Differential evolution (DE) algorithm with a small population size is called Micro-DE (MDE). A small population size decreases the computational complexity but also reduces the exploration ability of DE by limiting the population diversity. In this paper, we propose the idea of combining ensemble mutation scheme selection and opposition-based learning concepts to enhance the diversity of population in MDE at mutation and selection stages. The proposed algorithm enhances the diversity of population by generating a random mutation scale factor per individual and per dimension, randomly assigning a mutation scheme to each individual in each generation, and diversifying individuals selection using opposition-based learning. This approach is easy to implement and does not require the setting of mutation scheme selection and mutation scale factor. Experimental results are conducted for a variety of objective functions with low and high dimensionality on the CEC Black- Box Optimization Benchmarking 2015 (CEC-BBOB 2015). The results show superior performance of the proposed algorithm compared to the other micro-DE algorithms.
DE is about the difference of two or more vectors. Amplification of the difference vector is adjustable by the mutation scale factor @math which controls the evolving rate of the population. A small mutation scale factor decreases the exploration and may result in premature convergence. This is while a large mutation scale factor increases the exploration which results in a longer convergence time. The optimal value of @math is usually set based on the nature of the problem and experimental observations @cite_26 . MDEVM algorithm @cite_13 relaxes static selection of @math as a hyper-parameter by generating a random @math for each dimension of an individual in the population @cite_13 . This randomness adds diversity to the population by amplifying the differences vectors in various scales. The experimental results in @cite_12 have demonstrated outstanding performance of this technique for diversity enhancement.
@cite_26 @cite_13 @cite_12
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One of the main disadvantages of population-based evolutionary algorithms (EAs) is their high computational cost due to the nature of evaluation, specially when the population size is large. The micro-algorithms employ a very small number of individuals, which can accelerate the convergence speed of algorithms dramatically, while it highly increases the stagnation risk. One approach to overcome the stagnation problem can be increasing the diversity of the population. To do so, a microdifferential evolution with vectorized random mutation factor (MDEVM) algorithm is proposed in this paper, which utilizes the small size population benefit while preventing stagnation through diversification of the population. The proposed algorithm is tested on the 28 benchmark functions provided at the IEEE congress on evolutionary computation 2013 (CEC-2013). Simulation results on the benchmark functions demonstrate that the proposed algorithm improves the convergence speed of its parent algorithm. The differential evolution (DE) algorithm suffers from high computational time due to slow nature of evaluation. In contrast, micro-DE (MDE) algorithms employ a very small population size, which can converge faster to a reasonable solution. However, these algorithms are vulnerable to a premature convergence as well as to high risk of stagnation. In this paper, MDE algorithm with vectorized random mutation factor (MDEVM) is proposed, which utilizes the small size population benefit while empowers the exploration ability of mutation factor through randomizing it in the decision variable level. The idea is supported by analyzing mutation factor using Monte-Carlo based simulations. To facilitate the usage of MDE algorithms with very-small population sizes, new mutation schemes for population sizes less than four are also proposed. Furthermore, comprehensive comparative simulations and analysis on performance of the MDE algorithms over various mutation schemes, population sizes, problem types (i.e. uni-modal, multi-modal, and composite), problem dimensionalities, and mutation factor ranges are conducted by considering population diversity analysis for stagnation and trapping in local optimum situations. The studies are conducted on 28 benchmark functions provided for the IEEE CEC-2013 competition. Experimental results demonstrate high performance and convergence speed of the proposed MDEVM algorithm.