**1. Introduction**

The single objective global optimization problem involves finding a solution vector *x* = (*x1*, ... , *xD*) that minimizes the objective function *f*(*x*), where *D* is the dimension of the problem. The task of black box optimization is to solve the global optimization problem without clear objective function form or structure, that is, f is a "black box". This problem appears in many problems of engineering optimization, where complex simulations are used to calculate the objective function.

The differential evolution (DE) algorithm, proposed by Price and Storm in 1995, laid the foundation for a series of successful algorithms for continuous optimization. DE is a random black box search method, which was originally designed for numerical optimization problems [1], and it's also an evolutionary algorithm that ensures that every next generation has better solutions than the previous generation: a phenomenon known as elitism. The extensive study fields of DE are summarized lately in the references [2].

Studies on DE have yielded a number of improvements [3–17] to the classical DE algorithm, and the status of research on it can be easily obtained by noting the results of the Continuous Optimization Competition and the Evolutionary Computing Conference (CEC).

A popular variant of DE [18] is the algorithm proposed by Fukunaga and Tanabe called Success History-based Adaptive Differential Evolution (SHADE) [19]. In the optimization process, the scale factor *F* and the crossover rate *CR* of control parameters are adjusted to adapt to the given problem, and the "current to *p*best/1" mutation strategy and the external archive of poor quality solutions in JADE [20] are combined in SHADE. The SHADE algorithm ranked third in CEC2013. In the second year, the author proposed an improved scheme, adding a linear reduction to the population size called L-SHADE to improve the convergence rate of SHADE [21]. L-SHADE won the CEC2014 competition. The winners in the subsequent years were SPS-L-SHADE-EIG [22] (CEC2015), LSHADE-EpSin [23] (joint winner of CEC2016), and jSO [24] (CEC2017). These algorithms are all based on L-SHADE, which makes it one of the most effective variants of SHADE [25]. With the exception of the jSO, the other winners benefited from general enhancements in the area [26]. Consequently, this study applies an improved method to the SHADE, L-SHADE and jSO algorithms. LSHADE-ESP [27] came in second in CEC2018 and the jDE100 [28] won CEC2019. And the j2020 [29] algorithm, which was proposed on CEC2020 recently, is also within the reference range. Enhanced versions of these DE algorithms add new mechanisms or parameters for optimization, similar to those in other optimization algorithms [30], as described in substantive surveys of these areas [25,31–37]. Moreover, theoretical analysis supporting DE has also been provided, such as in [38–41].

The DE consists of three main steps: mutation, crossover, and selection. Many proposals [6,10,11,14, 24] have been made to improve the mutation process to improve optimization performance. For instance, four strategies of combining mutation and crossover was used in SHADE4 [6], SHADE44 [10] and L-SHADE44 [11] to create a new trial individual and realize an adaptive mechanism. A novel multi-chaotic framework was used in MC-SHADE [14] to generate random numbers for the parent selection process in mutation process. A new weighted mutation strategy with parameterization enhancement was used in jSO [24] to enhance adaptability. This paper also focuses on improving this process in the DE algorithm, especially SHADE-based algorithms.

The CEC2020 [42] single-objective boundary-constrained numerical optimization benchmark sets are designed to determine the improvement in performance obtained by increasing the number of the calculation of the fitness function of an optimization algorithm. There are thus two motivations for this study. First, we need to solve the problem of premature convergence of algorithms based on SHADE in high dimensional search spaces on CEC2020 benchmark sets, so that they can maintain a high population diversity and a longer exploration phase. Second, the improvement to the algorithm should be simple, should not excessively increase complexity, and should not render the proposed algorithm incomprehensible and less applicable, as discussed in [43]. We proposed a method using turning-based mutation, and apply it to the SHADE, L-SHADE, and jSO algorithms to yield good performance while using relatively simple algorithm structure. Through experimental analysis involving 10, 15, and 20 dimensions, the improved algorithms achieved better performance than the original algorithms as well as the advanced DISH [44] and jDE100 algorithms on CEC2020 benchmark sets, but were slightly worse than the j2020 algorithm. We also use population diversity measure and population clustering analysis to verify the effectiveness of the proposed method.

Section 2 describes the process of evolution from the DE algorithm to the SHADE, L-SHADE, and jSO algorithms, and turning-based mutation is introduced in Section 3. The experimental settings and results are described in Sections 4 and 5, respectively. Section 6 discusses the results, and the conclusion of this paper is given in Section 7.

#### **2. DE SHADE L-SHADE and jSO**
