Differential Evolution (DE) stands as one of the most esteemed algorithms within Evolutionary Computation (EC), initially introduced by Storn et al. [1]. Since its inception in 1995, DE has found wide-ranging applications in scientific and engineering domains [2], inspiring numerous EC approaches that have become pivotal benchmarks in the literature [3]. In the specialized literature, most approaches have been derived from schemes involving tailored modifications to DE's key stages, such as initialization, mutation, crossover, and selection mechanisms. Notably, mutation and crossover operators, along with the tuning of their parameters, have been the focal points of extensive research efforts.

In the last few years, a significant number of variations of the DE algorithm have been developed that have proven to be both novel and noteworthy [4]. These variations have demonstrated the algorithm's capacity for enhancement in varied ways, underscoring its adaptability and flexibility. This flexibility has enabled the DE to align with a substantial body of related research, making it challenging to comprehensively survey all relevant works in a single compendium, as evidenced by Pant et al. [5]. A notable example is the Chaotic Opposition-based-learning Differential Evolution (CODE) algorithm, which proposes the implementation of chaotic maps and an opposition-based learning strategy to develop a novel variant of DE featuring a modified initialization scheme [6]. Additionally, a proposal known as Dynamic Combination Differential Evolution (DCDE) [7] has emerged, which involves a variant of the mutation operator based on the dynamic combination and a two-level parameter regulation strategy. Additionally, the Adaptive Differential Evolution With Optional External Archive (JADE) algorithm [8] is noteworthy for implementing a novel mutation strategy with an optional external file and adaptively updating the control parameters.

Various proposals regarding mutation operators have surfaced in the literature, such as the triangular mutation [9] and the hemostasis-based mutation [10]. On the other hand, crossover operators have been presented, such as the parent-centered crossover [11] and the multiple exponential crossover [12]. However, the literature on selection stage modifications remains relatively sparse, with fewer related works emerging over the past decade. One prominent investigation in this domain is the new selection operator algorithm (NSO) proposed by Zeng et al. [13]. However, in general terms, investigations on selection mechanisms for DE represent a small fraction, approximately 2%, of the modifications reported in the specialized literature [5]. Conversely, advancements in evolutionary computation and machine learning have demonstrated substantial progress [14, 15, 16]. The development of these algorithms has created a range of possibilities for modifications, which can be implemented in specialized areas [17, 18].

To provide novel insights into the DE selection stage, this paper introduces Differential Evolution with Selection by Neighborhood (DESN), a new evolutionary algorithm with an improved selection mechanism. DESN employs a decision variable based on current function access to choose between two classical mutation strategies: DE/rand/2 and DE/best/1. Additionally, it features a selection stage where the current trial vector is compared against its individual and its neighborhood, which constitutes approximately 25% of the population. These modifications markedly enhance the algorithm's performance, improving exploration and exploitation phases compared to the original DE.

DESN's efficacy is evaluated on the benchmark functions of the 2017 Congress on Evolutionary Computation (CEC) against eight other established evolutionary and meta-heuristic algorithms. Computational experiments demonstrate DESN's superior performance in exploration and exploitation, with the Friedman test confirming its superiority. Furthermore, a Big O analysis affirms DESN's competitive computational cost, establishing it as a promising algorithm for future applications.

The paper is structured as follows: Section 2 provides a general background to understand the presented work, Section 3 details the proposed DESN, Section 4 presents computational and statistical results, and Section 5 discusses the conclusions.

The proposed DESN introduces a series of three targeted modifications to enhance the efficiency and efficacy of the standard DE algorithm. Firstly, an adjustment is made to the mutation stage, where traditional DE algorithms typically employ a predefined mutation strategy. In the proposed DESN, this stage undergoes refinement to accommodate a more nuanced approach, which aims to foster greater diversity and exploration within the population. Secondly, the selection stage, which governs the retention and propagation of candidate solutions, undergoes scrutiny and optimization in the proposed DESN. The selection process is fine-tuned through strategic adjustments and enhancements to identify better and prioritize promising candidate solutions, thereby enhancing the algorithm's ability to converge toward optimal or near-optimal solutions. Thirdly, DESN incorporates a novel control parameter adaptation scheme, leveraging the iterative process's dynamics to dynamically adjust key parameters such as scaling factor (F) and crossover rate (C⁢r). By monitoring the iterative process for signs of stagnation or diminishing returns, DESN adapts these parameters dynamically, thereby promoting sustained exploration and exploitation throughout the optimization process. Collectively, these above modifications synergistically contribute to the overall effectiveness and versatility of DESN, enabling it to tackle a wider range of optimization problems with improved efficiency and robustness compared to traditional DE algorithms.

The DE/rand/1 mutation, a cornerstone of the canonical differential evolution algorithm, is renowned for its adept exploration capabilities while exhibiting limitations in effectively exploiting promising regions. Conversely, though proficient in exploitation, the DE/best/1 mutation is prone to stagnation, particularly in multimodal problems. Recognizing the inherent trade-offs within these mutation operators, it becomes evident that no single strategy satisfactorily addresses the diverse challenges encountered across optimization landscapes.

In response to this realization, a strategic unification of mutation strategies is proposed. By leveraging the strengths of both DE/rand/2 and DE/best/1 strategies, as shown in Equation (1) and Equation (2) respectively. The aim is to synthesize a hybrid approach that capitalizes on the complementary advantages of each strategy. This unification approach seeks to balance the exploration- exploitation trade-off inherent in optimization processes, thereby enhancing the algorithm's ability to navigate complex and dynamic search spaces.

v i g + 1 = x r ⁢ 5 g + F ⁢ ( x r ⁢ 1 g − x r ⁢ 2 g ) + F ⁢ ( x r ⁢ 3 g − x r ⁢ 4 g )

v i g + 1 = x b ⁢ e ⁢ s ⁢ t g + F ⁢ ( x r ⁢ 1 g − x r ⁢ 2 g )

By integrating diverse mutation strategies, the proposed approach endeavors to harness the best capabilities of each strategy while mitigating their limitations. Through this synthesis, the algorithm aims to achieve a more robust and adaptive optimization framework capable of addressing a broader spectrum of optimization challenges with heightened efficacy and efficiency.

The classical DE/rand/1 mutation, renowned for its prowess in exploration, encounters challenges in fully exploiting promising regions within the search space. On the contrary, the DE/best/1 mutation excels in exploitation but often grapples with stagnation, particularly in complex multimodal problems. Acknowledging the inherent constraints of individual mutation operators, a compelling impetus arises to harness the collective strengths of the DE/rand/2 and DE/best/1 strategies.

Through a strategic integration of these mutation schemes, as described in Equation (1) and Equation (2), respectively, a balanced and synergistic approach to the search process emerges. By leveraging the exploratory capabilities of DE/rand/2 alongside the exploitation-enhancing features of DE/best/1, the consolidated strategy seeks to mitigate the shortcomings of each method.

This unification aims to foster a more robust and adaptive optimization framework capable of navigating the intricacies of diverse optimization landscapes with heightened efficacy and resilience. By orchestrating a harmonious interplay between exploration and exploitation, the proposed approach strives to transcend the limitations inherent in traditional mutation strategies, thereby advancing the frontiers of evolutionary optimization methodologies.

Determining which strategy to employ relies on the sigmoid function depicted in Equation (3). Essentially, this function governs the likelihood of utilizing the DE/rand2 mutation.

P ⁢ m ⁢ u ⁢ t = 1 1 + e 1.05 − ( F ⁢ E m ⁢ a ⁢ x / F ⁢ E ) 2

From Equation (3) F⁢E corresponds to the number of function evaluations. Figure 1 visually represents this function. Observation of the plot reveals that within the initial 25 % of function evaluations, the probability of selecting the exploration-oriented mutation stands at approximately 0.95. Subsequently, in the middle phase of the search process, this probability declines to around 0.82. Towards the concluding stages, it further diminishes to approximately 0.45, affording the algorithm greater opportunities to exploit the present region. As illustrated in Figure 1, this approach fosters a balanced integration of the diverse mutation options, meticulously calibrated in a strategic manner. The interchangeable behavior between the different mutations, with a variable probability percentage guided by the sigmoid function, generates a superior combination of exploration and exploitation, thereby enhancing the algorithm's performance.

Figure 1.

Indicator of the percentage of probability for the chosen mutation.

This study introduces a novel evolutionary strategy named DESN, which builds upon differential evolution principles. The algorithm features an iterative selection process that dynamically alternates between two popular mutation schemes based on the number of function evaluations. Additionally, it enhances the traditional greedy selection stage of DE by incorporating comparisons not only with the current individual but also with neighboring individuals, representing approximately 25% of the population. These adaptations have notably elevated the algorithm's performance, demonstrating superiority over DE and other approaches across all metric experiments using the CEC-2017 test problems.

The evaluation, which includes Friedman's non-parametric test and Big O analysis, underscores the remarkable competitiveness of the proposed algorithm. However, to further validate its efficacy, DESN necessitates evaluation across diverse benchmark function datasets and testing within high-dimensional problem domains. Furthermore, it is essential to explore its applicability in digital image processing and training machine learning algorithms to ascertain its real-world performance. These avenues, among others, constitute potential directions for future research on DESN. Despite these avenues for exploration, the findings of this study position DESN as a promising evolutionary strategy within the current landscape of optimization techniques.

However, we recognize the challenges associated with this algorithm in certain tasks. We acknowledge its potential in various domains, including multi-objective optimization, in real-world and high-dimensional engineering problems. Consequently, future work will focus on developing this methodology to make it more robust and applicable to different problems, such as implementations with machine learning models.