Enabling the Extended Compact Genetic Algorithm for Real-Parameter Optimization by Using Adaptive Discretization

Abstract

An adaptive discretization method, called split-on-demand (SoD), enables estimation of distribution algorithms (EDAs) for discrete variables to solve continuous optimization problems. SoD randomly splits a continuous interval if the number of search points within the interval exceeds a threshold, which is decreased at every iteration. After the split operation, the nonempty intervals are assigned integer codes, and the search points are discretized accordingly. As an example of using SoD with EDAs, the integration of SoD and the extended compact genetic algorithm (ECGA) is presented and numerically examined. In this integration, we adopt a local search mechanism as an optional component of our back end optimization engine. As a result, the proposed framework can be considered as a memetic algorithm, and SoD can potentially be applied to other memetic algorithms. The numerical experiments consist of two parts: (1) a set of benchmark functions on which ECGA with SoD and ECGA with two well-known discretization methods: the fixed-height histogram (FHH) and the fixed-width histogram (FWH) are compared; (2) a real-world application, the economic dispatch problem, on which ECGA with SoD is compared to other methods. The experimental results indicate that SoD is a better discretization method to work with ECGA. Moreover, ECGA with SoD works quite well on the economic dispatch problem and delivers solutions better than the best known results obtained by other methods in existence.

An introduction and survey of estimation of distribution algorithms

“…Finally, some regions of the search space are more densely covered with high quality solutions whereas others contain mostly solutions of low quality; this suggests that some regions require a more dense discretization than others. To deal with these difficulties, various approaches to adaptive discretization were developed using EDAs (Tsutsui, Pelikan, & Goldberg, 2001;Pelikan, Goldberg, & Tsutsui, 2003;Chen, Liu, & Chen, 2006;Suganthan, Hansen, Liang, Deb, Chen, Auger, & Tiwari, 2005;Chen & Chen, 2010). We discuss some of these next.…”

“…The resulting algorithm was shown to be successful on the two-peaks and deceptive functions. Another way to deal with discretization was proposed by Chen and Chen (2010). Their method uses the ECGA model and a split-on-demand (SoD) discretization to adjust on the fly how the realvalued variables are coded as discrete values.…”

Quality Analysis of Discretization Methods for Estimation of Distribution Algorithms

“…To enhance the applicability of EDAs over continuous domains, direct attempts to modify the type of decision variables have been made, including continuous population-based incremental learning with Gaussian distribution [31], real-coded variant of population-based incremental learning with interval updating [32], Bayesian evolutionary algorithms for continuous function optimization [33], real-coded extended compact genetic algorithm based on mixtures of models [18], and the real-coded Bayesian optimization algorithm [1]. Instead of modifying the infrastructure of the algorithm, such as the type of decision variables or the global program flow, as a more general, component-wise approach, discretization methods are employed to cooperate with EDAs [6], [7], [28], [35]. Discretization methods enable EDAs designed for discrete variables to solve continuous optimization problems without the need of altering algorithmic structures.…”

Use model building on discretization algorithms for discrete EDAs To work on real-valued problems

“…Existing discretization algorithms, such as the fixed-height histogram [6] (FHH) and the split-on-demand [7]- [9] (SoD), use only the densities of selected chromosomes to discretize the continuous region.…”

“…At the last section, we focus on comparing these newly formed discretization algorithms with other discretization algorithms, their original versions, FHH and SoD. The discretization algorithms are integrated with ECGA to be tested on the 25 benchmark problems [11] used in the SoD paper [7]. Results show that ev-FHH and dev-FHH outperforms FHH, and ev-SoD and dev-SoD outperforms SoD on about 20 out of 25 benchmark problems [11] with 95% confidence.…”