SOMGA for Large Scale Function Optimization and Its Application

Abstract

Self Organizing Migrating Genetic Algorithm (SOMGA) is a hybridized variant of Genetic Algorithm (GA) inspired by the features of Self Organizing Migrating Algorithm, presented by Deep and Dipti (IEEE Congr Evol Comput, pp 2796–2803, 2007) [1]. SOMGA extracts the features of binary coded GA and real coded SOMA in such a way that diversity of the solution space can be maintained and thoroughly exploited keeping function evaluation low. It works with very less population size and tries to achieve global optimal solution faster in less number of function evaluations. Earlier SOMGA has been used to solve problems up to 10 dimensions with population size 10 only. This chapter is brake into three sections. In first section a possibility of using SOMGA to solve large scale problem (dimension up to 200) has been analyzed with the help of 13 test problems. The reason behind extension is that SOMGA works with very small population size and to solve large scale problems (dimension 200) only 20 population size is required. On the basis of results it has been concluded that SOMGA is efficient to solve large scale global optimization problems with small population size and hence required lesser function evaluations. In second section, two real life problems from the field of engineering as an application have been solved using SOMGA. In third section, a comparison between two ways of hybridization has been analyzed. There can be two approaches to hybridize a population based technique. Either by incorporating a deterministic local search in it or by merging it with other population based technique. To see the effect of both the approaches on GA, the results of SOMGA on five test problems are compared with the results of MA (GA+ deterministic local search). Results clearly indicates that SOMGA is less expensive and effective to solve these problems.

Appendix

This Appendix contains the list of 13 benchmark test problems taken from literature, which are used to evaluate the performance of the algorithm. These problems are unconstrained nonlinear optimization problems having a number of local as well as global optimal solutions. All the problems have varying difficulty level and contain unimodal as well as multi modal problems.

Problem 1: (Cosine Mixture Problem)

This problem is Cosine Mixture Function. The global optimum of this function is at (0, 0,…, 0) with f_min = −0.1n. where n is the dimension of the problem. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } - 0.1\sum\limits_{i = 1}^{n} {\cos \left( {5\pi x_{i} } \right)} ,{\kern 1pt} \quad for\;x_{i} \in \left[ { - 1,{\kern 1pt} 1} \right]. $$

Problem 2: (Exponential Problem)

This problem is Exponential Function. The global optimum of this function is at (0, 0,…, 0) with f_min = −1. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \exp \left( { - 0.5\sum\limits_{i = 1}^{n} {x_{i}^{2} } } \right),\quad for\;x_{i} \in \left[ { - 1,{\kern 1pt} 1} \right]. $$

Problem 3: (Ackley Function)

This problem is the Ackley function. The surface of the Ackley function has numerous local minima due to its exponential terms. Any search algorithm based on the gradient information will be trapped in local optima, but any search strategy that analyzes a wider region will be able to cross the valley among the optima and achieve better results. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ f\left( x \right) = 20 + e - 20e^{{ - \left( {\frac{1}{5}\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right)}} - e^{{ - \left( {\frac{i}{n}\sum\limits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right)}} ,\quad for\;x_{i} \in \left[ { - 15,30} \right]. $$

Problem 4: (Sphere Function Problem)

The next problem is Sphere Function. This problem is continuous convex and unimodal. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } ,\quad for\;x_{i} \in \left[ { - 5.12,5.12} \right]. $$

Problem 5: (Griewank Function)

This problem is a widely employed test function for global optimization, the Griewank function. While this function has an exponentially increasing number of local minima as its dimension increases, it turns out that a simple Multistart algorithm is able to detect its global minimum more and more easily as the dimension increases. The optima of this function are regularly distributed. Number of local minima for arbitrary n is unknown, but in two dimensional case there are some 500 local minima. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ f(x) = \sum\limits_{i = 1}^{n} {\frac{{x_{i}^{2} }}{4000}} - \prod\limits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right)} - 1,\quad for\;x_{i} \in \left[ { - 5.12,5.12} \right]. $$

Problem 6: (Axis Parallel Hyper Ellipsoid)

This problem is Axis Parallel Hyper Ellipsoid Function. This test problem is similar to sphere problem function. It is also known as the weighted sphere model. It is continuous convex and unimodal. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {ix_{i}^{2} } ,\quad for\;x_{i} \in \left[ { - 5.12,5.12} \right]. $$

Problem 7: (Schwefel’s Double Sum)

This problem is Schwefel’s double sum Function. This function is an extension of axis parallel hyper ellipsoid function. It produces a rotated hype-ellipsoid. It is continuous convex and unimodal. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {\left( {\sum\limits_{j = 1}^{i} {x_{j} } } \right)^{2} } ,\quad for\;x_{i} \in \left[ { - 65.536,\,65.536} \right]. $$

Problem 8: (Rastrigin Function)

This problem is the Rastrigin Function. It is the extended form of the sphere function with a modulator term α · cos(2πx_i). This function consists of a large number of local minima (not exactly known) whose value increases with the distance to the global minimum. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ f\left( x \right) = 10n + \sum\limits_{i = 1}^{n} {\left( {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right)} \right)} ,\quad for\;x_{i} \in \left[ { - 5.12,5.12} \right]. $$

Problem 9: (Rosenbrock Function)

This problem is the Rosenbrock function, also known as the banana function. It is a continuous, differentiable, unimodal and non separable function. Its difficulty arises due to nonlinear interaction between parameters. The global optimum is inside a long narrow parabolic shaped flat valley. Its global minimum is at (1, 1,…, 1) with f_min = 0. The functional form is as follows:

$$ f\left( x \right) = \sum\limits_{i = 1}^{n - 1} {(100(x_{i + 1} - x_{i}^{2} )^{2} + \left( {x_{i}^{{}} - 1} \right)}^{2} ),\quad for\;x_{i} \in \left[ { - 2.048,2.048} \right]. $$

Problem 10: (Schwefel Function)

This problem is Schwefel Function. The contour of this function is made up of a great number of peaks and valleys. This function has a second best minimum far from the global minimum, so it is difficult for many algorithms to locate the global optimum of this function. Its global minimum is at (1, 1,…, 1) with f_min = 0. The functional form is as follows:

$$ f\left( x \right) = 418.9829n - \sum\limits_{i = 1}^{n} {\left( {x_{i} \sin \sqrt {\left| {x_{i} } \right|} } \right)} ,\quad for\;x_{i} \in \left[ { - 500,500} \right]. $$

Problem 11: (Zakharov’s Problem)

This problem is Zakharov Function. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{n} {\frac{i}{2}x_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{n} {\frac{i}{2}x_{i} } } \right)^{4} ,\quad for\;x_{i} \in \left[ { - 5.12,5.12} \right]. $$

Problem 12: (Ellipsoidal Function)

This problem is Ellipsoidal Function. Its global minimum is at (1, 2,…, n) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \sum\limits_{i = 1}^{n} {\left( {x_{i} - i} \right)^{2} } ,\quad for\;x_{i} \in \left[ { - n,n} \right]. $$

Problem 13: (Schwefel Problem 4)

This problem is Schwefel Problem 4 Function. Its global minimum is at (0, 0,…, 0) with f_min = 0. The functional form is as follows:

$$ \mathop {\hbox{min} }\limits_{x} f\left( x \right) = \mathop {\hbox{max} \left\{ {\left| {x_{i} } \right|,1 \le i \le n} \right\}}\limits_{i} ,\quad for\;x_{i} \in \left[ { - 100,100} \right]. $$