Evolutionary boundary constraint handling scheme

Abstract

The performance of an optimization tool is largely determined by the efficiency of the search algorithms used in the process as well as the proper handling of complex constraints. From the implementation point of view, an important part of task ensuring an efficient algorithm to work to its best capability is to handle the boundary constraints properly and effectively. As most studies in the literature have focused on the development of algorithms and performance evaluation and comparison of optimization algorithms, this crucial step has not been explored very well, and consequently only limited studies have been carried out in this field. This paper intends to propose a simple and yet efficient evolutionary scheme for handling boundary constraints. The simplicity of this approach means that the proposed scheme is very easy to implement and thus can be suitable for many applications. We demonstrate this approach with an efficient algorithm, differential evolution, and we also compare it with other boundary constraint handling approaches for a wide set of benchmark problems. Based on statistical parameters and especially mean values, the results obtained by the evolutionary scheme are better than the best known solutions obtained by the existing methods.

Appendices

Appendix 1: Detailed formulation of the benchmark problems

F1: Ackley Function

$$ f(X) = - a\,\exp \left( { - 0.02\sqrt {n^{ - 1} \sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \exp \left( {n^{ - 1} \sum\limits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + a + e,\,a = 20 $$

F2: Becker and Lago Function

$$ f(X) = \left( {\left| {x_{1} } \right| - 5} \right)^{2} + \left( {\left| {x_{2} } \right| - 5} \right)^{2} $$

F3: Branin Function

$$ f(X) = a\left( {x_{2} - bx_{1}^{2} + cx_{1} - d} \right)^{2} g\left( {1 - h} \right)\cos \left( {x_{1} } \right) + g $$

where

$$ a = 1,\,b = 1.25\pi^{ - 2} ,\,c = 5\pi^{ - 1} ,\,d = 6,\,g = 10,\,h = 0.125\pi^{ - 1} $$

F4: Dekkers and Aarts Function

$$ f(X) = \left( {x_{1} - 1} \right)^{2} + \sum\limits_{i = 2}^{n} i \left( {2x_{i}^{2} - x_{i - 1} } \right)^{2} $$

F5: Easom Function

$$ f(X) = - \cos \left( {x_{1} } \right)\cos \left( {x_{2} } \right)\exp \left( { - \left( {x_{1} - \pi } \right)^{2} - \left( {x_{2} - \pi } \right)^{2} } \right) $$

F6: Goldstein and Price Function

$$ \begin{aligned} f(X) = & \left( {1 + \left( {x_{1} + x_{2} + 1} \right)^{2} \left( {19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} - 6x_{1} x_{2} + 3x_{2}^{2} } \right)} \right) \\ & \times \left( {30 + \left( {2x_{1} - 3x_{2} } \right)^{2} \left( {18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} } \right)} \right) \\ \end{aligned} $$

F7: Griewank Function

$$ f(X) = 1 + \frac{1}{4,000}\sum\limits_{i = 1}^{n} {x_{i}^{2} } - \prod\limits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right)} $$

F8: Hartman 3 Function (Table 3)

Table 3 Constants for Hartman 3 Function

$$ f(X) = - \sum\limits_{i = 1}^{4} {c_{i} \exp } \left( { - \sum\limits_{j = 1}^{4} {a_{ij} \left( {x_{j} - p_{ij} } \right)^{2} } } \right) $$

F9: Hartman 6 Function (Table 4)

Table 4 Constants for Hartman 6 Function

$$ f(X) = - \sum\limits_{i = 1}^{4} {c_{i} \exp } \left( { - \sum\limits_{j = 1}^{6} {a_{ij} \left( {x_{j} - p_{ij} } \right)^{2} } } \right) $$

F10: Hosaki Function (Table 5)

Table 5 Constants for Kowalik Function

$$ f(X) = x_{2}^{2} \left( {1 - 8x_{1} + 7x_{1}^{2} - \frac{{7x_{1}^{3} }}{3} + \frac{{x_{1}^{4} }}{4}} \right)\exp \left( { - x_{2} } \right) $$

F11: Kowalik Function

$$ f(X) = \sum\limits_{i = 1}^{11} {\left( {a{}_{i} - \frac{{x{}_{1}\left( {1 + x{}_{2}b{}_{i}} \right)}}{{\left( {1 + x{}_{3}b{}_{i} + x{}_{4}b{}_{i}^{2} } \right)}}} \right)}^{2} $$

F12: Levy and Montalvo 1 Function

$$ \begin{aligned} f(X) = & \frac{\pi }{n}\left( {10\sin^{2} \left( {\pi y_{1} } \right)) + \sum\limits_{i = 1}^{n - 1} {\left( {y_{i} - 1} \right)^{2} \left( {1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)} \right)} + \left( {y_{n} - 1} \right)^{2} } \right) \\ y_{i} = & 1 + \frac{{x_{i} + 1}}{4} \\ \end{aligned} $$

F13: Levy and Montalvo 2 Function

$$ f(X) = \frac{1}{10}\left( {\sin^{2} \left( {3\pi x_{1} } \right)) + \sum\limits_{i = 1}^{n - 1} {\left( {x_{i} - 1} \right)^{2} \left( {1 + \sin^{2} \left( {3\pi x_{i + 1} } \right)} \right)} + \left( {x_{n} - 1} \right)^{2} \left( {1 + \sin^{2} \left( {2\pi x_{i + 1} } \right)} \right)} \right) $$

F14: Modified Langerman Function (Table 6)

Table 6 Constants for Langerman Function

$$ f(X) = - \sum\limits_{i = 1}^{5} {c_{j} \cos \left( {\frac{{d_{j} }}{\pi }} \right)} \exp \left( { - \pi d_{j} } \right) $$

F15: Neumaier 3 Function

$$ f(X) = \sum\limits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} - \sum\limits_{i = 2}^{n} {x_{i} } } x_{i - 1} $$

F16: Paviani Function

$$ f(X) = \sum\limits_{i = 1}^{10} {\left( {\left( {\ln \left( {x_{i} - 2} \right)} \right)^{2} + \left( {\ln \left( {10 - x_{i} } \right)} \right)^{2} } \right) - \left( {\prod\limits_{i = 1}^{10} {x_{i} } } \right)^{\frac{1}{5}} } $$

F17: Rastrigin Function

$$ f(X) = 10n + \sum\limits_{i = 1}^{n} {\left( {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right)} \right)} $$

F18: Rosenbrock Function

$$ f(X) = \sum\limits_{i = 1}^{n - 1} {\left( {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right)} $$

F19: Shekel’s Foxholes Function (Table 7)

Table 7 Constants for Langerman Function

$$ f(X) = - \sum\limits_{j = 1}^{30} {\frac{1}{{c_{j} + \sum\limits_{i = 1}^{n} {\left( {x_{i} - a_{ji} } \right)^{2} } }}} $$

F20: Wood Function

$$ \begin{aligned} f(X) = & 100\left( {x_{2} - x_{1}^{2} } \right)^{2} + \left( {x_{1} - 1} \right)^{2} + 90\left( {x_{4} - x_{3}^{2} } \right)^{2} + \left( {x_{3} - 1} \right)^{2} \\ + & 10.1\left( {\left( {x_{2} - 1} \right)^{2} + \left( {x_{4} - 1} \right)^{2} } \right) + 19.8\left( {x_{2} - 1} \right)\left( {x_{4} - 1} \right) \\ \end{aligned} $$

Appendix 2

See Table 8.

Table 8 Statistical features of the best solutions obtained by existing and proposed boundary constraint schemes