A novel enhanced cuckoo search algorithm for global optimization

Abstract

Cuckoo search algorithm (CS) is an excellent nature-inspired algorithm that has been widely introduced to solve complex, multi-dimensional global optimization problems. However, the traditional CS algorithm has a low convergence speed and a poor balance between exploration and exploitation. In other words, the single search strategy of CS may make it easier to trap into local optimum and end in premature convergence. In this paper, we proposed a new variant of CS called Novel Enhanced CS Algorithm (NECSA) to overcome these drawbacks mentioned above inspired by the cuckoos’ behaviors in nature and other excellent search strategies employed in intelligent optimization algorithms. NECSA introduces several enhancement strategies, namely self-evaluation operation and modified greedy selection operation, to improve the searchability of the original CS algorithm. The former is proposed to enhance the exploration ability and ensure population diversity, and the latter is employed to enhance the exploitation ability and increase search efficiency. Besides, we introduced adaptive control parameter settings based on the fitness and iteration number to increase the convergence speed and the accuracy of the search process. The experimental results and analysis on the CEC2014 test have demonstrated the reliable performance of NECSA in comparison with the other five CS algorithm variants.

Keywords

Cuckoo search self-evaluation mechanism greedy selection

Nomenclature

The population scale

The problem dimension

The current iteration number

The maximum number of iterations

The switch parameter

Pa _max

The maximum value of the switch parameter Pa

The weight value of evaluation

The control parameter of greedy selection

Step scale

L (s, λ)

Characteristic scale

Power coefficient

⊕

The entry-wise multiplications

The gamma function

S ₀

Initial step of Lévy flight

H (u)

Heaviside functio

random number

Upper bound of the solution

Lower bound of the solution

Upper bound of the step scale

Lower bound of the step scale

List of abbreviations

NFL

‘No free lunch’ theory

MGS

Modified Greedy Selection operation

ACS

Adaptive CS

ACSA

Adaptive CS

GBCS

Gaussian bare-bone CS

ECS

Enhanced CS

NECSA

Novel enhanced CS

QOBL

Quasi-opposition based learning

BHCS

Biogeography-based heterogeneous CS (BHCS)

GBDE Gaussian bare-bones DE algorithm

Andersen-Tawil syndrome

LSAL

Solis and Wets local search method and augmented Lagrangian

HCSNM

Hybrid cuckoo search and Nelder–Mead method

1 Introduction

Meta-heuristic algorithms have been extensively applied to deal with nonlinear and complex global optimization problems. Compared with heuristic algorithms, which are primarily problem-specific algorithms, meta-heuristic algorithms generally do not require specific auxiliary information so that they can be applied to a wider range of practice. Besides, meta-heuristic algorithms are usually nature-inspired, some of them have become popular for their excellent global optimization ability, such as Genetic Algorithm (GA) [1] inspired by the theory of evolution, Simulated Annealing algorithm (SA) [2] inspired by the natural random phenomenon, Particle Swarm Optimization algorithm (PSO) [3] inspired by the predatory behavior of fish or birds, Artificial Bee Colony algorithm (ABC) [4] inspired by honey-collecting behavior of bee colony, Dragonfly algorithm (DA) [5] inspired by the static and dynamic swarming behaviors of dragonflies, Ant Lion optimization algorithm (ALO) [6] imitates the hunting mechanism of antlions, and Cuckoo Search algorithm (CS) [7] inspired by cuckoos’ parasitic reproduction and random walk.

CS is a new meta-heuristic algorithm proposed by Yang and Deb [7]. Inspired by natural phenomena, CS imitates the cuckoos’ breeding behavior and employs Lélight to implement the combination of rules and randomness. The core of CS is evolution operation, which contains two periods, namely exploration and exploitation, to improve the ability to search global optimum. The general idea of CS is to generate new individuals through mutation and crossover operations and then compare the fitness of previous and fresh individuals. The best one will be selected as offspring, and the iteration operation will continue unless achieving the max-generation or stop criterion. CS has the advantages of simple coding, few control parameters, and robust optimization results. CS algorithm has been applied for solving not only theoretical research but also various practical global optimization problems, such as clustering and data mining [8, 9], image processing [10 –12], engineering design [13, 14], path planning [15 –17] and so on.

However, because of the unitary search strategy, CS is likely to fall into local optimum or premature convergence when processing highly complex problems. Many CS variants have been developed to overcome the two weaknesses mentioned above, which can be roughly divided into the following three aspects: the improvement of control parameters (i.e., switch parameter Pa, scaling factor α and so on), the adjustment of search strategy (i.e., replacing Lévy flight, using multi-strategies and so on), and the hybrid algorithms based on CS and other intelligent optimization algorithms, such as GA, ABC and so on.

Many works have focused on the improvement of control parameters. Valian et al. [18] presented a new method to adaptively control the values of Pa and α according to the current iteration number to improve the accuracy and convergence rate. Walton et al. [19] designed an adaptive Lévy flight with a non-linearly decreasing method based on iteration number to enhance local search. Ong et al. [20] proposed an adaptive step size adjustment operation based on current individuals’ fitness to improve the convergence speed. Naik et al. [21] applied a new equation based on provisional evaluation results to update step size without Lévy distribution to enhance the performance. Guerrero [22] et al. introduced a fuzzy system to Lévy flight to adapt the parameters dynamically. The simulation results proved this algorithm was better than the original CS algorithm. Mareli and Twala et al. [23] designed three updating formulas to increase switching parameters dynamically, and then compared the experimental results of this new variant with other improved CS algorithms, which employed constant or dynamically decreasing switch parameters to prove the robustness of this algorithm. To avoid falling into local optimum and low convergence rate, Liu et al. [24] employed variational parameter and logistic map to modify original CS, and the proposed algorithm can effectively deal with both high and low dimensional problems.

Some significant works have focused on modifying search strategies to improve searchability and convergence speed. To fully discover the potential of CS, Zhang et al. [25] introduced grouping, parallel, incentives, information sharing, and adaptive operations to CS algorithm, and the simulation results illustrated its remarkable ability to deal with multi-objective, dynamic problems. In Ref [26], Li et al. introduced an orthogonal learning strategy to the CS algorithm to enhance the local searchability. Liu and Fu et al. [27] utilized chaos theory to increase the diversity of the initial population and employed inertia weight and local mechanism of frog leaping algorithm to enhance both global and local search. For preventing premature convergence, Cheung [28] et al. introduced a quantum-based strategy to the CS algorithm. Suresh et al. [29] proposed a series of enhancement methods, including chaotic initialization, adaptive Lévy flight, and mutative randomization to realize the enhancement of various satellite images. To increase the convergence speed and better balance the local and global search, Kamoona et al. [30] employed the Gaussian random walk to replace the Lévy distribution and designed the greedy selection approach to guarantee the global optimum. Inspired by PSO, Peng et al. [31] proposed a novel CS algorithm, which generates new solutions via Lévy flight or Gaussian bare-bones method randomly, and the obtained results had been proved promising. Gao et al. [32] developed a new kind of search strategy consisting of five sub-strategies to achieve less computational expense and better performance; these sub-strategies were selected based on the effect of the previous generation. Salgotra et al. [33] used adaptive parameters to avoid parameter tuning, then designed proportional population reduction based on the fitness of current best and previous best solution to decrease computational complexity. Then, to improve the tendency of global and local search, the Gaussian sampling mechanism was employed. They also utilized the concept of Weibull distributed probability switching for even better balance exploration and exploitation. In Ref [34], the author established a search framework consisting of three learning strategies, and these strategies were dynamically regulated according to the value of predefined parameters; the experimental results showed that the obtained solutions were more stable. Kang et al. [35] introduced a quasi-opposition based learning (QOBL) scheme to update the initial population and a dynamic adaptation strategy to generate new populations instead of Lévy flight.

Apart from the modifications mentioned above, plenty of works also focuses on the hybridization of CS and other intelligent optimization algorithms to increase the efficiency of dealing with different optimization problems. Kanagaraj et al. [36] proposed a hybrid algorithm of CS and GA called CS–GA for solving reliability–redundancy allocation problems. In Ref [37], the author presented a hybrid CS and GA algorithm called HCSGA to solve engineering design optimization problems involving specific constraints and various mixed variables. Long et al. [38] attempted to hybridize the augmented Lagrangian method, Solis and Wets local search and CS called HCS-LSAL for constrained global optimization, based on an augmented Lagrangian function for constraint-handling. Ali and Tawhid et al. [39] combined Nelder–Mead method with original CS, this proposed algorithm was called HCSNM. Mlakar et al. [40] introduced three features to CS, including a balancing the exploration search strategies, self-adaptation of control parameters, and a linear population reduction. Alkhateeb and Abed-Alguni et al. [41] utilized SA to hybridize CS to strengthen the solutions and designed four variations based on this idea. Zhang et al. [42] developed a hybrid algorithm based on CS and Differential Evolution (DE) called CSDE. This algorithm would divide the population into two subgroups in advance and apply CS and DE, respectively. The division operation could improve the information sharing and stick to these two meta-heuristic algorithms’ strengths for solving constrained engineering problems. Chen and Yu et al. [43] proposed an algorithm called BHCS, a hybridization of biogeography-based optimization (BBO) and CS, to improve the parameter estimation of solar photovoltaic models. BHCS had a good balance between exploration and exploitation. Abed-alguni et al. [44] proposed a CSBHC algorithm based on CS and β-hill climbing algorithm (BHO) [45] to achieve a better balance between computational time and search accuracy.

In the CS algorithm, the parameters are typically set constant. However, it has been proved that by introducing adaptive parameters to control the search process dynamically, the searchability of the CS algorithm can be guaranteed in terms of both convergence rate and accuracy [46]. The modifications of the parameters usually focus on the values of P_a and α. In the preliminary search phase, the values of P_a and α are typically set big enough to enhance the diversity of the solutions at the early phase to strengthen the global search. With the continuation of the search process, P_a and α will gradually decrease to ensure the implementation of local search. This idea excellently improves the balance between exploration and exploitation. Wang and Zhou et al. [47] introduced new equations based on randomization operation and predefined initial values to update the value of P_a at different iteration phases, and the experimental results obtained from the Lorenz system and Chen system were promising and effective when compared with other CS variants. Zendaoui and Layeb et al. [48] proposed an improved cuckoo search based on integer permutations-based Lévy flight and a decoding mechanism to optimize Bin Packing Problem. Suresh et al. [49] presented an improved CS using adaptive α based on the number of generations and randomly selected values to handle the multi-spectral satellite image denoising problems. Pankaj et al. [50] proposed a self-adaptive CS algorithm, which could dynamically change the values of P_a and α based on the iteration number and the fitness of current best and worst individuals.

Besides, the idea of modifying the search strategy of the original CS algorithm also attracted much attention. The single search strategy of the CS algorithm, Lévy flight, which obeys heavy tail distribution, may lead to the new solutions generated in the late-phase are more likely to be far from the potential optimum. In other words, the search ability of a single Lévy flight may be less than the expectation when dealing with complex or multi-dimensional problems. Different search strategies have been employed to modify the original meta-heuristic algorithms already. For example, Yelghi et al. [51] brought a new strategy, namely Tidal Force formula, to modify the Firefly Algorithm (FA) [52] and obtained outstanding performance. In Ref [53], Gao et al. proposed a modified ABC algorithm using several new search mechanisms and a novel initialization operation, and the results demonstrate its promising performance drawn from multiple improvement measures.

In several modifications of search strategies, Gaussian distribution is one of the most frequently adopted ones, and the experimental results are promising. In Ref [54], Wang et al. introduced Gaussian distribution to the DE algorithm and proposed a Gaussian bare-bones DE algorithm (GBDE). This algorithm was almost parameter-free. For improving the ABC algorithm, Zhou et al. [55] proposed a new strategy to generate the new food source; this strategy would randomly select the update method based on randomization and predefined CR values.

Despite the poor search ability, some drawbacks of original CS deserve more study. In CS, the exploration is implemented using the Lévy flight, and the exploitation is realized through randomly screening and regenerating some nests with a certain probability. Despite CS has obtained promising performance, it still has some shortcomings, such as the poor balance between global and local search and low convergence speed [56]. Specifically, because the search strategy relies on stochastic selection entirely, the convergence speed and the assignment of different search phases in the CS algorithm cannot be well ensured.

To overcome this, we can draw some ideas from some well-established modifications of CS. In Ref [19], Walton et al. proposed an information-sharing mechanism to dynamically balance the exploration and exploitation based on the search results. In Ref [57], Tuba et al. introduced the sorted fitness matrix to replace the only permuted one to generate solutions and had obtained better performance. In this paper, we decide to focus on improving the balance in terms of the poor balance between exploration and exploitation, and the convergence speed of the CS algorithm. Based on the contributions aforementioned, we can conclude that some ideas are utilized in our improved algorithm. Specifically speaking, the adaptive parameters and the information-sharing based enhancement strategy can enhance the convergence trend and speed up the convergence speed very well; the Gaussian random walk can ensure the balance between global and local search because it is more likely to generate more proper step size in comparison with the Lévy flight.

In this paper, we employed several methods to improve the original CS algorithm. Specifically, to improve the convergence ability, we introduced adaptive parameters P_a and α to replace the constant one, and an information-sharing based enhancement strategy, namely the modified greedy selection operation, to enhance the newly-generated solutions after the search periods. Besides, we proposed a nature-inspired self-evaluation operation, namely, a balanced search strategy consisting of Gaussian random walk and Lévy flight, to achieve the trade-off between exploration and exploitation. The search strategies of the proposed algorithm in this study are designed to not only simplify the complexity but also obtain higher efficiency. The proposed algorithm is called the Novel Enhanced CS Algorithm (NECSA), and the main contributions of this study are as follows:

Adaptive parameters are introduced to enhance the population diversity and convergence rate.

Gaussian distribution is introduced to cooperate with Lévy flight, and new generations will be generated using different search strategies dynamically based on the evaluation results.

Modified greedy operation is employed in the search process to enhance the convergence trend.

Experiments of NECSA and other competitive algorithms demonstrate the effectiveness of robustness in optimization problems.

The rest of this paper is organized as follows. The original CS algorithm is briefly introduced in Section 2. Section 3 introduces the novel CS variant, NECSA in detail. In Section 4, the experimental results and analysis are presented. In the last section, Section 5, we conclude this paper.

2 Cuckoo Search Algorithm

Cuckoo Search Algorithm is a novel nature-inspired meta-heuristic algorithm Yang and Deb proposed in 2009 [7]. This algorithm is based on swarm intelligence. CS is inspired by the brood parasitism behavior of some cuckoo species. These cuckoos usually lay their eggs in the nests of other bird species, and as a result, the host birds are possible to discover the foreign eggs and choose to either discard the eggs or the nest and build new ones. This has led to the evolution of parasitic cuckoos, that some female cuckoos can imitate the colors and patterns of the host birds’ eggs to increase the survival possibility of baby birds. For the purpose of better simulating the complex behaviors of cuckoos, three ideal rules have been proposed in the original CS.

Each cuckoo selects only one nest each time and lays only one egg in it.

The nest with the best quality will be reserved for the next generation.

The number of available hosts is fixed, and the host birds can discover the foreign eggs with a probability P_a (P_a ∈ [0, 1]). At this point, the host will either throw away the egg or abandon the nest and rebuild a new one.

Based on the rules proposed above, each egg in the nest represents a solution, and the egg of cuckoos represents the solution. The main purpose is to replace worse solutions with new or potential better solutions. Additionally, the CS algorithm introduces Lévy flight to implement the global search. Lévy flight is a stochastic equation for the random walk. It simulates the foraging process of animals and has characteristics of heavy-tail. This kind of random walk belongs to the classification of the Markov chain. The next state is determined by the current state and transition probability.

In implement process, the CS algorithm is composed of local random walk and global random walk. And the balance between them is controlled by the switch parameter P_a (P_a ∈ [0, 1]). The schematic diagram of the original CS is presented in Fig. 1.

Fig. 1

The schematic diagram of CS.

The global random walk realized by Lévy flight is defined as the following Equation (1):

$\begin{matrix} x_{i}^{t + 1} = x_{i}^{t} + α L (s, λ) . \end{matrix}$ (1)

where $\begin{matrix} L (s, λ) = \frac{λ Γ (λ) sin (π λ / 2)}{π} \frac{1}{s^{1 + λ}}, (s ⪢ s_{0} > 0) . \end{matrix}$ (2)

where $x_{i}^{t}$ denotes the i^th individual of t generation, $α = α_{0} (x_{i}^{t} - x_{best}^{t}) (α > 0)$ is the step scale, and $x_{best}^{t}$ is the best individual of t generation and a₀ is usually taken as 0.01, L (s, λ) represents the characteristic scale, λ (1 < λ < 3) represents the power coefficient, ⊕ denotes the entry-wise multiplications, and Γ is the gamma function. Besides, S₀ is the initial step of Lévy flight and |s₀| ⪢ 0.

Mantegna’s algorithm [58] is introduced to implement the Lévy flight in the CS algorithm. The step size of Lévy flight is named as s in Eq.(3) and designed as follows: $\begin{matrix} s = \frac{U}{{| V |}^{1 / λ}} . \end{matrix}$ (3)

U and V are two different values taken from the normal distribution: $\begin{matrix} U \sim N (0, σ^{2}), V \sim N (0, 1) . \end{matrix}$ (4)

where $\begin{matrix} σ^{2} = {[\frac{Γ (1 + λ)}{λ Γ ((1 + λ) / 2)} \cdot \frac{sin (π λ / 2)}{2^{(λ - 1) / 2}}]}^{1 / λ} . \end{matrix}$ (5)

In the CS algorithm, the local search strategy can be described as follows: $\begin{matrix} x_{i}^{t + 1} = {\begin{matrix} x_{i}^{t} + α s \otimes H (p_{a} - ɛ) \otimes (x_{j}^{t} - x_{k}^{t}), r > P_{a} \\ x_{i}^{t}, others \end{matrix} \end{matrix}$ (6) where H (u) is the Heaviside function, ɛ and r ∈ (0, 1) are two random numbers drawn from the uniform distribution, p_a is a constant switch parameter that is normally taken as 0.25.

Based on the formulas and description mentioned above, the pseudo-code of the CS algorithm is shown as bellow:

Algorithm 1

CS algorithm
Input: N: the population scale
D: the problem dimension
t: the current iteration number
T: the maximum number of iterations
Pa: the switch parameter
1:t=1;
2: Randomly generate an initial population x of N solutions;
3: Evaluate all the solutions →f_min;
4: whilet < Tdo
5: fori = 1 : Ndo
6: Generate new population x_new using Lévy flight as shown
in Equation (2);
7: Evaluate all the solutions →f_new;
8: iff_new < f_minthen
9: x = x_new;
10: f_min = f_new_;
11: end if
12: end for
13: Discard a fraction (Pa) of worst nests and generate new
solutions as shown in Equation (6);
14: Update the global optimal solution;
15: t = t + 1
16: end while
Output: Optimal solution

3 Novel enhanced CS algorithm

3.1 Motivation

Plenty of modifications of CS merely focused on the improvement of adaptive control parameters. Although this method helps improve the search performance to a certain extent, a single introduction of adaptive parameters is hard to obtain global optimum all the time, especially in the face of multi-dimensional and highly nonlinear problems. Therefore, numerous CS variants based on strategy have been proposed to enhance the searchability and convergence rate. According to the ‘No free lunch’ theory (NFL) [59], algorithms cannot consistently deal with all kinds of problems very well. When an algorithm is good at dealing with one type of problem, it may likely be insufficient in a different type of problem. In another world, simple replacement of search strategy or modification through improving parameters concerning the Lévy flight still cannot ensure the performance. So, research on multi-strategy CS algorithms has become popular. Moreover, the complexity of these algorithms is a problem worth consideration [34]. Overly complex coding may result in poor efficiency and computational burden. Besides, most variants concentrate on modifying mutation operation, Lévy flight. In the CS algorithm, both Lévy flight and greedy selection are necessary for balancing the exploration and exploitation phases.

In nature, birds evaluate their livability based on the current location or related nest information before choosing a nest location. Similarly, cuckoos will take several methods to increase their offspring’s survival rate. When cuckoos implement parasitic behavior, they may evaluate the current alternative nest first and then determine whether to choose it according to the evaluation results. And when evaluating, they usually adopt stricter standards. Besides, they typically fine-tune the nest at the beginning to obtain a more promising future and at the end to achieve the effect of reconfirmation. Inspired by these behaviors, we proposed several enhancement strategies to improve the performance of the CS algorithm in this paper.

Based on the consideration mentioned above, a novel enhanced CS algorithm called NECSA is proposed for optimization problems. NECSA has relatively simple coding and robust performance. The NECSA employs adaptive parameters, self-evaluation operation, and modified greedy selection operation to improve the convergence speed, balance the exploration and exploitation and obtain more reliable performance. Then, a well-established algorithm is proposed.

Besides, a selection mechanism based on Lévy flight and Gaussian distribution is employed to enhance the exploration ability. This paper tries to combine different random walk strategies to make the best of their characteristics. Furthermore, the modified greedy selection operation is introduced to improve the global search and local search ability, respectively. Again, this excellent algorithm has simple coding.

3.2 Novel enhanced CS algorithm (NECSA)

NECSA adopts a similar framework with CS, namely initialization and switch operation, as described in Section 2. However, NECSA has various modifications in terms of search strategies and control parameters. Detailed descriptions are as follows.

3.2.1 Adaptive control parameters

In the original CS, the switch parameter is typically set as a constant value of 0.25, which results in low convergence speed and poor local searchability. According to the previous research, adaptive parameters can improve the convergence trend and strengthen the exploitation phase. Choosing an appropriate adaptive strategy is of great significance. In this study, the P_a, which determines the possibility of discarding cuckoo eggs, is set as a gradually decreasing parameter based on the current iteration number and total iteration number. The parameter range setting of P_a is similar to the one applied in [18]. The value of P_a is fine-tuned by the following Equation (7): $\begin{matrix} P_{a} = P_{a}_\max - (t / T) \times P_{a}_\max \end{matrix}$ (7) where Pa _ max is the maximum of parameter Pa, t, and Max_iter are the number of current and total iteration, respectively.

3.2.2 Self-evaluation operation

Concerning the CS modifications in terms of search strategies, the application of Gaussian distribution plays a significant role. Many variants of CS have been developed to overcome the drawbacks, such as poor performance when dealing with multi-dimensional, complex, and highly nonlinear problems. Improvement could be realized by generating new solutions by adopting both Gaussian distribution and Lévy flight. Some researchers have tried a lot in this term. In Ref [60], Fan et al. introduced Gaussian distribution into the iterative process, and the experimental results proved its promising efficiency. Zheng et al. [61] replaced Lévy flight with Gaussian distribution to generate new solutions, and the modified CS algorithm has a higher convergence rate. Although it has been proved that Gaussian distribution has proper step size compared with Lévy flight, a single search strategy cannot guarantee the efficiency in facing different kinds of optimization problems [34]. It can be concluded that the combination of Gaussian distribution, Lévy flight, and other strategies can provide better solutions.

Inspired by the phenomenon, cuckoos usually lay their eggs quickly when host birds leave the nest before they start to hatch their eggs. Because cuckoos only lay 2–10 eggs per year and neither build nests nor take care of their offspring, the method cuckoos choose parasitic nests determines their offspring’s survival rate to a great extent. In other words, the nest cuckoos selected for their offspring play a significant role. Therefore, we proposed an evaluation called self-evaluation operation to imitate the behavior that cuckoos may keep the current selection or reselect for avoiding being discovered by hosts and finding possible better nests to ensure the survival rate.

In the implementation phase, the main idea is to compare the fitness of temporarily selected nests and potentially better solution for the current population. The stricter criterion is realized by introducing coefficient ω. Based on the comparison results, the cuckoos will determine whether continue seeking better locations or reserve current nests and discard temporarily selected nests. In NECSA, the individuals will firstly employ Gaussian random walk, which is well at global search to generate new solutions and has a long step size compared to Lévy flight. Then, the new best solution will be weighed and compared with the previous best one. If it fails to meet psychological expectations, the cuckoos will generate new solutions using Lévy flight instead of the Gaussian random walk. In other words, if Gaussian random walk cannot obtain well performance, Lévy flight will be applied once to get a possible better solution in NECSA. $\begin{matrix} x_{i}^{t + 1} = Gaussian (x_{best}^{t}, σ) + (r_{1} \cdot x_{best}^{t} - r_{2} \cdot x_{i}^{t}) . \end{matrix}$ (8) $\begin{matrix} σ = log (t / T) \times (x_{i}^{t} - x_{best}^{t}) . \end{matrix}$ (9) where $Gaussian (x_{best}^{t}, σ)$ is Gaussian distribution takes $x_{best}^{t}$ as exception and $σ = log (t / T) \times (x_{i}^{t} - x_{best}^{t})$ as standard deviation, r₁ and r₂ are stochastic values subject to the uniform distribution, t and T represent the number of current and total generations, respectively.

In addition, the standard version of Lévy flight applied in CS also has some drawbacks that need to be improved as mentioned in Section 1. Given this consideration, despite the fact we have introduced a novel mechanism based on Gaussian distribution and Lévy flight already, we choose to further improve it by introducing some modifications to Lévy flight. In the novel proposed algorithm, adaptive parameter α is introduced to the original Lévy flight as shown in Eq.(10), which represents the step size scale factor and is defined as follows: $\begin{matrix} α = r \times \frac{1}{exp (\frac{f_{current}}{f_{best} - f_{worst}})} . \end{matrix}$ (10) where f_current is the fitness of the i^th solution in t^th population, f_best and f_worst represent the best and worst solutions of t generation, respectively, r implies a stochastic number obeys that uniform distribution.

3.2.3 Modified greedy selection operation

Inspired by the greedy selection (GS) mechanism employed in the ABC algorithm and its variants [4 , 63], NECSA introduced a Modified Greedy Selection operation (MGS) to enhance the results obtained from the Self-evaluation operation. In NECSA, the modified greedy selection operation can be divided into early-MGS and later-MGS, namely two strategies applied at the early and later search phase.

In the early search phase, the algorithm is supposed to focus on the exploration to ensure population diversity and avoid premature. The first one, which is more conducive to global search, is applied in the early search process. The early-MGS generates new solutions based on the information of best, randomly selected and worst solutions of different dimensions of the t generation to mutate the solution vectors. It is formulated as Equation (11): $\begin{matrix} x_{i, j}^{t + 1} = x_{r_{1}, m}^{t} + r_{2} \times (x_{worst, n}^{t} - x_{best, n}^{t}) \end{matrix}$ (11) where r₁ is a randomly selected dimension among the D dimensions of nests, and r₂ ∈ [- 1, 1] denotes a random value subject to the uniform distribution, m, n, and j (m ≠ n ≠ j) are integers randomly selected in [1, D], D represents the number of dimensions of population, x_best and x_worst are the best and worst nests of the t generation.

When it comes to the later search phase, exploitation deserves more attention. In other words, we need to take measures to increase the convergence rate and enhance the convergence trend to the global optimum. Based on the consideration mentioned above, the later-MGS, which is better at local search, is employed in the later search phases. It updates solutions according to the best, random and worst individuals of the same dimension. Later-MGS is modeled as Equation (12): $\begin{matrix} x_{i, j}^{t + 1} = x_{r_{1}, j}^{t} + r_{2} \times (x_{worst, j}^{t} - x_{best j}^{t}) \end{matrix}$ (12) where D is the problem’s dimension, r₁ is an integer selected from [1, D], r₂ is a number that obeys uniform distribution and belongs to [- 1, 1], j is an integer randomly selected from the D dimensions.

The framework of the modified greedy selection operation can be organized as follows: $\begin{matrix} x_{i, j}^{t + 1} = {\begin{matrix} x_{r_{1}, m}^{t} + r_{2} \times (x_{worst, n}^{t} - x_{best, n}^{t}), 1 < t ⩽ c \times T \\ x_{i, j}^{t}, c \times T < t ⩽ (1 - c) \times T \\ x_{r_{1}, j}^{t} + r_{2} \times (x_{worst, j}^{t} - x_{best j}^{t}), (1 - c) \times T < t ⩽ T \end{matrix} \end{matrix}$ (13)

where c (c ∈ (0, 0.5)) is a predefined constant limitation control parameter of modified greedy selection operation, and in this paper, this value is set as 0.3 to divide the iteration phase roughly. The modified greedy selection operation only works at the early and late search phases. Besides, t and T represent the current and maximum iteration numbers.

The schematic diagram of the original CS is presented in Fig. 2.

Fig. 2

The schematic diagram of NECSA.

The proposed NECSA is proposed by introducing adaptive control parameters, self-evaluation, and the modified greedy selection operation we designed. The performance of NECSA is excellent, and the pseudo-code is shown in Algorithm 2:

3.3 Complexity analysis

In order to show the results more intuitively, we set N as the number of populations, D as the dimension of the solution space, and the parameter T as the number of maximum generations. The complexity of the NECSA algorithm is presented as follows:

The random initialization operation requires O (1) time.

The execution of Gaussian random walk and Lévy flight is both O (N × D) time.

The time of evaluation operation is O (N).

The operations such as comparing fitness value of new and old solutions and the random replacement, etc. require O (1) time.

Algorithm 2

NECSA algorithm

Input: N: the population scale;

   D: the problem dimension;

   Pa_max: the maximum value of the switch parameter Pa;

   t: the current iteration number;

   T: the maximum number of iterations;

   ω: the weight value of evaluation;

   c: the control parameter of greedy selection;

1:t = 1;

2: Randomly generate an initial population of N solutions x;

3: Evaluate all the solutions →f_min;

4: whilet < Tdo

5: fori = 1 : Ndo

6: Generate new solutions x_new using Gaussian random

walk as formulated in Equation (8);

7: Evaluate all the solutions →f_new;

8: iff_new < ω · f_minthen

9: x = x_new;

10: f_min = f_new;

11: else

12: Generate new solutions x_new using adaptive Lévy

       flight shown in Equation (1) and Equation (8);

11: end if

12: Enhance the generated solutions using the modified

       greedy selection as shown in Equation (12) →x_new;

13: Evaluate all the solutions →f_new;

14: iff_new < f_minthen

15: x = x_new;

16: f_min = f_new;

17: end if

18: end for

19: Discard a fraction (adaptive Pa as shown in

    Equation (7)) of worst nests and generate new

    solutions using Equation (6);

20: Update the global optimal solution;

21: t = t + 1

20: end while

Output: Optimal solution

NECSA algorithm
Input: N: the population scale;
D: the problem dimension;
Pa_max: the maximum value of the switch parameter Pa;
t: the current iteration number;
T: the maximum number of iterations;
ω: the weight value of evaluation;
c: the control parameter of greedy selection;
1:t = 1;
2: Randomly generate an initial population of N solutions x;
3: Evaluate all the solutions →f_min;
4: whilet < Tdo
5: fori = 1 : Ndo
6: Generate new solutions x_new using Gaussian random
walk as formulated in Equation (8);
7: Evaluate all the solutions →f_new;
8: iff_new < ω · f_minthen
9: x = x_new;
10: f_min = f_new;
11: else
12: Generate new solutions x_new using adaptive Lévy
flight shown in Equation (1) and Equation (8);
11: end if
12: Enhance the generated solutions using the modified
greedy selection as shown in Equation (12) →x_new;
13: Evaluate all the solutions →f_new;
14: iff_new < f_minthen
15: x = x_new;
16: f_min = f_new;
17: end if
18: end for
19: Discard a fraction (adaptive Pa as shown in
Equation (7)) of worst nests and generate new
solutions using Equation (6);
20: Update the global optimal solution;
21: t = t + 1
20: end while
Output: Optimal solution

Overall, assuming that probability of Lévy flight is P, the complexity of NECSA is approximately equal to O (((1 + P) × N × D + (2 × c + 2) × N) × T) ≈ O (N × D × T), where the coefficient c is the limitation control parameter of MGS.

4 Experimental results

4.1 Benchmark functions

Thirty benchmark functions of the CEC2014 special session on real-parameter optimization were used to test the performance of NECSA [64]. These benchmark functions can be divided into four groups:

Unimodal Functions(F1 –F3)

Simple Multimodal Functions(F4 –F16)

Hybrid Function 1(F17 –F22)

Composition Functions(F23 –F30)

4.2 Parameter settings

To prove the excellent performance of NECSA, we compare it with the standard CS algorithm and other four clever variants. These variants are: ACS [21], ACSA [20], GBCS [31], ECS [30]. The parameter settings of this paper are: the population size N is equal to 30 or 50, the dimension of the populations D is set to 30 or 50, the maximum number of iterations T is set to 300000 or 500000. More details of the parameter settings of different algorithms have been presented in Table 1.

Table 1
Parameter settings of the algorithms involved in the control experiment

Algorithm Parameter Settings

CS λ = 1.5; Pa= 0.25

ACS λ = 1.5; Pa= 0.25

ACSA λ = 1.5; xL = –600; xU = 600; aL = 0.02D;

aU = 0.08D; Pa = 0.25;

GBCS λ = 1.5; Pa = 0.25

ECS λ = 1.5; Pa = 0.25

NECSA λ = 1.5; Pa_max = 0.35; c = 0.3; ω = 0.8

Algorithm	Parameter Settings
CS	λ = 1.5; Pa= 0.25
ACS	λ = 1.5; Pa= 0.25
ACSA	λ = 1.5; xL = –600; xU = 600; aL = 0.02*D;
	aU = 0.08*D; Pa = 0.25;
GBCS	λ = 1.5; Pa = 0.25
ECS	λ = 1.5; Pa = 0.25
NECSA	λ = 1.5; Pa_max = 0.35; c = 0.3; ω = 0.8

4.3 Comparison of NECSA with other algorithms

During the experiments, the number of runs per function is set to 51 to avoid the fortuitous large deviation. Each algorithm runs 51 times per benchmark function and reserves the mean value of 51 optimums. The mean error and standard deviation of the algorithms are presented in Table 2.

Table 2
Experimental results of CS, ACS, ACSA, GBCS, ECS at D = 30

Functions Num Mean/Std CS ACS ACSA GBCS ECS NECSA

Unimodal F1 Mean Error 1.13E+06 1.48E+06 1.80E+06 1.54E+06 3.59E+05 3.30E+04

Functions Std Dev 5.58E+05 4.70E+05 7.70E+05 6.18E+05 2.32E+05 2.81E+04

F2 Mean Error 1.00E+10 1.00E+10 1.00E+10 1.00E+10 1.00E+10 1.00E+10

Std Dev 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F3 Mean Error 0.00E+00 1.56E-07 7.82E-08 6.51E-08 0.00E+00 0.00E+00

Std Dev 0.00E+00 2.02E-07 7.38E-08 6.43E-08 0.00E+00 0.00E+00

Simple F4 Mean Error 1.58E+01 1.60E+01 4.74E+01 1.28E+01 6.96E+00 8.44E+00

Multimodal Std Dev 2.65E+01 2.83E+01 3.52E+01 2.58E+01 2.04E+01 2.15E+01

F5 Mean Error 2.08E+01 2.05E+01 2.09E+01 2.08E+01 2.02E+01 2.03E+01

Std Dev 6.65E-02 7.18E-02 5.42E-02 6.12E-02 3.49E-02 5.23E-02

F6 Mean Error 2.50E+01 2.40E+01 2.72E+01 1.76E+01 1.75E+01 1.53E+01

Std Dev 1.67E+00 2.71E+00 4.14E+00 4.93E+00 3.27E+00 2.45E+00

F7 Mean Error 2.01E-07 5.66E-05 3.19E-08 0.00E+00 1.60E-07 2.59E-08

Std Dev 6.10E-07 2.86E-04 1.71E-07 0.00E+00 1.13E-06 6.76E-08

F8 Mean Error 3.00E+01 3.09E+01 4.91E+01 2.33E+01 6.49E+01 1.95E-02

Std Dev 5.84E+00 6.46E+00 1.21E+01 8.92E+00 2.60E+01 1.39E-01

F9 Mean Error 1.34E+02 8.68E+01 1.40E+02 7.60E+01 1.17E+02 7.49E+01

Std Dev 2.06E+01 1.42E+01 2.75E+01 1.66E+01 2.45E+01 1.49E+01

F10 Mean Error 1.12E+03 1.81E+03 1.87E+03 1.39E+03 1.90E+02 2.87E+00

Std Dev 2.61E+02 3.17E+02 4.91E+02 5.80E+02 2.53E+02 2.01E+00

F11 Mean Error 3.57E+03 3.51E+03 4.26E+03 4.28E+03 2.15E+03 2.43E+03

Std Dev 2.61E+02 4.06E+02 2.75E+02 5.99E+02 3.55E+02 3.26E+02

F12 Mean Error 7.44E-01 9.69E-01 9.10E-01 1.43E+00 2.26E-01 4.33E-01

Std Dev 1.08E-01 1.46E-01 1.34E-01 2.63E-01 4.91E-02 8.12E-02

F13 Mean Error 3.15E-01 3.19E-01 3.41E-01 3.81E-01 4.75E-01 3.34E-01

Std Dev 5.35E-02 5.03E-02 4.51E-02 5.54E-02 1.21E-01 3.97E-02

F14 Mean Error 2.49E-01 2.47E-01 2.38E-01 2.81E-01 2.96E-01 2.36E-01

Std Dev 3.19E-02 3.60E-02 3.26E-02 3.06E-02 2.07E-01 2.61E-02

F15 Mean Error 1.15E+01 1.31E+01 1.31E+01 1.01E+01 1.55E+01 7.27E+00

Std Dev 2.23E+00 1.84E+00 1.98E+00 9.92E-01 9.67E+00 9.62E-01

F16 Mean Error 1.24E+01 1.25E+01 1.24E+01 1.20E+01 1.02E+01 1.07E+01

Std Dev 1.76E-01 2.38E-01 3.62E-01 3.48E-01 5.68E-01 3.30E-01

Hybrid F17 Mean Error 2.02E+03 2.68E+03 3.06E+03 3.03E+03 5.30E+03 1.01E+03

Function 1 Std Dev 2.40E+02 4.37E+02 9.72E+02 4.74E+02 5.37E+03 1.79E+02

F18 Mean Error 7.53E+01 1.05E+02 3.14E+02 1.02E+02 5.01E+01 2.40E+01

Std Dev 1.67E+01 1.66E+01 7.64E+02 1.78E+01 1.82E+01 6.28E+00

F19 Mean Error 8.39E+00 7.54E+00 6.64E+00 6.99E+00 1.58E+01 6.11E+00

Std Dev 8.19E-01 6.74E-01 1.53E+00 8.67E-01 2.06E+01 6.10E-01

F20 Mean Error 6.31E+01 5.89E+01 7.02E+01 5.54E+01 3.26E+01 1.68E+01

Std Dev 1.76E+01 1.42E+01 5.21E+01 1.12E+01 9.63E+00 3.50E+00

F21 Mean Error 1.02E+03 1.16E+03 1.29E+03 1.23E+03 7.44E+02 4.78E+02

Std Dev 1.70E+02 1.90E+02 1.86E+02 1.88E+02 1.96E+02 1.27E+02

F22 Mean Error 2.61E+02 2.44E+02 2.53E+02 1.61E+02 3.99E+02 1.48E+02

Std Dev 1.07E+02 1.02E+02 1.28E+02 8.27E+01 1.40E+02 9.29E+01

Composition F23 Mean Error 3.15E+02 3.15E+02 3.14E+02 3.15E+02 3.15E+02 3.15E+02

Functions Std Dev 0.00E+00 0.00E+00 6.17E-02 0.00E+00 0.00E+00 0.00E+00

F24 Mean Error 2.26E+02 2.25E+02 2.25E+02 2.25E+02 2.06E+02 2.17E+02

Std Dev 1.26E+00 1.98E+00 1.53E+00 1.47E+00 1.23E+01 1.11E+01

F25 Mean Error 2.06E+02 2.06E+02 2.07E+02 2.06E+02 2.06E+02 2.03E+02

Std Dev 1.17E+00 1.44E+00 2.01E+00 1.11E+00 5.28E+00 4.95E-01

F26 Mean Error 1.00E+02 1.00E+02 1.00E+02 1.00E+02 1.08E+02 1.00E+02

Std Dev 4.88E-02 4.76E-02 6.66E-02 6.09E-02 3.95E+01 6.12E-02

F27 Mean Error 4.08E+02 4.06E+02 6.33E+02 4.50E+02 7.75E+02 5.64E+02

Std Dev 7.13E+00 2.38E+00 2.86E+02 9.54E+01 1.71E+02 1.80E+02

F28 Mean Error 9.75E+02 9.99E+02 7.68E+02 8.83E+02 1.38E+03 8.67E+02

Std Dev 5.02E+01 6.57E+01 1.91E+02 4.98E+01 4.28E+02 3.34E+01

F29 Mean Error 1.01E+03 1.26E+03 2.88E+02 1.66E+05 3.91E+06 8.40E+05

Std Dev 8.78E+01 2.47E+02 3.82E+01 1.18E+06 5.90E+06 2.57E+06

F30 Mean Error 1.76E+03 2.05E+03 1.47E+03 1.88E+03 2.11E+03 1.30E+03

Std Dev 4.29E+02 4.48E+02 2.21E+02 5.14E+02 1.05E+03 4.10E+02

+ -≈ 24∖4∖2 24∖3∖3 26∖3∖1 25∖3∖2 22∖6∖2

Functions	Num	Mean/Std	CS	ACS	ACSA	GBCS	ECS	NECSA
Unimodal	F1	Mean Error	1.13E+06	1.48E+06	1.80E+06	1.54E+06	3.59E+05	3.30E+04
Functions		Std Dev	5.58E+05	4.70E+05	7.70E+05	6.18E+05	2.32E+05	2.81E+04
	F2	Mean Error	1.00E+10	1.00E+10	1.00E+10	1.00E+10	1.00E+10	1.00E+10
		Std Dev	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	F3	Mean Error	0.00E+00	1.56E-07	7.82E-08	6.51E-08	0.00E+00	0.00E+00
		Std Dev	0.00E+00	2.02E-07	7.38E-08	6.43E-08	0.00E+00	0.00E+00
Simple	F4	Mean Error	1.58E+01	1.60E+01	4.74E+01	1.28E+01	6.96E+00	8.44E+00
Multimodal		Std Dev	2.65E+01	2.83E+01	3.52E+01	2.58E+01	2.04E+01	2.15E+01
	F5	Mean Error	2.08E+01	2.05E+01	2.09E+01	2.08E+01	2.02E+01	2.03E+01
		Std Dev	6.65E-02	7.18E-02	5.42E-02	6.12E-02	3.49E-02	5.23E-02
	F6	Mean Error	2.50E+01	2.40E+01	2.72E+01	1.76E+01	1.75E+01	1.53E+01
		Std Dev	1.67E+00	2.71E+00	4.14E+00	4.93E+00	3.27E+00	2.45E+00
	F7	Mean Error	2.01E-07	5.66E-05	3.19E-08	0.00E+00	1.60E-07	2.59E-08
		Std Dev	6.10E-07	2.86E-04	1.71E-07	0.00E+00	1.13E-06	6.76E-08
	F8	Mean Error	3.00E+01	3.09E+01	4.91E+01	2.33E+01	6.49E+01	1.95E-02
		Std Dev	5.84E+00	6.46E+00	1.21E+01	8.92E+00	2.60E+01	1.39E-01
	F9	Mean Error	1.34E+02	8.68E+01	1.40E+02	7.60E+01	1.17E+02	7.49E+01
		Std Dev	2.06E+01	1.42E+01	2.75E+01	1.66E+01	2.45E+01	1.49E+01
	F10	Mean Error	1.12E+03	1.81E+03	1.87E+03	1.39E+03	1.90E+02	2.87E+00
		Std Dev	2.61E+02	3.17E+02	4.91E+02	5.80E+02	2.53E+02	2.01E+00
	F11	Mean Error	3.57E+03	3.51E+03	4.26E+03	4.28E+03	2.15E+03	2.43E+03
		Std Dev	2.61E+02	4.06E+02	2.75E+02	5.99E+02	3.55E+02	3.26E+02
	F12	Mean Error	7.44E-01	9.69E-01	9.10E-01	1.43E+00	2.26E-01	4.33E-01
		Std Dev	1.08E-01	1.46E-01	1.34E-01	2.63E-01	4.91E-02	8.12E-02
	F13	Mean Error	3.15E-01	3.19E-01	3.41E-01	3.81E-01	4.75E-01	3.34E-01
		Std Dev	5.35E-02	5.03E-02	4.51E-02	5.54E-02	1.21E-01	3.97E-02
	F14	Mean Error	2.49E-01	2.47E-01	2.38E-01	2.81E-01	2.96E-01	2.36E-01
		Std Dev	3.19E-02	3.60E-02	3.26E-02	3.06E-02	2.07E-01	2.61E-02
	F15	Mean Error	1.15E+01	1.31E+01	1.31E+01	1.01E+01	1.55E+01	7.27E+00
		Std Dev	2.23E+00	1.84E+00	1.98E+00	9.92E-01	9.67E+00	9.62E-01
	F16	Mean Error	1.24E+01	1.25E+01	1.24E+01	1.20E+01	1.02E+01	1.07E+01
		Std Dev	1.76E-01	2.38E-01	3.62E-01	3.48E-01	5.68E-01	3.30E-01
Hybrid	F17	Mean Error	2.02E+03	2.68E+03	3.06E+03	3.03E+03	5.30E+03	1.01E+03
Function 1		Std Dev	2.40E+02	4.37E+02	9.72E+02	4.74E+02	5.37E+03	1.79E+02
	F18	Mean Error	7.53E+01	1.05E+02	3.14E+02	1.02E+02	5.01E+01	2.40E+01
		Std Dev	1.67E+01	1.66E+01	7.64E+02	1.78E+01	1.82E+01	6.28E+00
	F19	Mean Error	8.39E+00	7.54E+00	6.64E+00	6.99E+00	1.58E+01	6.11E+00
		Std Dev	8.19E-01	6.74E-01	1.53E+00	8.67E-01	2.06E+01	6.10E-01
	F20	Mean Error	6.31E+01	5.89E+01	7.02E+01	5.54E+01	3.26E+01	1.68E+01
		Std Dev	1.76E+01	1.42E+01	5.21E+01	1.12E+01	9.63E+00	3.50E+00
	F21	Mean Error	1.02E+03	1.16E+03	1.29E+03	1.23E+03	7.44E+02	4.78E+02
		Std Dev	1.70E+02	1.90E+02	1.86E+02	1.88E+02	1.96E+02	1.27E+02
	F22	Mean Error	2.61E+02	2.44E+02	2.53E+02	1.61E+02	3.99E+02	1.48E+02
		Std Dev	1.07E+02	1.02E+02	1.28E+02	8.27E+01	1.40E+02	9.29E+01
Composition	F23	Mean Error	3.15E+02	3.15E+02	3.14E+02	3.15E+02	3.15E+02	3.15E+02
Functions		Std Dev	0.00E+00	0.00E+00	6.17E-02	0.00E+00	0.00E+00	0.00E+00
	F24	Mean Error	2.26E+02	2.25E+02	2.25E+02	2.25E+02	2.06E+02	2.17E+02
		Std Dev	1.26E+00	1.98E+00	1.53E+00	1.47E+00	1.23E+01	1.11E+01
	F25	Mean Error	2.06E+02	2.06E+02	2.07E+02	2.06E+02	2.06E+02	2.03E+02
		Std Dev	1.17E+00	1.44E+00	2.01E+00	1.11E+00	5.28E+00	4.95E-01
	F26	Mean Error	1.00E+02	1.00E+02	1.00E+02	1.00E+02	1.08E+02	1.00E+02
		Std Dev	4.88E-02	4.76E-02	6.66E-02	6.09E-02	3.95E+01	6.12E-02
	F27	Mean Error	4.08E+02	4.06E+02	6.33E+02	4.50E+02	7.75E+02	5.64E+02
		Std Dev	7.13E+00	2.38E+00	2.86E+02	9.54E+01	1.71E+02	1.80E+02
	F28	Mean Error	9.75E+02	9.99E+02	7.68E+02	8.83E+02	1.38E+03	8.67E+02
		Std Dev	5.02E+01	6.57E+01	1.91E+02	4.98E+01	4.28E+02	3.34E+01
	F29	Mean Error	1.01E+03	1.26E+03	2.88E+02	1.66E+05	3.91E+06	8.40E+05
		Std Dev	8.78E+01	2.47E+02	3.82E+01	1.18E+06	5.90E+06	2.57E+06
	F30	Mean Error	1.76E+03	2.05E+03	1.47E+03	1.88E+03	2.11E+03	1.30E+03
		Std Dev	4.29E+02	4.48E+02	2.21E+02	5.14E+02	1.05E+03	4.10E+02
+ -≈			24∖4∖2	24∖3∖3	26∖3∖1	25∖3∖2	22∖6∖2

4.3 Comparison of NECSA with other algorithms

In this section, experimental results of NECSA and other algorithms have been presented. When the dimension of problem D is 30, the population scale N is set to 30, and the maximum of generation T is 10000. When D is set to 50, N and T are assigned as 50 and 10000, respectively. During the experiments, the number of runs per function is set to 51 to avoid the fortuitous large deviation. Each algorithm runs 51 times per benchmark function and reserves the mean value of 51 optimums. The mean error and standard deviation of the algorithms are presented in Tables 2 3. Besides, the Wilcoxon signed-rank test at α = 0.05 is adopted to analyze the performance of NECSA, and the result is also included in Tables 2 3. In addition, the Friedman test is applied to calculate the rankings, and the results have been presented in these Tables.

Table 3
Experimental results of CS, ACS, ACSA, GBCS, ECS at D = 50

Functions Num Mean/Std CS ACS ACSA GBCS ECS NECSA

Unimodal F1 Mean Error 7.11E+06 9.21E+06 1.36E+07 1.36E+07 9.26E+06 8.67E+05

Functions Std Dev 1.69E+06 1.18E+06 3.74E+06 3.73E+06 2.09E+06 2.80E+05

F2 Mean Error 1.00E+10 1.00E+10 1.00E+10 1.00E+10 1.00E+10 1.00E+10

Std Dev 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F3 Mean Error 1.09E+03 2.91E+03 3.76E+03 2.18E+03 7.19E-02 2.38E-04

Std Dev 2.84E+02 9.70E+02 8.27E+03 6.66E+02 3.86E-02 2.69E-04

Simple F4 Mean Error 7.89E+01 8.37E+01 7.89E+01 9.17E+01 6.93E+01 6.64E+01

Multimodal/ Std Dev 2.22E+01 2.50E+01 3.40E+01 1.14E+01 3.06E+01 3.08E+01

F5 Mean Error 2.11E+01 2.08E+01 2.11E+01 2.11E+01 2.03E+01 2.05E+01

Std Dev 4.79E-02 4.88E-02 5.40E-02 4.32E-02 4.17E-02 3.28E-02

F6 Mean Error 4.90E+01 5.24E+01 5.51E+01 4.66E+01 3.16E+01 3.69E+01

Std Dev 1.65E+00 2.65E+00 1.69E+00 4.50E+00 2.19E+00 2.69E+00

F7 Mean Error 1.21E-04 1.47E-05 1.50E-03 6.97E-05 3.95E-05 3.40E-04

Std Dev 3.40E-04 5.85E-05 3.10E-03 2.59E-04 7.00E-05 1.50E-03

F8 Mean Error 1.22E+02 1.22E+02 1.80E+02 1.03E+02 8.15E+00 9.00E-03

Std Dev 1.27E+01 2.01E+01 2.24E+01 2.74E+01 2.02E+00 3.48E-02

F9 Mean Error 2.82E+02 1.95E+02 3.45E+02 2.11E+02 2.38E+02 1.74E+02

Std Dev 3.08E+01 1.75E+01 3.12E+01 2.91E+01 2.53E+01 1.96E+01

F10 Mean Error 3.83E+03 6.09E+03 5.35E+03 5.99E+03 3.83E+01 2.60E+01

Std Dev 3.33E+02 4.93E+02 8.28E+02 1.16E+03 4.19E+01 1.95E+01

F11 Mean Error 6.96E+03 7.45E+03 8.34E+03 9.63E+03 4.61E+03 5.73E+03

Std Dev 2.73E+02 4.72E+02 4.31E+02 7.25E+02 3.96E+02 3.59E+02

F12 Mean Error 1.23E+00 1.56E+00 1.72E+00 2.18E+00 2.95E-01 5.73E-01

Std Dev 1.51E-01 1.97E-01 2.29E-01 3.32E-01 3.23E-02 7.79E-02

F13 Mean Error 4.10E-01 4.20E-01 4.48E-01 4.41E-01 4.03E-01 4.05E-01

Std Dev 4.66E-02 5.23E-02 4.34E-02 6.26E-02 5.71E-02 4.80E-02

F14 Mean Error 2.97E-01 2.88E-01 2.90E-01 2.93E-01 2.75E-01 2.72E-01

Std Dev 2.11E-02 2.59E-02 2.60E-02 2.76E-02 2.63E-02 2.32E-02

F15 Mean Error 2.81E+01 3.22E+01 3.19E+01 2.47E+01 1.86E+01 1.77E+01

Std Dev 2.99E+00 3.34E+00 2.56E+00 2.13E+00 2.35E+00 1.72E+00

F16 Mean Error 2.19E+01 2.21E+01 2.20E+01 2.18E+01 1.89E+01 1.95E+01

Std Dev 3.07E-01 2.53E-01 2.96E-01 3.24E-01 4.08E-01 3.50E-01

Hybrid F17 Mean Error 1.93E+05 2.34E+05 4.45E+05 4.45E+05 5.88E+05 4.39E+03

Function 1/ Std Dev 6.40E+04 7.03E+04 1.72E+05 1.54E+05 2.01E+05 5.81E+02

F18 Mean Error 6.92E+09 8.24E+09 3.98E+09 5.86E+09 3.45E+09 8.04E+09

Std Dev 4.61E+09 3.85E+09 4.89E+09 4.95E+09 4.71E+09 4.01E+09

F19 Mean Error 2.19E+01 1.82E+01 1.87E+01 2.25E+01 1.68E+01 1.53E+01

Std Dev 4.88E+00 1.79E+00 4.35E+00 8.57E+00 2.24E+00 2.48E+00

F20 Mean Error 4.68E+02 6.07E+02 5.14E+02 4.95E+02 1.70E+02 7.22E+01

Std Dev 8.11E+01 1.20E+02 8.40E+01 7.54E+01 3.10E+01 1.38E+01

F21 Mean Error 2.40E+04 3.60E+04 4.84E+04 6.11E+04 8.01E+04 2.02E+03

Std Dev 7.16E+03 8.68E+03 2.67E+04 1.79E+04 2.63E+04 2.51E+02

F22 Mean Error 8.96E+02 8.50E+02 1.15E+03 6.79E+02 6.54E+02 7.13E+02

Std Dev 1.33E+02 1.54E+02 5.28E+02 1.56E+02 1.86E+02 1.51E+02

Composition F23 Mean Error 3.44E+02 3.44E+02 3.40E+02 3.44E+02 2.00E+02 3.44E+02

Functions Std Dev 4.50E-13 2.37E-11 1.21E+00 4.53E-13 0.00E+00 4.18E-13

F24 Mean Error 2.67E+02 2.70E+02 2.68E+02 2.64E+02 2.00E+02 2.57E+02

Std Dev 4.72E+00 3.24E+00 5.31E+00 5.26E+00 5.20E-08 2.04E+00

F25 Mean Error 2.17E+02 2.19E+02 2.21E+02 2.20E+02 2.00E+02 2.09E+02

Std Dev 2.39E+00 2.91E+00 5.60E+00 2.98E+00 0.00E+00 1.40E+00

F26 Mean Error 1.00E+02 1.00E+02 1.01E+02 1.00E+02 1.63E+02 1.00E+02

Std Dev 5.67E-02 3.67E-02 1.90E+00 5.49E-02 4.87E+01 4.70E-02

F27 Mean Error 9.07E+02 1.11E+03 1.69E+03 1.41E+03 2.00E+02 1.21E+03

Std Dev 4.40E+02 5.53E+02 3.96E+01 1.57E+02 0.00E+00 1.88E+02

F28 Mean Error 1.46E+03 1.49E+03 6.56E+02 1.28E+03 2.00E+02 1.25E+03

Std Dev 1.44E+02 1.39E+02 1.27E+02 7.20E+01 0.00E+00 3.72E+01

F29 Mean Error 7.03E+03 2.22E+04 4.18E+02 9.76E+06 4.06E+02 2.63E+07

Std Dev 2.20E+03 8.76E+03 8.42E+01 1.60E+07 6.02E+02 1.55E+07

F30 Mean Error 9.88E+03 1.36E+04 2.38E+03 8.81E+03 2.38E+03 8.16E+03

Std Dev 6.17E+02 1.99E+03 2.11E+02 8.14E+02 3.76E+03 2.28E+02

+ -≈ 23∖5∖2 24∖4∖2 24∖5∖1 24∖4∖2 22∖6∖2

Functions	Num	Mean/Std	CS	ACS	ACSA	GBCS	ECS	NECSA
Unimodal	F1	Mean Error	7.11E+06	9.21E+06	1.36E+07	1.36E+07	9.26E+06	8.67E+05
Functions		Std Dev	1.69E+06	1.18E+06	3.74E+06	3.73E+06	2.09E+06	2.80E+05
	F2	Mean Error	1.00E+10	1.00E+10	1.00E+10	1.00E+10	1.00E+10	1.00E+10
		Std Dev	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	F3	Mean Error	1.09E+03	2.91E+03	3.76E+03	2.18E+03	7.19E-02	2.38E-04
		Std Dev	2.84E+02	9.70E+02	8.27E+03	6.66E+02	3.86E-02	2.69E-04
Simple	F4	Mean Error	7.89E+01	8.37E+01	7.89E+01	9.17E+01	6.93E+01	6.64E+01
Multimodal/		Std Dev	2.22E+01	2.50E+01	3.40E+01	1.14E+01	3.06E+01	3.08E+01
	F5	Mean Error	2.11E+01	2.08E+01	2.11E+01	2.11E+01	2.03E+01	2.05E+01
		Std Dev	4.79E-02	4.88E-02	5.40E-02	4.32E-02	4.17E-02	3.28E-02
	F6	Mean Error	4.90E+01	5.24E+01	5.51E+01	4.66E+01	3.16E+01	3.69E+01
		Std Dev	1.65E+00	2.65E+00	1.69E+00	4.50E+00	2.19E+00	2.69E+00
	F7	Mean Error	1.21E-04	1.47E-05	1.50E-03	6.97E-05	3.95E-05	3.40E-04
		Std Dev	3.40E-04	5.85E-05	3.10E-03	2.59E-04	7.00E-05	1.50E-03
	F8	Mean Error	1.22E+02	1.22E+02	1.80E+02	1.03E+02	8.15E+00	9.00E-03
		Std Dev	1.27E+01	2.01E+01	2.24E+01	2.74E+01	2.02E+00	3.48E-02
	F9	Mean Error	2.82E+02	1.95E+02	3.45E+02	2.11E+02	2.38E+02	1.74E+02
		Std Dev	3.08E+01	1.75E+01	3.12E+01	2.91E+01	2.53E+01	1.96E+01
	F10	Mean Error	3.83E+03	6.09E+03	5.35E+03	5.99E+03	3.83E+01	2.60E+01
		Std Dev	3.33E+02	4.93E+02	8.28E+02	1.16E+03	4.19E+01	1.95E+01
	F11	Mean Error	6.96E+03	7.45E+03	8.34E+03	9.63E+03	4.61E+03	5.73E+03
		Std Dev	2.73E+02	4.72E+02	4.31E+02	7.25E+02	3.96E+02	3.59E+02
	F12	Mean Error	1.23E+00	1.56E+00	1.72E+00	2.18E+00	2.95E-01	5.73E-01
		Std Dev	1.51E-01	1.97E-01	2.29E-01	3.32E-01	3.23E-02	7.79E-02
	F13	Mean Error	4.10E-01	4.20E-01	4.48E-01	4.41E-01	4.03E-01	4.05E-01
		Std Dev	4.66E-02	5.23E-02	4.34E-02	6.26E-02	5.71E-02	4.80E-02
	F14	Mean Error	2.97E-01	2.88E-01	2.90E-01	2.93E-01	2.75E-01	2.72E-01
		Std Dev	2.11E-02	2.59E-02	2.60E-02	2.76E-02	2.63E-02	2.32E-02
	F15	Mean Error	2.81E+01	3.22E+01	3.19E+01	2.47E+01	1.86E+01	1.77E+01
		Std Dev	2.99E+00	3.34E+00	2.56E+00	2.13E+00	2.35E+00	1.72E+00
	F16	Mean Error	2.19E+01	2.21E+01	2.20E+01	2.18E+01	1.89E+01	1.95E+01
		Std Dev	3.07E-01	2.53E-01	2.96E-01	3.24E-01	4.08E-01	3.50E-01
Hybrid	F17	Mean Error	1.93E+05	2.34E+05	4.45E+05	4.45E+05	5.88E+05	4.39E+03
Function 1/		Std Dev	6.40E+04	7.03E+04	1.72E+05	1.54E+05	2.01E+05	5.81E+02
	F18	Mean Error	6.92E+09	8.24E+09	3.98E+09	5.86E+09	3.45E+09	8.04E+09
		Std Dev	4.61E+09	3.85E+09	4.89E+09	4.95E+09	4.71E+09	4.01E+09
	F19	Mean Error	2.19E+01	1.82E+01	1.87E+01	2.25E+01	1.68E+01	1.53E+01
		Std Dev	4.88E+00	1.79E+00	4.35E+00	8.57E+00	2.24E+00	2.48E+00
	F20	Mean Error	4.68E+02	6.07E+02	5.14E+02	4.95E+02	1.70E+02	7.22E+01
		Std Dev	8.11E+01	1.20E+02	8.40E+01	7.54E+01	3.10E+01	1.38E+01
	F21	Mean Error	2.40E+04	3.60E+04	4.84E+04	6.11E+04	8.01E+04	2.02E+03
		Std Dev	7.16E+03	8.68E+03	2.67E+04	1.79E+04	2.63E+04	2.51E+02
	F22	Mean Error	8.96E+02	8.50E+02	1.15E+03	6.79E+02	6.54E+02	7.13E+02
		Std Dev	1.33E+02	1.54E+02	5.28E+02	1.56E+02	1.86E+02	1.51E+02
Composition	F23	Mean Error	3.44E+02	3.44E+02	3.40E+02	3.44E+02	2.00E+02	3.44E+02
Functions		Std Dev	4.50E-13	2.37E-11	1.21E+00	4.53E-13	0.00E+00	4.18E-13
	F24	Mean Error	2.67E+02	2.70E+02	2.68E+02	2.64E+02	2.00E+02	2.57E+02
		Std Dev	4.72E+00	3.24E+00	5.31E+00	5.26E+00	5.20E-08	2.04E+00
	F25	Mean Error	2.17E+02	2.19E+02	2.21E+02	2.20E+02	2.00E+02	2.09E+02
		Std Dev	2.39E+00	2.91E+00	5.60E+00	2.98E+00	0.00E+00	1.40E+00
	F26	Mean Error	1.00E+02	1.00E+02	1.01E+02	1.00E+02	1.63E+02	1.00E+02
		Std Dev	5.67E-02	3.67E-02	1.90E+00	5.49E-02	4.87E+01	4.70E-02
	F27	Mean Error	9.07E+02	1.11E+03	1.69E+03	1.41E+03	2.00E+02	1.21E+03
		Std Dev	4.40E+02	5.53E+02	3.96E+01	1.57E+02	0.00E+00	1.88E+02
	F28	Mean Error	1.46E+03	1.49E+03	6.56E+02	1.28E+03	2.00E+02	1.25E+03
		Std Dev	1.44E+02	1.39E+02	1.27E+02	7.20E+01	0.00E+00	3.72E+01
	F29	Mean Error	7.03E+03	2.22E+04	4.18E+02	9.76E+06	4.06E+02	2.63E+07
		Std Dev	2.20E+03	8.76E+03	8.42E+01	1.60E+07	6.02E+02	1.55E+07
	F30	Mean Error	9.88E+03	1.36E+04	2.38E+03	8.81E+03	2.38E+03	8.16E+03
		Std Dev	6.17E+02	1.99E+03	2.11E+02	8.14E+02	3.76E+03	2.28E+02
+ -≈			23∖5∖2	24∖4∖2	24∖5∖1	24∖4∖2	22∖6∖2

In the following two tables, “Mean Error” and “Std Dev” denote the average and standard deviation of the function error values obtained 51 runs, respectively. “+”, “–”, “≈” represent that the performance of our approach is better than, worse than, and the same as that of its competitors according to the Wilcoxon signed-rank test at α = 0.05, respectively.

The performance of NECSA is obviously better than the other five algorithms according to the Wilcoxon signed-rank test at α = 0.05 and the Friedman test. The details are discussed as follows:

Unimodal Functions(F₁ - F₃). On the three unimodal functions, except the second one, NECSA is significantly better than the others no matter D = 30 or D = 50. More specifically, for D = 30, NECSA outperforms the other five algorithms on F1, and performs better than ACS, ACSA, and GBCS on F3. Besides, for D = 50, NECSA significantly performs the best on F1 and F3 among the six algorithms involved in the comparison. On the one hand, the application of self-evaluation operation can well improve searchability and increase population diversity. On the other hand, the algorithm applies the greedy selection method and result in a high convergence rate when dealing with unimodal problems. Hence, NECSA has obtained the best performance.

Simple Multimodal Functions (F4 - F16). According to the performance shown in Tables 2 3, NECSA is the most robust among the six algorithms at D = 30 and D = 50. For D = 30, NECSA is significantly better than CS, ACS, ACSA, GBCS and ECS on 12, 12, 13, 12, 8 benchmark functions, respectively. CS, ACS, and GBCS outperform NECSA on only one function, and ACSA cannot obtain better results on any functions. For D = 50, NECSA outperforms CS, ACS and GBCS on twelve functions. ACSA cannot outperform NECSA on any functions. Furthermore, ECS is better than NECSA merely on five benchmark functions. As mentioned above, NECSA has been proved likely to trap into local optimum, the series of enhancement measures have been confirmed effective.

Hybrid Function 1(F17 - F22). For the hybrid functions in this section, the variables have been randomly divided into several subcomponents, and these subcomponents are composed of different basic functions. When dealing with these challenging functions, NECSA is significantly better than CS, ACS, ACSA, GBCS and ECS on all functions at D = 30, only inferior to CS, ACS, ACSA, GBCS and ECS on 1, 0, 1, 2, 1 function at D = 50. Obviously, NECSA is the best one.

Composition Functions (F23 - F30). Each composition function sets a local optimum to the origin as a trap. In light of the results shown in Tables 2 3, NECSA is still the winner in six comparative algorithms. For functions involved in this section, NECSA has similar optimization efficiency with other comparative algorithms, does not achieve overwhelming performance. While, generally speaking, NECSA remains robust when compared with other algorithms.

Figure 3 shows the logarithm of mean function error values of CS, ACS, ACSA, GBCS, ECS, and NECSA over 51 independent runs on some selected benchmark functions of CEC2014. They are made into convergence curves. The horizontal ordinate represents the logarithm values, and the longitudinal coordinates represent the number of iterations. According to the figure, the searchability of NECSA is better. In conclusion, NECSA can always obtain the most satisfactory results when dealing with various kinds of global optimization problems.

Fig. 3

Convergence graph of CS, ACS, ACSA, GBCS, ECS and NECSA on 30 test functions.

In short, according to the experimental results and analysis presented above, the enhancement strategies applied in NECSA can well improve the performance for various kinds of continuous global optimization problems. The modifications imposed on the original CS algorithm, including self-evaluation, modified greedy selection, adaptive control parameters, and so on, can sufficiently increase the population diversity and convergence speed, and strengthen the balance between exploration and exploitation.

5 Discussion

The control parameter c of the modified greedy selection operation and the weight coefficient ω of the self-evaluation operation may influence the performance of NECSA. The value of ω determines whether using Gaussian random walk or Lévy flight. On the one hand, if it is set too large, it may cause redundant calculation and the waste of feasible solutions. On the other hand, if it is set too small, the possibility of using Lévy flight will decrease. Besides, whether adopting enhancement methods or not is a problem worth considering. In this paper, we adopt 0.8 as the value of weight coefficient ω and 0.3 as the value of limitation control parameter c.

To demonstrate that NECSA is not parameter sensitive, we adopt different parameter settings to examine it. The experiments are carried out on nineteen functions of CEC2014, screen out F1, F2, F7, F8, F10, F12, F17, F24, F26, F27, F29 for that according to the experiments mentioned above, optimization ability on these functions is not significantly different.

Table 4 shows the mean error of different NECSA algorithms that take different parameter settings over 51 independent runs on nineteen functions. No significant result has occurred when NECSA takes similar parameter settings. So, we take the parameter settings as mentioned above.

Table 4
Discussion of mean error obtained from different c and ω

Function Unchanged c = 0.2 ω = 0.6

F3 1.11E-15 4.35E-14 0.00E+00

F4 8.44E+00 9.80E+00 9.75E+00

F5 2.03E+01 2.04E+01 2.03E+01

F6 1.53E+01 1.54E+01 1.57E+01

F9 7.49E+01 7.57E+01 7.64E+01

F11 2.43E+03 2.57E+03 2.48E+03

F13 3.34E-01 3.30E-01 3.21E-01

F14 2.36E-01 2.54E-01 2.46E-01

F15 7.27E+00 8.19E+00 7.60E+00

F16 1.07E+01 1.10E+01 1.08E+01

F18 2.40E+01 2.46E+01 2.37E+01

F19 6.11E+00 5.85E+00 6.21E+00

F20 1.68E+01 1.79E+01 1.62E+01

F21 4.78E+02 4.28E+02 4.10E+02

F22 1.48E+02 1.47E+02 1.33E+02

F23 3.15E+02 3.15E+02 3.15E+02

F25 2.03E+02 2.03E+02 2.03E+02

F28 8.67E+02 8.69E+02 8.67E+02

F30 1.30E+03 1.23E+03 1.26E+03

Function	Unchanged	c = 0.2	ω = 0.6
F3	1.11E-15	4.35E-14	0.00E+00
F4	8.44E+00	9.80E+00	9.75E+00
F5	2.03E+01	2.04E+01	2.03E+01
F6	1.53E+01	1.54E+01	1.57E+01
F9	7.49E+01	7.57E+01	7.64E+01
F11	2.43E+03	2.57E+03	2.48E+03
F13	3.34E-01	3.30E-01	3.21E-01
F14	2.36E-01	2.54E-01	2.46E-01
F15	7.27E+00	8.19E+00	7.60E+00
F16	1.07E+01	1.10E+01	1.08E+01
F18	2.40E+01	2.46E+01	2.37E+01
F19	6.11E+00	5.85E+00	6.21E+00
F20	1.68E+01	1.79E+01	1.62E+01
F21	4.78E+02	4.28E+02	4.10E+02
F22	1.48E+02	1.47E+02	1.33E+02
F23	3.15E+02	3.15E+02	3.15E+02
F25	2.03E+02	2.03E+02	2.03E+02
F28	8.67E+02	8.69E+02	8.67E+02
F30	1.30E+03	1.23E+03	1.26E+03

6 Conclusion

The adaptive parameters can significantly improve convergence and exploitation ability. However, it may result in the local optimum. The employment of self-evaluation operations can ensure the diversity of solutions. Besides, the modified greedy selection operation can not only further improve the exploration but also enhance the exploitation capability. Inspired by these ideas, we proposed a novel enhanced CS algorithm to improve the efficiency of the CS algorithm. The innovations of NECSA are listed as follows:

The adaptive control parameters based on fitness and iteration number are employed to speed up the convergence and ensure the balance between exploration and exploitation.

NECSA introduces the global search strategy combined with Lévy flight and Gaussian distribution, increasing the global searchability and the diversity of populations.

Compared with the standard CS algorithm, NECSA adds a modified greedy selection operation to enhance the search process in both early and later search phases.

The performance testing experiments on NECSA are conducted on 30 benchmark functions proposed by the CEC2014 special session. NECSA was compared with five CS and CS variants algorithms. Several enhancement methods have been introduced to NECSA, including adaptive control parameters, self-evaluation operation, and modified greedy selection operation, i.e., The experimental results demonstrate the efficiency in dealing with various kinds of global optimization problems.

In summary, we designed multiple enhancement methods to improve the searchability of the NECSA algorithm in this paper. Despite NECSA obtained promising improvement, there remains much improvement space of the proposed algorithm. For example, the parameter setting of the self-evaluation operation may deserve more consideration when dealing with different optimization problems. Besides, the balance between three different search strategies requires more flexible control mechanism. In future study, relevant work will be carried out not only on the problem mentioned before but also on the application of several novel technologies, including the Chaos theory, opposition learning, and so on.

Footnotes

Acknowledgments

This research was funded by the China Natural Science Foundation (No.71974100), Major Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu province (2019SJZDA039), Natural Science Foundation in Jiangsu Province (No. BK20191402), Qing Lan Project (R2019Q05).

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