSIONS

In this paper, we present a novel methodology to generate new complex test functions for testing, comparing and tuning global optimization algorithms. Our methodology is based in the combination of two two-dimensional classical test functions using a weighted sum. These weights are generated by means of a two-dimensional fuzzy set. Thus, using only two initial test functions, many complex surfaces can be generated changing the parameters and composition operators used for generating the two-dimensional fuzzy sets. Although several topologies could be generated, the characteristics of the function are still known, so the new function remains a good benchmark.

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Recibido: 28/01/2010

REFERENCES

[1] P. M. Pardalos, y M. G. C. Resende, ed, "Handbook of Applied Optimization," New York: Oxford University Press, 2002

[2] W. G. Macready, y D. H. Wolpert, "What makes an optimization problem hard?," Complexity, vol. 5, pp. 40-46, 1996.

[3] S. Das y P. N. Suganthan. "Problem Definitions and Evaluation Criteria for CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems". Singapore, 2010. Available: http://web. mysites.ntu.edu.sg/epnsugan/PublicSite/Shared%20 Documents/CEC%202011-%20RWP/Tech -Rep.pdf

[4] W. Hock y K. Schittkowski. "Test Examples for Nonlinear Programming Codes", Lecture Notes in Economics and Mathematical Systems, vol. 187, Springer 1981.

[5] J.J. More, et al. "Testing Unconstrained Optimization Software", ACM Transactions on Mathematical Software, vol. 7, no. 1, pp. 17-41, 1981.

[6] I. Bongartz, et al. "CUTE: Constrained and Unconstrained Testing Environment", ACM Transactions on Mathematical Software, vol. 21, no. 1, pp. 123-160, 1995.

[7] N. I. M. Gould, et al. "CUTEr (and SifDec): a Constrained and Unconstrained Testing Environment, revisited", ACM Transactions on Mathematical Software, vol. 29, no. 4, pp. 373-394, 2003.

[8] K. A. De Jong. "Analysis Of The Behavior Of A Class Of Genetic Adaptive System". University of Michigan. Michigan, Technical Report No. 185, 1975

[9] X. Yao, Y. Liu, y G. Lin. "Evolutionary programming made faster", IEEE Transactions on Evolutionary Computation. Vol. 3, no. 2, pp. 82-102,1999

[10] A. D. Bukin. "New Minimization Strategy For Non-Smooth Functions", Budker Institute of Nuclear Physics preprint Novosibirsk, BUDKER-INP-1997-79, 1997

[11] J.-S.R, Jang, et al., Neuro-Fuzzy and Soft Computing: A computational approach to learning and machine intelligence. Prentice Hall, 1996.

[12] N. Kasabov. Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering, 2nd edn, Massachusetts Institute of Technology, 1998.

* Articulo de investigacion cientifica y tecnologica. Es producto de la investigacion realizada por el grupo de Optimizacion Heuristica y Algoritmos Bio-Inspirados. Patrocinado por la Facultad de Minas, Universidad Nacional de Colombia, Medellin, Colombia.

Eddy Mesa, Ingeniera Mecanica, Universidad de Antioquia, Colombia, 2006; Miembro del grupo de Optimizacion Heuristica y Algoritmos Bio-Inspirados, Facultad de Minas, Universidad Nacional de Colombia. Direccion de correspondencia: Cra 80 65-223, Bloque M8A, Of. 201. Medellin, Colombia. Tel. +57 +4 425 5350. Correo electronico: ejmesad@unal.edu.co

Juan David Velasquez, Autor para correspondencia. Doctor en Ingenieria-Area Sistemas Energeticos, Universidad Nacional de Colombia, sede Medellin, Colombia, 2009; Magister en Ingenieria de Sistemas, Ingeniero Civil, Profesor Asociado de la Escuela de Sistemas, Facultad de Minas, Universidad Nacional de Colombia, sede Medellin. Miembro del grupo de Optimizacion Heuristica y Algoritmos Bio-Inspirados, Facultad de Minas, Universidad Nacional de Colombia. Direccion de correspondencia: Cra 80 65-223, Bloque M8A, Of. 206. Medellin, Colombia. Tel. +57 +4 425 5370. Correo electronico: jdvelasq@unal.edu.co

Patricia Jaramillo, Doctora en Planificacion y Gestion de Recursos Hidraulicos, Universidad Politecnica de Valencia, Valencia, Espana. Profesora Asociado de la Escuela de Sistemas, Facultad de Minas, Universidad Nacional de Colombia, sede Medellin. Directora del grupo de Optimizacion Heuristica y Algoritmos Bio-inspirado, Facultad de Minas, Universidad Nacional de Colombia. Direccion de correspondencia: Cra 80 65-223, Bloque M8A, Of. 213. Medellin, Colombia. Tel. +57 +4 425 5343. Correo electronico: gpjarami@unal.edu.co