Controlling search magnitude using a beta distribution for bounded continuous optimization

Abstract

Evolutionary algorithm (EA), mimicking biological evolution in nature, is a well-known optimization algorithm for solving a black-box optimization problem. Different from gradient-based methods, EA does not require any prior knowledge about the problem. Different from EA, swarm intelligence (SI), mimicking collective behavior in nature, is another optimization algorithm. Generally, EA and SI start with a randomly distributed population and update the population by reproducing offspring with unique operators. For the performance improvement, additional components are combined with EA and SI or numerous reproduction variants are proposed in EA and SI. With regard to the additional components, opposition-based learning (OBL), introduced as a new scheme for machine intelligence, has been broadly combined with EA and SI due to the simplicity and adaptability. In OBL, estimates and counter estimates are considered simultaneously to accelerate the search or learning process. Mathematical and experimental proofs show that opposite points are more beneficial than random points. With regard to the reproduction variants, parameter-oriented reproduction variants which change the search step magnitude and/or direction to take effective steps and individual distribution-oriented reproduction variants which distribute the individuals in the population using distribution center and distribution degree have been proposed. Generally, the former group (parameter-oriented) requires parameters that control the search parameters of the original algorithms, whereas the latter group (individual distribution-oriented) appears to be almost parameter-free. Also, in many proposed algorithms, randomness is added to the algorithms to accelerate the search process, make the algorithm robust, and escape the local optima. A beta distribution, which is rarely used, is a probability distribution whose characteristic is defined by two parameters. The beta distribution has an advantage over other probability distributions in that its domain is bounded and it provides a variety of shapes depending on its parameters. However, because the existing researches use mean and standard deviation when adopting the beta distribution, there exist some limitations. Therefore, in this thesis, new parameters are defined to use a beta distribution without limitation and two beta distribution utilizations to control the search magnitude of the algorithm are proposed: one for adding randomness to OBL and the other for individual distribution-oriented reproduction. The first utilization is OBL using a beta distribution with changes in partial dimensions and selection scheme (BetaCOBL). BetaCOBL controls degree of opposition through various shapes of the beta distribution, changes a subset of dimensions into opposite values, and switches the selection scheme. The second utilization is a beta distribution-based bare bones reproduction (B³R). B³R controls the search magnitude through the beta distribution and mixes the information of the original individual with the reproduced offspring. BetaCOBL and B³R are developed into DE embedding BetaCOBL (BetaCODE) and two-phase B³R optimization (TPBO) respectively. The proposed algorithms are tested on various test functions and two real world applications and compared with other algorithms with respect to the performance criteria. The results indicate that the proposed algorithms outperform or perform comparatively to the comparison group especially in terms of solution accuracy and reliability.