FSB-EA: Fuzzy search bias guided constraint handling technique for evolutionary algorithm

Highlights

• Constraint handling techniques (CHTs) are classified according to the search biases.

• CHTs are discussed on five state-of-the-art evolutionary algorithms respectively.

• The dynamic fuzzy search biases of individuals are defined by fuzzy theory.

Abstract

Constraint handling technique (CHT) is a critical task for the evolutionary algorithms (EAs) on solving the constrained optimization problems (COPs). In the CHTs development, the definition of the search bias in the search process of an EA is one of the important issues. Based on the different search bias definitions, this study divides the CHTs into two categories: deterministic and non-deterministic techniques. The deterministic techniques define the search bias with an artificial rule. These techniques generally perform well for several scenarios. However, they may have poor performance for some others. This is mainly due to the difficulty to establish an accurate mapping between the complex search scenarios and the uncertain search biases. In contrast with the deterministic techniques, the non-deterministic techniques apply stochastic based methods to escape this mapping establishment which will improve the robustness of the algorithm. However, the probability-based blind random selection can affect negatively on the search efficiency of the algorithm. To compensate the deficiencies of these two categories, this paper proposes a fuzzy search bias guided CHT, namely FSB. The search bias in the FSB is guided by a membership function. It is used to meet the fuzzy knowledge that a solution with large constraint violation biases the search toward solutions with smaller constraint violation and a solution with small constraint violation biases the search toward solutions with better objective function value. The experimental results show that FSB is more robust and more efficient than the other compared techniques.

Introduction

Most of optimization problems have constraints of different types (e.g. physical, time, geometric, etc). These problems are generally called constrained optimization problems (COPs). The feasible regions in a COP decision space may be small or disjoint which will largely affect on the optimization process using the search based algorithms. The optimization of the COPs is both meaningful and challenging. The COPs can be defined as follows (Deb, 2000):

$$\begin{array}{cc}\hfill \mathrm{Minimize}& f\left(x\right),x=(x_1,x_2,\cdots ,x_n)\hfill \\ \hfill \mathrm{Subject}\text{to}:& g_j\left(x\right)\le 0,j=1,2,\cdots ,J,\hfill \\ & h_k\left(x\right)=0,k=1,2,\cdots ,K,\hfill \\ & l_i^{\min }\le x_i\le l_i^{\max }\hfill \end{array}$$

where f(x) is the objective function, g_j(x) is the jth inequality constraint, h_k(x) is the kth equality constraint. The ith variable is varied on the range $[l_i^{\min },l_i^{\max }]$.

EAs combined with CHTs are widely used to optimize COPs (Deb, 2000, Jordehi, 2015, Li, Li, Nguyen, Chen, 2015, Runarsson, Yao, 2000, Takahama, Sakai, 2006). In the optimization process of the EAs on solving the COPs, the feasible and infeasible solutions may exist simultaneously. In fact, one of the major issues on solving the constraint optimization problems is the selection process of the appropriate solutions which will continue evolving throughout the search process. More precisely, if an algorithm biases the search toward solutions having small constraint violation, it may trap on a local optimum. Whereas, if an algorithm biases the search toward solutions having good objective values, it may fail to find a feasible solution. Furthermore, the search bias is changed with the different search phases and it also depends on the COP to be solved and the used EA. In order to address these problems, many CHTs are proposed such as the feasible rule method (Deb, 2000), the stochastic ranking approach (Runarsson & Yao, 2000), the ε-constraint handling technique (Takahama & Sakai, 2006), the penalty function method (Farmani & Wright, 2003) and the multi-objective approach (CARLOS, 2000).

Based on the different definitions of the search bias, in this paper, we divide the CHTs into two categories deterministic and non-deterministic techniques. In the deterministic techniques (e.g. Self-adaptive penalty function approach (Melo, Iacca, 2014, Tessema, Yen, 2009), feasibility rules (Deb, 2000) and ε-constraint handling approach (Takahama & Sakai, 2006)), if the objective function and the constraint violation values of the candidate solutions are given, the search bias could be determined. The deterministic techniques are effective for some search scenarios but they may be not suitable for some others, This is generally due to the difficulty of developing an accurate mapping between the complicated dynamic search scenarios and the uncertainties preferences of the search process. For each algorithm or problem, designing the corresponding CHT will be a laborious work.

In contrast with the deterministic techniques, the non-deterministic techniques such as the stochastic ranking method (Runarsson, Yao, 2000, Runarsson, Yao, 2005, Zhang, Luo, Wang, 2008) use a stochastic based method to escape the establishment of the mapping. However, it is worth noticing that in the recently proposed stochastic mode of non-deterministic techniques, the information of the solutions and the search environment are not considered. This can have a negative impact on the efficiency of algorithms that use this type of technology. Therefore, it is essential to redefine the search bias in order to make it both efficient and robust for the complicated search scenarios caused by the dynamic search processes or by the different problems and algorithms.

Learning knowledge from the deterministic techniques to improve the stochastic mode of the non-deterministic techniques may be a promising way to achieve the above mentioned objective. Besides, fuzzy theory is a good tool to represent knowledge as a mathematical model (Zedeh, 1989). Based on these remarks, a fuzzy search bias guided CHT, named FSB is proposed in this study. In this paper, first the performance of five state-of-the-art CHTs on EAs, namely differential evolution (DE), evolution strategies (ES), genetic algorithm (GA), particle swarm optimization (PSO), and evolutionary programming (EP) are tested and discussed. Then, the knowledge that a solution with large constraint violation bias the search toward solutions with smaller constraint violation, and a solution with small constraint violation bias the search toward solutions with better function value is extracted. Finally, a concept called membership function is used to define this fuzzy knowledge.

Globally, the main contributions of this study can be summarized as follows:

• CHTs classification. To the best of our knowledge, this is the first time that the classification of CHTs based on the search biases definition is presented. It provides new perspective for solving the COPs.

• CHTs discussion. In our study, CHTs are tested and discussed using five state-of-the-art EAs. This could give an impartial evaluation of the tested CHTs without being influenced by the performance of the EA itself. Note that, this is usually overlooked in the previous studies.

• Fuzzy search bias guided CHT. Although some studies have been already proposed using fuzzy theory to improve CHTs such as Liu, Fernandez, and Gao (2009) and Saha, Das, and Pal (2014). This is the first time that the fuzzy theory is used to define the search bias in the optimization process. The experimental results demonstrate that the proposed FSB is more robust than the other compared techniques.

The rest of this paper is organized as follows. Various state-of-the-art CHTs are explained and discussed in Section 2. Section 3 describes the motivation and the detailed content of the proposed FSB. The experimental results and discussion are given in Section 4. Finally, Section 5 concludes the paper.