Probabilistic Analysis of the Convergence of the Differential Evolution Algorithm

Abstract

Differential evolution algorithms represent an efficient framework to tackle complicated optimization problems with many variables and involved constraints. Nevertheless, the classic differential evolution algorithms in general do not ensure the convergence to the global minimum of the cost function. Therefore, the authors of the article designed a modification of these algorithms that guarantees the global convergence in the asymptotic and probabilistic sense. The modification consists in adding a certain ratio of random individuals to each generation formed by the algorithm. The random individuals limit the premature convergence to the local minimum and contribute to more thorough exploration of the search space. This article concentrates specifically on the role of random individuals in the identification of the global minimum of the cost function. Besides, the paper also contains some useful estimates of the probability of finding the global minimum of the corresponding cost function.

ECMS 2021 Proceedings edited by Khalid Al-Begain, Mauro Iacono, Lelio Campanile, Andrzej Bargiela

“…Gradient methods, genetic algorithms and also CDEA found only a local minimum of the corresponding cost function (this optimization problem is described in more detail in (Mlýnek and Knobloch 2018) and (Mlýnek et al 2016). MDEA has also proved successful in optimizing the fibre winding procedures using a fibre-processing head and a nonbearing frame moved by an industrial robot (for more details see (Mlýnek et al 2020)).…”

“…Evolutionary algorithms are in general more computationally demanding and they are therefore suitable for calculations that are not time limited (e.g. offline calculations of trajectories of an industrial robot, see (Mlýnek et al 2020)). Their use in time critical calculations is rather limited.…”

Differential Evolution Algorithm in Models of Technical Optimization

“…The asymptotic convergence of MDEA is proved in detail in (Knobloch et al 2017), see also (Hu et al 2013). Probability estimates of reaching the global minimum after performing G generations of MDEA are given in (Knobloch and Mlýnek 2020). These estimates help to decide after how many generations to finish the MDEA calculation.…”