FSB-EA: Fuzzy search bias guided constraint handling technique for evolutionary algorithm
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):where f(x) is the objective function, gj(x) is the jth inequality constraint, hk(x) is the kth equality constraint. The ith variable is varied on the range .
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:
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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.
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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.
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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.
Section snippets
Related work
The constraint violation of solution x can be defined as follows:where δ is a tolerance parameter for equality constraints. The differences between different CHTs can be reflected in the definition of search bias. Let (ϕ1, f1) and (ϕ2, f2) be the function value and the constraint violation set of solutions x1 and x2, respectively. (ϕ1, f1)≺(ϕ2, f2) means that solution x1 biases the search towards solution x2. In this section, we present five
Proposed method
From the above CHTs review, we can conclude that there is no CHT that can outperform all the others CHTs in all aspects. In fact, there are many factors that can affect on the definition of the search bias. Usually, we tend to consider some factors while ignoring some others. Moreover, the search bias is uncertain. Thus, it is too difficult to accurately define the search bias. More precisely, using exact expressions to define an uncertain concept may be the main reason that makes most of the
Experiment and discussion
It is worth noticing that experiments in many studies often do not distinguish between the contribution of the CHT and the search strategy to problem optimization. In contrast with those studies, we test the performance of CHTs for both different EAs as well as different problems. The details of the experimental settings with the results are described below.
Conclusion
CHT is significant and critical task for solving COP. We assume that the use of deterministic expressions to define non-deterministic concepts is the main reason why most existing CHTs are difficult to improve robustness. It is the purpose of this paper in which a more robust CHT for different EAs and COPs is developed. In our study, the robustness of the different state-of-the-art CHTs with different EAs on solving different types of problems is discussed and verified by experiments.
Acknowledgment
We are particularly grateful to Deb K, P. Runarsson, Suganthan P N and Yong Wang who share their source code to us that help us a lot. And the work reported in this paper was supported by the National Natural Science Foundation of China (No.61672215, U1613209, No. 61702180), the Postgraduate Research and Innovation Project of Hunan Province (CX2016B067), the Doctoral Scientific Research Foundation of Hunan University of Science and Technology (No.E51790), the Open Research Fund of Hunan
References (38)
- et al.
Simulated binary crossover for continuous search space
Complex Systems
(1994) - et al.
Dynamic multi-swarm particle swarm optimizer with a novel constraint-handling mechanism
Proceedings of IEEE international conference on evolutionary computation, California, USA
(2006) Knowledge representation in fuzzy logic
IEEE Transactions on Knowledge & Data Engineering
(1989)- et al.
Handling constraints using multiobjective optimization concepts
International Journal for Numerical Methods in Engineering
(2004) - et al.
A critical review of adaptive penalty techniques in evolutionary computation
Evolutionary constrained optimization
(2015) - et al.
A multiobjective optimization-based evolutionary algorithm for constrained optimization
IEEE Transactions on Evolutionary Computation
(2006) COELLO COELLO. Constraint-handling using an evolutionary multiobjective optimization technique
Civil Engineering & Environmental Systems
(2000)- et al.
Constraint-handling using an evolutionary multiobjective optimization technique
Civil Engineering & Environmental Systems
(2000) - et al.
Quantum-inspired hyper-heuristics for energy-aware scheduling on heterogeneous computing systems
IEEE Transactions on Parallel and Distributed Systems
(2016) - et al.
Probabilistic constraint handling in the framework of joint evolutionary-classical optimization with engineering applications
Proceedings of IEEE congress on evolutionary computation
(2012)
An efficient constraint handling method for genetic algorithms
Computer methods in applied mechanics and engineering
GA with a new multi-parent crossover for constrained optimization
Proceedings of IEEE congress on evolutionary computation, LA, USA
Self-adaptive fitness formulation for constrained optimization
IEEE Transactions on Evolutionary Computation
System identification through simulated evolution: A machine learning approach to modeling
Performance comparison of local search operators in differential evolution for constrained numerical optimization problems
Proceedings IEEE symposium on differential evolution
A review on constraint handling strategies in particle swarm optimisation
Neural Computing & Applications
Variants of an adaptive penalty scheme for steady-state genetic algorithms in engineering optimization
Engineering Computations
Orthogonal chemical reaction optimization algorithm for global numerical optimization problems
Expert Systems with Applications
Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization
International Journal of Computer Assisted Radiology & Surgery
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