An improved Crow Search Algorithm for high-dimensional problems

Abstract

Crow search algorithm (CSA) is a recently proposed metaheuristic optimizer inspired by the intelligent behaviour of crows with attributes like simplicity and ease of implementation. CSA is claimed superior and more effective in optimizing a variety of constrained engineering design problems in comparison to other state-of-art algorithms. In the present work, CSA is applied to high dimensional optimization problems and it is found that CSA suffers from premature convergence which leads to lower precision and less accuracy in optimization or sometimes failure. Therefore an improvement in CSA (ICSA) is suggested to solve high-dimensional global optimization problems efficiently. The balance between exploitation and exploration capabilities of CSA is improved by introducing experience factor, adaptive adjustment operator and Lévy flight distribution in position updating mechanism of crows. Lévy flight distribution promotes continuous exploration of search space and prevents premature convergence by escaping from local optimum at any stage. The performance of ICSA is validated on high-dimensional nonlinear scalable benchmark test functions. The proposed improvement in CSA makes it highly competitive and less sensitive to function dimensions. ICSA is also found superior to other well established optimizers.

Keywords

Crow search algorithm Lévy flight Unconstrained optimization

1 Introduction

In the current competitive world, resources are depleting day by day which forces people to make maximum profit at minimal cost and resources. Mathematical modelling of these issues evolves into maxima and minima problems, generally known as optimization problems. Several methods available to solve these optimization problems are broadly divided into two categories: classical approaches and modern metaheuristic algorithms. Classical approaches generally follow a step by step procedure and provide same solution every time for the same initial point. On the other hand, modern metaheuristic algorithms use stochastic distributions and generally do not provide same solution under same circumstances. In last few decades, a revolutionary change is observed in the field of optimization due to remarkable development of nature inspired metaheuristic search algorithms. This is due to the reason that classical methods make the task of optimization quite tedious and cumbersome especially when the problem is highly complex as well as non-linear. In contrary to this, nature inspired metaheuristics offer effortless handling and less time consumption to solve any complex engineering design problem due to their global search capability. As an illustration, knapsack problem is solved by novel global harmony search algorithm [1]. S. Talatahari et al. proposed the optimum design of truss structures using multi-stage particle swarm optimization (MSPSO) [2]. Task assignment problem is another complex problem and effectively solved by an improved differential evolution algorithm (IDE) [3]. Moreover, several other complicated engineering design problems have also been efficiently solved by metaheuristic search algorithms [4, 5]. Generally these metaheuristics mimic the natural behaviour of living things [6] and take benefit of meaningful information obtained from the complete population of solutions.

Genetic Algorithm (GA) is one of the most popular heuristic search algorithms inspired by nature’s evolutionary behaviour [7]. After the development of GA, numerous heuristic algorithms were proposed for optimization problems. In 1995 Kennedy et al. proposed Particle Swarm Optimization (PSO) based on foraging and navigation behaviour of bird flocks [8]. The algorithm was tested for optimization of continuous nonlinear functions and multidimensional problem of adjusting weights to train a feedforward multilayer perceptron neural network (NN). Dorigo et al. investigated Ant Colony Optimization (ACO) algorithm [9], which simulates the intelligent food searching behaviour of ants. ACO demonstrated excellent performance in solving symmetric and asymmetric travelling salesman problems. Karaboga [10] introduced Artificial Bee Colony (ABC) algorithm based on clever foraging behaviour of honey bee swarm. The author claimed promising performance of the algorithm while evaluating unimodal and multimodal benchmark functions. Some of the recent developments are Bat Algorithm (BA) [11], Firefly Algorithm (FA) [6], Flower Pollination Algorithm (FPA) [12], Cuckoo Search Algorithms (CS) [13], Gravitational Search Algorithm (GSA) [14], Ant Lion Optimizer (ALO) [15], Water Cycle Algorithm (WCA) [16], Dragonfly Algorithm (DA) [17] etc.

Till now researchers have utilized only limited characteristics found in nature and therefore Alireza Askarzadeh [18] proposed a new metaheuristic optimizer based on the intelligent behaviour of crows called as Crow Search Algorithm (CSA). The performance of CSA was claimed superior and more effective in optimizing a variety of constrained engineering design problems in comparison to other state-of-art algorithms. Unlike the gradient search methods which require derivative information, CSA uses stochastic random search which is an advantage especially when function is not differentiable, or is not continuous. Apart from this CSA needs only two control variables i.e. awareness probability and flight length. Generally each metaheuristic technique adopts best features available in nature. Two eminent features are adaptability and choice of the fittest which may be interpreted as exploration and exploitation respectively. During exploration the algorithm is directed far away from the current best solution i.e. unvisited search regions are explored whereas in case of exploitation the search is focused around the neighbourhood of previously visited solutions with an expectation of more fit solution.

Even though CSA seems to be a promising optimization technique but it is observed in the present work that sometimes it may get trapped in local optimum regions and fail to achieve the global optimal solution due to its simple solution updating mechanism. This problem is observed especially while solving large scale global optimization problems that involve high-dimensions. The possible causes behind this may be its low global exploration efficiency and lack of balance between exploration and exploitation phases. These issues are well addressed in the present work by modifying the solution updating mechanism of CSA. In this study global exploration efficiency of CSA is strengthened, by incorporating Lévy flight [13] behaviour of birds and an adaptive adjustment operator to enhance the process of randomization. Lévy flight is a generalized Brownian motion which comprises non-Gaussian randomly distributed step sizes. It helps in finding new candidate solutions far away from the current best solution. Further the adjustment operator initially motivates the global search at large scale but as the algorithm progresses its effect is minimized in order to stabilize the search. Apart from this, exploitation phase is made (i.e. local search) more efficient by incorporating experience factor in solution updating mechanism of standard CSA. This helps in seeking the required balance between exploration and exploitation phases. The experience factor correlates the previous experiences of crow regarding robbery. The improved CSA (ICSA) is tested on 15 high-dimensional nonlinear scalable benchmark functions and compared with some well-known optimizers.

Moreover, CSA offers attributes like simplicity and ease of implementation due to less adjustable parameters in comparison to popular optimizers like GA, PSO and Harmony Search (HS) [18] which motivates for the present work. Apart from this “No free lunch theorem” [19] also encourages the researchers to develop new and improved algorithms which can solve various optimization problems. Main contributions of the proposed work are as follows:

A maiden attempted has been made to explore and improve the CSA for high-dimensional optimization problems.

Basic framework of CSA is extended for unconstrained optimization problems. An improved balance between exploration and exploitation is achieved through experience factor, Lévy distribution and adaptive adjustment operator.

Rigorous comparative analysis is performed on the basis of statistical measures as well as average CPU time.

Rest of the paper is organized as follows. Section 2 presents the details of crow search algorithm. Section 3 explains the problem formulation. Section 4 describes the proposed modification in CSA. Section 5 discusses the comparative analysis of ICSA with existing methods on high-dimensional benchmark test functions. Finally, the work is concluded in Section 6.

2 Overview of Crow Search Algorithm

Crows are known for their cleverness and can communicate in a sophisticated manner, remember faces and use tools. That is why they have been recognized as one of the most intelligent animals found on the planet. The basic concept behind the algorithm is that crows store their surplus food secretly at hiding locations and retrieve it whenever needed. They can recall their food hiding places even after several months. Crows observe food hiding places of other birds and steal their food. To find the food hiding locations of crows is a tough task as they can make fool by going to other location in case they know that someone is following. CSA is formulated on the following assumptions [18]:

Crows are generally found in flocks.

They memorize their food hiding locations.

They follow each other to rob food.

They guard their food stores from being robbed by using probability.

It is considered that crows hide their food in an n-dimensional search environment. Let us consider N_c number of crows i.e. flock size is N_c. The current location of crow k at i^th iteration can be defined as vector x^k,i: $x^{k, i} = [x_{1}^{k, i}, x_{2}^{k, i}, x_{3}^{k, i} . . ., x_{n}^{k, i}]$ (1) where k = 1, 2, …, N_c, i = 1, 2 …, i_m and i_m is the maximum number of iterations. Each crow has an associated memory at which the information about its food hiding place is stored. The food hiding location of crow k at i^th iteration is represented by M^k,i. This is the best location found by crow k till now. In fact the information about the best experience of hiding place of each crow is stored. Crows visit and observe different locations in n-dimensional search space in order to find better food hiding places.

2.1 Position updating mechanism of crows

Suppose at ith iteration, crow k needs to visit its food hiding location, M^k,i. At the same time (iteration) crow q decides to chase crow k in order to access the food hiding location of crow k. In this situation, two cases may be considered:

Case 1: Crow k may not be aware that it is being followed by crow q. As a consequence, crow q will see the food hiding location of crow k and the new position of crow q will be obtained by the following equation: $x^{q, i + 1} = x^{q, i} + r_{q} \times {FL}^{q, i} \times (M^{k, i} - x^{q, i})$ (2) where r_q is a uniformly distributed random number between 0 and 1 and FL^q,i represents the flight length of crow q at i^th iteration. FL^q,i significantly affects the searching capability of algorithm i.e. lower values of FL promotes local search (closer to x^q,i) while higher values of FL lead to global search (far away from x^q,i).

Case 2: Crow k is aware that crow q is following. As an outcome, crow k will try to defend its reserved food source from being robbed and it will go to some random location in search space, in order to mislead crow q.

On the basis of these cases, the position updating mechanism of crows can be formulated as follows: $x^{q, i + 1} = {\begin{matrix} x^{q, i} + r_{q} \times {FL}^{q, i} \\ \times (M^{k, i} - x^{q, i}) & r_{k} \geq {ap}^{k, i} (3 a) \\ a random location & otherwise (3 b) \end{matrix}$ where r_k is uniformly distributed random number in the range [0,1] and ap^k,i represents the awareness probability of crow k at i^th iteration [18].

3 Problem formulation

Standard CSA was tested on five unconstrained optimization problems i.e. Sphere, Rosenbrock, Griewank, Schwefel and Ackley function with 10 dimensions [18]. In order to solve the challenging real world problems a metaheuristic must be tested for high-dimensional benchmark functions. Therefore, in the present work CSA is tested on five previously mentioned test functions [18] with higher dimension size up to 100. To perform this experimentation CSA is executed for 30 independent runs and in each run it is supposed to perform 40000 number of function evaluations. Table 1 shows the recorded mean values and standard deviation (SD) for every test function. It is observed from the results that increase in problem dimension, increases the mean and standard deviation and CSA could not find the optimal solution in most of the cases. Hence the performance of CSA depends on dimension, which indicates that CSA has low global exploration efficiency. The rise in standard deviation indicates that there is a lack of balance between exploration and exploitation phases and hence motivates to improve the algorithm.

Table 1
Performance of CSA on benchmark functions^* for different dimensions

Dim Parameters Ackley (F4) Griewank (F5) Sphere (F6) Schwefel (F8) Rosenbrock (F11)

2 Mean 4.22181E-14 0.002320232 6.00648E-27 5.95315E-15 4.75297E-27

SD 3.26077E-14 0.003424384 1.43912E-26 5.01582E-15 8.724E-27

5 Mean 0.054874129 0.118582293 2.43105E-18 9.09857E-10 1.048224336

SD 0.300557938 0.073699426 5.16149E-18 1.0872E-09 1.767998688

10 Mean 1.709234009 0.174942752 2.27554E-11 0.020545132 6.234959868

SD 1.000449226 0.10074841 4.24591E-11 0.070164652 1.809413112

20 Mean 3.749700427 0.02348376 3.82544E-06 1.053115209 20.48773447

SD 1.109900578 0.021584118 2.11146E-06 0.904176806 10.85729092

30 Mean 4.202282219 0.026421056 0.000420164 1.965363273 29.77942782

SD 1.058015896 0.012337384 0.004436245 0.84697725 9.96008744

50 Mean 5.095712641 0.302431738 0.255781239 3.898791967 51.42966691

SD 0.994693617 0.060795357 0.151407813 1.101076828 11.19938789

100 Mean 5.556669771 1.180326697 21.09227544 10.73035768 183.7286116

SD 5.556669771 0.036965183 5.298098882 1.425867498 32.48228078

Dim	Parameters	Ackley (F4)	Griewank (F5)	Sphere (F6)	Schwefel (F8)	Rosenbrock (F11)
2	Mean	4.22181E-14	0.002320232	6.00648E-27	5.95315E-15	4.75297E-27
	SD	3.26077E-14	0.003424384	1.43912E-26	5.01582E-15	8.724E-27
5	Mean	0.054874129	0.118582293	2.43105E-18	9.09857E-10	1.048224336
	SD	0.300557938	0.073699426	5.16149E-18	1.0872E-09	1.767998688
10	Mean	1.709234009	0.174942752	2.27554E-11	0.020545132	6.234959868
	SD	1.000449226	0.10074841	4.24591E-11	0.070164652	1.809413112
20	Mean	3.749700427	0.02348376	3.82544E-06	1.053115209	20.48773447
	SD	1.109900578	0.021584118	2.11146E-06	0.904176806	10.85729092
30	Mean	4.202282219	0.026421056	0.000420164	1.965363273	29.77942782
	SD	1.058015896	0.012337384	0.004436245	0.84697725	9.96008744
50	Mean	5.095712641	0.302431738	0.255781239	3.898791967	51.42966691
	SD	0.994693617	0.060795357	0.151407813	1.101076828	11.19938789
100	Mean	5.556669771	1.180326697	21.09227544	10.73035768	183.7286116
	SD	5.556669771	0.036965183	5.298098882	1.425867498	32.48228078

^*See Appendix A for benchmark functions.

4 Improved Crow Search Algorithm

Equilibrium between exploitation and exploration phases is always required for global optimization so as to solve high-dimensional optimization problems [20, 21]. Besides this, a complete search space must be explored to obtain a guaranteed optimal solution. A careful observation of position updating mechanism in standard CSA (Equations (3a) & (3b)) gives the complete insight for the required modification. In case 1 (Equation (3a)), CSA tends to search in local optimal region where the current best solution exists. The boundaries of local search regions can be expanded slightly which may provide better exploitation. On the other hand, case 2 (Equation (3b)) diverts the search far away from the current best solution which means CSA tries to find other best solutions (global search). As per Equation (3b) the global search solely depends on the random walk and fast convergence cannot be guaranteed. Hence there is a scope of improvement in solution updating strategy without any loss in the attributes i.e. simplicity and effortless implementation of standard CSA.

4.1 Modified position updating mechanism

In order to design any potentially acclaimed meta-heuristic algorithm there should be a proper balance between exploitation and exploration. There is no thumb rule to solve this issue [22], therefore it becomes purely an experimental or observational exercise. The standard CSA also suffers with this problem. The improvement in exploitation phase refers to intensifying search in the vicinity of best possible solutions by increasing search area in the neighbourhood of best possible solutions. This is achieved by the introduction of experience factor. Further the increase in problem dimension shows premature convergence in CSA which indicates lack of diversity in new candidate solutions. Therefore diversification in generating new candidate solutions is improved by considering Lévy flight.

Experience factor

A precise observation of Equation (3a) reveals that search boundary can be extended by introducing a fractional constant called as experience factor (ef) which leads to a modified rule for updating the position of crow q in case 1 (Equation (4)):

$\begin{matrix} x^{q, i + 1} & = & ef \times x^{q, i} + r_{q} \times {FL}^{q, i} \\ \times (M^{k, i} - x^{q, i}) \end{matrix}$ (4) This term can be related to the fact that crows use their experience of being a thief in order to predict the behaviour of other crows [18]. The experience factor basically shifts the current search location (x^q,i) to a small distance and then through random perturbations new solutions are generated near the current best solutions. The effect of ef may be understood by the following mathematical computation:

Consider a 3-dimensional optimization problem to minimize $F 1 (x) = x_{1}^{2} + 10^{6} \sum_{j = 2}^{n} x_{j}^{2}$ . The initial parameter values are assumed as $\begin{matrix} r_{q} = 0.3198, {FL}^{q, 1} = 2, \\ M^{k, 1} = [\begin{matrix} 62.2524 & 97.8031 & 92.8466 \end{matrix}], \\ x^{q, 1} = [\begin{matrix} 56.9657 & 50.6516 & 51.8849 \end{matrix}] . \end{matrix}$ The updated position of crow q after first iteration using Equation (3a) will be as follows: $x^{q, 2} = [\begin{matrix} 60.3471 & 80.8097 & 78.0840 \end{matrix}]$ and the corresponding objective function value is 1.2627 × 10¹⁰. On the other hand considering the effect of ef = 0.5, using Equation (4), the new position of crow q will be: $x^{q, 2} = [\begin{matrix} 31.8642 & 55.4839 & 52.1416 \end{matrix}]$ and the objective function value is 5.7972 × 10⁹. It is observed that, F1 (x) is minimum using ICSA in comparison to standard CSA as the solution generated is quite close to the global optimum ( $F 1 (x^{*}) = [\begin{matrix} 0 & 0 & 0 \end{matrix}]$ ). The graphical interpretation of ef is provided in Fig. 1. It is observed from the figure that standard CSA searches in small regions around the best solutions whereas the proposed version (ICSA) enhances the nearby search region up to an acceptable limit. This increases the probability of finding global optimal solution near the best solutions. Thus case 1 of standard CSA represented by Equation (2) is modified by incorporating experience factor (ef) and leads to Equation (4). The value of ef is obtained after rigorous experimentations on the benchmark functions under consideration. Some observations of this experimental analysis are presented in Table 2. A significant effect of ef is revealed on the performance of ICSA. Higher values of ef i.e. ef > 1 causes unstable performance of the algorithm whereas 0 < ef < 1 provides acceptable performance of the algorithm. In the present work, best fractional value of ef is found to be 0.5 as it provides stable as well as accurate solutions for most of the test cases.

Lévy Flight

The second case of standard CSA is designed to produce any random location in search space using uniform distribution (Equation (3b)) and may be represented by the following expression: $x^{q, i + 1} = x_{l} - (x_{l} - x_{u}) \times r$ (5) where r is a random number in the range of [0,1], x_l is lower bound and x_u is upper bound of the variable. The proposed work introduces the advantage of Lévy distribution instead of uniform distribution for better exploration while maintaining the integrity of CSA. Lévy flight is a powerful mathematical tool used by the researchers for improving the global exploration capability of various meta-heuristics algorithms [23 –27]. Lévy flights help to find new candidate solutions far away from the current best solution. It is a kind of random walk in which step length is drawn from a Lévy distribution. This distribution is often expressed by a power-law formula L (s) ∼ |s|^-1-β where 0 < β ≤ 2 is an index. Lévy distribution is stated mathematically as follows:

$\begin{matrix} L (s, γ, μ) \\ = {\begin{matrix} \sqrt{\frac{γ}{2 π}} \exp [- \frac{γ}{2 (s - μ)}] \\ \times \frac{1}{(s - μ)^{3 / 2}} & 0 < μ < s < \infty \\ 0 & otherwise \end{matrix} \end{matrix}$ (6) where μ, γ > 0. γ is scale parameter and μ is shift parameter.

Fig.1

Improvement in exploitation using experience factor (ef).

Table 2

The effect of different ef values on the performance of ICSA on three benchmark functions^* with 50 dimensions

Function	Parameters	ef = 0.1	ef = 0.3	ef = 0.5	ef = 0.7	ef = 0.9	ef = 1.5	ef = 2
F1	Mean	1.7751E-18	1.5655E-22	7.3699E-24	1.4117E-23	1.5974E-19	1.2081E+11	1.2573E+11
	SD	5.9851E-18	3.7348E-22	1.4731E-23	2.2757E-23	1.6548E-19	1.4041E+10	8.6617E+09
F2	Mean	1.5235E-20	2.2558E-24	5.5289E-26	8.2919E-26	2.9027E-22	2.5215E+07	2.4229E+07
	SD	5.1033E-20	4.2689E-24	1.1383E-25	1.4178E-25	3.1029E-22	4.6343E+07	3.8602E+07
F3	Mean	1.0856E-74	6.5251E-85	1.1529E-90	4.9631E-89	1.5289E-77	7.7525E+00	8.0422E+00
	SD	4.0611E-74	2.4464E-84	3.2301E-90	2.5619E-88	5.4132E-77	1.5508E+00	1.3192E+00

*See Appendix A for benchmark functions.

The following equation can be used to generate new positions of crows: $x^{q, i + 1} = ef \times x^{q, i} + Lévy (n) + α_{1} (1 - i / i_{m})$ (7) where Lévy flight is calculated as follows: $Lévy (x) = 0.01 \times \frac{r_{a} \times σ}{| r_{b} |^{\frac{1}{β}}}$ (8) where r_a and r_b are two normally distributed random numbers in [0,1], β is a constant considered to be 1.5 in the present work, and σ is calculated as: $σ = {(\frac{Γ (1 + β) \times \sin (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{(\frac{β - 1}{2})}})}^{1 / β}$ (9) where Γ (x) = (x - 1) !. Fig. 2 shows the exploration of different positions in search space by crow q while following crow k in 10 Lévy flights. The third term in Equation (7) is used to adjust the flight lengths adaptively and hence it is adaptive flight adjustment operator, where α₁ is adjustment coefficient having value 0.5 for the present work. This operator has a linearly decreasing variation and depends on maximum iteration value. In the beginning it provides large change in Lévy flights but as the algorithm proceeds its value decreases in order to stabilize the algorithm in global optimal region. The equations Equation (3b) and Equation (7) both generate a new random location of crow q. Hence in improved version of CSA i.e. ICSA uses Equation (7) in place of Equation (3b). Summarizing the above discussion, following improved position updating mechanism is proposed in this work: $x^{q, i + 1} = {\begin{matrix} ef \times x^{q, i} + r_{q} \times {FL}^{q, i} \\ \times (M^{k, i} - x^{q, i}) & r_{k} \geq {ap}^{k, i} \\ ef \times x^{q, i} + Lévy (n) \\ + α_{1} (1 - i / i_{m}) & otherwise \end{matrix}$ (10)

Fig.2

Exploration of search space with Lévy flight inspired behaviour of crows.

The changes in position updating mechanism of crow search algorithm leads to an improved CSA i.e. ICSA. The flow chart of the proposed ICSA discussed above is presented in Fig. 3 and the results obtained are given in next section.

Fig.3

Flowchart of ICSA.

5 Experimental results and discussion

To investigate the effectiveness and stability of the proposed ICSA, experimental tests are designed and conducted on fifteen well-known high-dimensional nonlinear scalable benchmark functions taken from [28 –30]. Two types of functions are considered i.e. unimodal and multimodal, listed in Appendix A (Table 12). Unimodal functions have only one optimum and are used to examine the exploitation capabilities of optimization algorithms. On the other hand, multimodal functions have several local optima and are suitable for investigating the exploration capabilities of optimization algorithms [31]. The simulation is carried out in MATLAB on PC having 32-bit operating system and Intel(R) Core(TM) 2 Duo CPU T5870 processor operated on 2.00 GHz frequency with 3 GB RAM.

5.1 Experimental test 1

In this experimental test the performance of ICSA is compared with standard CSA for fifteen benchmark functions with high dimensions i.e. 50, 100 and 500. For fair comparison, common parameters of the two algorithms are taken to be identical i.e. awareness probability (ap) is 0.1, flight length (FL) is taken as 2 while maximum number of function evaluations are considered to be 40000. For an effective statistical analysis, both the algorithms are executed for 30 independent runs for every benchmark function. The results in terms of mean and standard deviation of solutions are recorded in Table 3. For the sake of clarity in comparison, results with values 10E-14 or less are treated as optimal value. It is observed from Table 3 that ICSA outperforms the standard CSA on all 15 benchmark functions with 50, 100 and 500 dimension size. In terms of optimization accuracy, ICSA could achieve global optimum for functions F1, F2, F3, F4, F5, F6, F8, F9, F10, F12 and F14 i.e. on 11 cases out of 15, while CSA could not find global optimum in any case. Both the algorithms show premature convergence on function F7, F11, F13 and F15, however ICSA is still closer to the optimal region as compared to CSA. Apart from this ICSA is also found highly stable and consistent as it offers very low SD in almost every case except for function F11. The results of ICSA for F11 function are quite deviated, but far better than the standard CSA. It is also observed that ICSA shows consistent performance on increasing the dimensions as variation in mean value and standard deviation is very less. This indicates that the balance between exploration and exploitation phases is improved and dimensional dependency is also reduced.

Table 3
Comparison of CSA and ICSA for high-dimensional problems

Function Dim CSA ICSA

Mean SD Mean SD

F1 50 141958.2162 69524.92668 3.43627E-24 4.4588E-24

100 18062202.4 4726344.982 3.15442E-23 9.66126E-23

500 2139212705 138386265.1 1.65842E-22 2.50778E-22

F2 50 2926.512167 652.1193795 1.00302E-25 2.13434E-25

100 6443.745568 811.9103454 7.67352E-26 1.46875E-25

500 27716.57384 2237.974773 6.20423E-26 1.29045E-25

F3 50 2.4979E-07 3.23345E-07 1.87004E-89 1.02138E-88

100 1.19875E-06 6.96429E-07 1.70603E-91 6.36266E-91

500 5.80625E-06 1.4361E-06 2.38551E-97 1.26702E-96

F4 50 4.981097279 1.050753896 3.01981E-15 2.00104E-15

100 5.804424346 0.760781254 2.90138E-15 2.22422E-15

500 5.720030358 0.258512255 2.54611E-15 1.8027E-15

F5 50 0.305763318 0.072410225 0 0

100 1.163487856 0.033195617 0 0

500 20.35014553 1.323013656 0 0

F6 50 0.147827933 0.056330129 4.06476E-30 8.2005E-30

100 20.32349631 5.19410872 1.3545E-29 2.29772E-29

500 2157.146401 142.6462666 5.82914E-29 8.86089E-29

F7 50 0.152991839 0.078241512 0.00399163 0.001036104

100 19.46786416 5.239417138 0.010070079 0.002316065

500 2208.156315 136.3091798 0.075965953 0.017115234

F8 50 4.104335819 1.28057424 1.02742E-14 9.67138E-15

100 10.85525469 1.624299239 1.98625E-14 1.1994E-14

500 92.77481937 3.177149042 9.68641E-14 8.39009E-14

F9 50 335.8360486 155.0045404 1.36416E-26 2.18875E-26

100 2.25198E+12 4.73553E+12 3.52279E-25 6.7018E-25

500 9.56532E+19 2.40556E+19 1.09284E-22 2.40725E-22

F10 50 3.58604544 3.518307216 1.21946E-15 1.15752E-15

100 12.77378123 8.572248045 2.83389E-15 3.51716E-15

500 22.27940249 1.79747184 9.09609E-15 8.49992E-15

F11 50 55.49929929 14.23209521 7.984789234 13.75765982

100 175.1087461 36.61534588 9.612760441 2.455924066

500 3095.420702 233.4311319 299.6839544 198.2459915

F12 50 0.024895209 0.00719464 1.37551E-28 2.58363E-28

100 0.105757624 0.020125995 1.11941E-28 2.57752E-28

500 2.609311183 0.244512374 2.10095E-31 2.96623E-31

F13 50 4.339363039 3.269991168 0.007530664 0.002410572

100 6.84161321 3.007670743 0.02005451 0.004675741

500 37.4004739 1.255614047 0.146182762 0.0441013

F14 50 54.61905894 15.99670943 0 0

100 119.7417043 38.02783264 0 0

500 2161.589245 79.92338891 0 0

F15 50 1.570624159 0.132468179 0.006658223 0.02533868

100 3.266561444 0.245063243 1.1591E-13 6.23717E-13

500 10.09085252 0.36275396 3.6674E-06 2.00872E-05

Function	Dim	CSA	ICSA
F1	50	141958.2162	69524.92668	3.43627E-24	4.4588E-24
	100	18062202.4	4726344.982	3.15442E-23	9.66126E-23
	500	2139212705	138386265.1	1.65842E-22	2.50778E-22
F2	50	2926.512167	652.1193795	1.00302E-25	2.13434E-25
	100	6443.745568	811.9103454	7.67352E-26	1.46875E-25
	500	27716.57384	2237.974773	6.20423E-26	1.29045E-25
F3	50	2.4979E-07	3.23345E-07	1.87004E-89	1.02138E-88
	100	1.19875E-06	6.96429E-07	1.70603E-91	6.36266E-91
	500	5.80625E-06	1.4361E-06	2.38551E-97	1.26702E-96
F4	50	4.981097279	1.050753896	3.01981E-15	2.00104E-15
	100	5.804424346	0.760781254	2.90138E-15	2.22422E-15
	500	5.720030358	0.258512255	2.54611E-15	1.8027E-15
F5	50	0.305763318	0.072410225	0	0
	100	1.163487856	0.033195617	0	0
	500	20.35014553	1.323013656	0	0
F6	50	0.147827933	0.056330129	4.06476E-30	8.2005E-30
	100	20.32349631	5.19410872	1.3545E-29	2.29772E-29
	500	2157.146401	142.6462666	5.82914E-29	8.86089E-29
F7	50	0.152991839	0.078241512	0.00399163	0.001036104
	100	19.46786416	5.239417138	0.010070079	0.002316065
	500	2208.156315	136.3091798	0.075965953	0.017115234
F8	50	4.104335819	1.28057424	1.02742E-14	9.67138E-15
	100	10.85525469	1.624299239	1.98625E-14	1.1994E-14
	500	92.77481937	3.177149042	9.68641E-14	8.39009E-14
F9	50	335.8360486	155.0045404	1.36416E-26	2.18875E-26
	100	2.25198E+12	4.73553E+12	3.52279E-25	6.7018E-25
	500	9.56532E+19	2.40556E+19	1.09284E-22	2.40725E-22
F10	50	3.58604544	3.518307216	1.21946E-15	1.15752E-15
	100	12.77378123	8.572248045	2.83389E-15	3.51716E-15
	500	22.27940249	1.79747184	9.09609E-15	8.49992E-15
F11	50	55.49929929	14.23209521	7.984789234	13.75765982
	100	175.1087461	36.61534588	9.612760441	2.455924066
	500	3095.420702	233.4311319	299.6839544	198.2459915
F12	50	0.024895209	0.00719464	1.37551E-28	2.58363E-28
	100	0.105757624	0.020125995	1.11941E-28	2.57752E-28
	500	2.609311183	0.244512374	2.10095E-31	2.96623E-31
F13	50	4.339363039	3.269991168	0.007530664	0.002410572
	100	6.84161321	3.007670743	0.02005451	0.004675741
	500	37.4004739	1.255614047	0.146182762	0.0441013
F14	50	54.61905894	15.99670943	0	0
	100	119.7417043	38.02783264	0	0
	500	2161.589245	79.92338891	0	0
F15	50	1.570624159	0.132468179	0.006658223	0.02533868
	100	3.266561444	0.245063243	1.1591E-13	6.23717E-13
	500	10.09085252	0.36275396	3.6674E-06	2.00872E-05

Besides the statistical measures like mean and standard deviation values, commonly used non-parametric statistical test i.e. Wilcoxon’s rank test is also performed to compare the results of ICSA and CSA. The test is conducted with 5% significance level i.e. α = 0.05 and the obtained p-values are presented in Table 4 for ICSA vs CSA for 50, 100 and 500 dimensions. All p values are less than 0.05 which strongly indicates that the results of ICSA are statistically significant and not occurred just by coincidence. The sign ‘+’ indicates that the difference is of statistical interest while sign ‘-’ shows that difference is not statistically significant. The behaviour of search agents over the course of iterations is studied to analyse the reason of improved performance. The search history of ICSA is recorded for this purpose while solving a highly complex multimodal function F14 (Fig. 4 (a)). It is revealed from the results (Fig. 4 (c)-(f)) that in starting phase, ICSA explores the promising regions of search space and as the algorithm progresses it starts exploitation near the global optima. Trajectory of the first variable for first search agent (Fig. 4 (b)) also confirms that ICSA first explores and then exploitation is done. In other words the search agent faces abrupt fluctuations in early stage of optimization and these variations are suppressed gradually over the course of iterations. The above facts show that ICSA maintains a proper balance between exploration and exploitation phases and hence provides the improved performance.

Table 4

Results of Wilcoxon’s test for ICSA vs CSA

Dim	p-value	α = 0.05
50	6.53698E-06	+
100	3.86804E-07	+
500	5.80205E-07	+

Fig.4

(a) 2-D version of function F14, (b) Trajectory of the first variable for first search agent while solving F14 and the positions of individual crow after (c) 1^st iteration, (d) 10^th iteration, (e) 20^th iteration and (f) 30^th iteration.

5.2 Experimental test 2

The effectiveness of ICSA is demonstrated by comparing it with other existing state-of-art algorithms like DA, BA, FA, PSO, GA including CSA. The test is performed on the same benchmark functions with dimension size 30 which is relatively low in comparison to the experimental test 1. The commonly defined parameters used for various algorithms and their details are presented in Table 5. All the algorithms are executed for 30 independent runs with 40000 number of function evaluations for fair comparison. The best, mean, worst, standard deviation and successful runs (SR) offered by the algorithms are recorded for each test function and presented in Tables 6 and 7. It is observed that ICSA shows superior performance in comparison to the other six optimization algorithms on all benchmark functions except F13. ICSA successfully found global optimum for 13 cases out of 15. However other algorithms show premature convergence in almost all cases. FA could achieve global optimum for functions F3 and F7 while CSA achieves global optimum only for function F3. None of the algorithms could find global optimum for functions F11 and F13. In case of function F11, ICSA is quite close to optimal region while for function F13, FA defeats the other algorithms but could not achieve the global optimum. Quantitative analysis of the algorithms is carried out by computing the mean absolute error (MAE) for all 15 cases. MAE is an effective and valid performance index used in statistical analysis which indicates the difference of results from actual values [32]. The MAE can be defined as follows: $MAE = \frac{\sum_{k = 1}^{n} | m_{k} - o_{k} |}{n}$ (11)

where m_k denotes the mean of optimal results produced by an algorithm, o_k is actual value of global optimum for the function under optimization and n indicates the total number of benchmark functions considered for study. Average error rates offered by the algorithms on 15 benchmark functions are presented in Table 8. The algorithms are ranked on the basis of MAE as shown in Table 9. ICSA offered minimum MAE and hence ranked 1. The consistency of any metaheuristic can be measured by calculating the total number of successful runs and therefore each algorithm is executed 450 times i.e. 30 runs for each benchmark function. More number of successful runs means that a particular metaheuristic obtained global optimum solution almost every time and is highly consistent or reproducible. It is revealed from the results (Fig. 5) that ICSA has a very good reproducibility as it achieved the global optimum solution 381 times out of 450 runs and hence ranked one. FA is ranked second as it could find 60 global optimums while CSA achieves 7 global optimums. The DA, BA, PSO and GA failed to achieve global optimums in 450 runs. To show the statistical significance between the proposed ICSA and other existing algorithms, Wilcoxon’s ranks test is also performed and the results are given in Table 10. The sign ‘+’ in Table 10 indicates that null hypothesis is rejected which means the first algorithm outperforms the second one. Hence it can be concluded that ICSA performs better than all six algorithms by considering the 5% level of significance. Apart from statistical analysis, average CPU time consumption of any optimization algorithm is also an important performance measure. Therefore average CPU time (Avg. time) for all the algorithms is evaluated for 30 trial runs and results are recorded in Tables 6 and 7. It is observed that standard CSA takes least average CPU time for most of the optimization problems (13 cases out of 15) as compared to other algorithms. On the other hand, the proposed ICSA is slightly slower as compared to CSA for same number of runs due to Lévy function calculation. However, ICSA offers consistency and accuracy for all the test cases.

Fig.5

Comparison of algorithms in finding the global optimal solution out of 450 runs.

Table 5

Parameter settings used for DA, BA, FA, PSO and GA during experimental test 2

Algorithm	Parameter settings
DA	Inertia weight w = 0.9 and this is adaptively decreasing up to 0.4
BA	Loudness = 0.5, Pulse rate = 0.5, f_min = 0, f_max = 2
FA	α = 0.25, β = 0.20, γ = 1
PSO	c1 = 1.5, c2 = 2.0, w = 0.99
GA	Crossover fraction =0.8, Tournament selection, arithmetic crossover, adaptive feasible mutation

Table 6

Results obtained by ICSA, CSA, DA, BA, FA, PSO and GA on 15 benchmark functions

Function	Algorithm	Best	Mean	Worst	SD	Avg. time (s)	SR
F1	ICSA	3.82088E-27	2.07649E-24	1.6043E-23	3.53486E-24	2.3319	30
	CSA	122.5139647	584.7759207	1628.436165	341.7069286	0.8959	0
	DA	229311057.7	697649238.6	1400545063	293359153.2	65.7876	0
	BA	16327145925	40167366979	55123228237	10296955291	0.8229	0
	FA	800.7938888	3672.857654	9008.248203	2301.733428	5.0514	0
	PSO	3.154477577	26202.86437	635821.6603	116053.4237	3.6371	0
	GA	4631588.228	5761532.421	7235807.917	567790.5835	6.8006	0
F2	ICSA	2.16354E-28	1.54778E-25	2.42004E-24	4.40321E-25	1.9372	30
	CSA	852.1439094	1551.884228	2474.041355	431.3228076	0.6011	0
	DA	8393.180523	54004.06484	147152.8717	29976.53015	48.8142	0
	BA	46524.30019	290740.1174	4627473.974	848256.5344	0.8372	0
	FA	13537.67164	22704.25684	40029.80303	6208.112837	3.8201	0
	PSO	5.49691E-05	1.653068306	23.06507371	5.260038126	3.1036	0
	GA	6.727647356	754.6156638	7358.354288	1609.365942	5.5105	0
F3	ICSA	5.32885E-97	1.35422E-90	2.82513E-89	5.18855E-90	3.8936	30
	CSA	7.54542E-16	8.17011E-09	1.23538E-07	2.44634E-08	1.4928	7
	DA	3.67971E-06	5.34062E-05	0.000161343	4.8001E-05	51.8643	0
	BA	1.89302E-06	3.80555E-06	6.06857E-06	1.03549E-06	1.3543	0
	FA	3.97299E-23	3.314E-21	7.14931E-20	1.29007E-20	4.4883	30
	PSO	2.01926E-09	1.237E-07	4.91259E-07	1.62214E-07	3.7918	0
	GA	0.018155541	0.218968714	0.546296402	0.155445323	6.3689	0
F4	ICSA	8.88178E-16	2.19084E-15	4.44089E-15	1.7413E-15	2.5835	29
	CSA	2.408366549	4.408719534	7.004611355	1.142272704	0.9177	0
	DA	5.797367218	7.545816548	9.311505001	0.942274446	49.9735	0
	BA	16.71880335	19.8554813	19.96676885	0.592425501	1.2073	0
	FA	0.006784197	0.008647631	0.011013161	0.000906503	4.0885	0
	PSO	0.019958821	2.049234212	3.681431877	0.855205218	4.2431	0
	GA	2.751206131	3.44136285	3.706600974	0.176737866	5.3549	0
F5	ICSA	0	0	0	0	3.0515	30
	CSA	0.005935103	0.023683373	0.049540807	0.010653558	0.9823	0
	DA	2.961152572	7.756107591	19.57317972	3.451558102	45.1195	0
	BA	204.9391182	421.4736255	689.2638495	118.6788558	1.2659	0
	FA	0.001597635	0.004618614	0.027380932	0.005404559	4.1802	0
	PSO	0.00065835	0.114288924	1.210520005	0.244910662	3.9651	0
	GA	0.182549437	0.265509465	0.340646392	0.034423253	6.4789	0
F6	ICSA	1.49304E-32	2.59851E-30	4.01754E-29	7.6581E-30	1.9962	30
	CSA	0.00021559	0.00066698	0.001216751	0.000250768	0.6787	0
	DA	238.8775235	719.3233919	1540.912693	313.3878719	45.2059	0
	BA	20459.05549	41355.03441	63243.37502	12499.35133	0.9000	0
	FA	0.000664305	0.001231892	0.002020177	0.00031834	3.8583	0
	PSO	2.67264E-06	0.078541282	1.996917481	0.363703515	3.0892	0
	GA	4.920929056	5.890235216	7.269705051	0.61245105	5.4214	0
F7	ICSA	0	0	0	0	2.0483	30
	CSA	5	11.8	26	5.108478416	0.7032	0
	DA	203	905.4666667	2124	437.8671451	45.2648	0
	BA	25406	43586.4	83155	13356.26758	0.8230	0
	FA	0	0	0	0	3.1890	30
	PSO	1	18.43333333	120	23.78449703	3.0610	0
	GA	4	7.933333333	11	1.529780993	5.4731	0
F8	ICSA	6.40781E-16	6.10908E-15	3.0306E-14	5.7429E-15	2.3781	24
	CSA	0.513689068	1.959627583	3.421028613	0.847925939	0.7397	0
	DA	9.17135089	18.89253945	36.93556155	5.894305909	44.0437	0
	BA	78.09174335	972445.8112	13590574.21	2889021.955	0.7422	0
	FA	0.016996455	0.04031065	0.127594461	0.027355171	3.8252	0
	PSO	0.027610162	0.368761657	1.145109186	0.308212045	3.0094	0
	GA	9.518337459	11.61257964	13.57878212	0.866680065	5.6553	0

Table 7

Results obtained by ICSA, CSA, DA, BA, FA, PSO and GA on 15 benchmark functions

Function	Algorithm	Best	Mean	Worst	SD	Avg. time (s)	SR
F9	ICSA	1.67293E-30	5.03138E-27	1.07697E-25	1.95256E-26	2.2048	30
	CSA	14.62282697	38.71011662	72.88519555	13.5604579	0.6798	0
	DA	41.72968392	320.4699812	644.1093793	147.9483173	43.1496	0
	BA	151.4110614	395.396481	808.697423	126.9463849	0.8057	0
	FA	0.002609606	0.282054262	1.399579539	0.362778384	3.8208	0
	PSO	5.06783E-05	0.019158737	0.214749423	0.049261397	3.0674	0
	GA	7.266581145	7369593.027	68826352.63	16794275.28	5.9603	0
F10	ICSA	9.32009E-17	7.94153E-16	2.45082E-15	6.45514E-16	2.3234	30
	CSA	0.021591085	1.72211973	6.103619989	1.570388095	0.9797	0
	DA	1.386779516	19.55243011	37.67136129	8.469284377	44.8401	0
	BA	4.548751203	16.13887222	31.54008412	5.701209324	1.2565	0
	FA	0.002258863	0.027548365	0.172619936	0.038240647	4.0881	0
	PSO	0.003582595	0.081545804	0.380379661	0.10361814	3.3830	0
	GA	5.100479619	6.679953109	7.73180244	0.609629592	6.2519	0
F11	ICSA	0.709631721	1.416004564	2.863838285	0.537199449	2.0546	0
	CSA	21.32141437	33.11608896	79.15947296	14.72471061	0.6721	0
	DA	392.452622	1784.94971	4846.65462	1125.542145	45.7772	0
	BA	18587.14986	200742.6436	772546.7548	165693.1842	0.8006	0
	FA	26.13187702	42.36573681	170.0843006	35.44450304	3.9884	0
	PSO	18.80390768	45.82788865	126.9581601	28.6664207	3.1751	0
	GA	4.578337265	13.23259029	21.77661265	4.857310119	5.3411	0
F12	ICSA	8.75803E-32	5.33282E-29	6.12952E-28	1.19155E-28	2.0806	30
	CSA	0.002554447	0.008131471	0.018929228	0.003506422	0.8213	0
	DA	0.301232349	1.741527818	4.70751246	1.245743193	44.2593	0
	BA	0.004257415	0.007082975	0.009960995	0.001436884	0.9322	0
	FA	0.014806292	0.079617562	0.217042764	0.047081078	4.8005	0
	PSO	9.68063E-05	0.001672578	0.008153235	0.00184366	3.3739	0
	GA	16.44049716	62.314256	126.5932597	26.48902336	5.6241	0
F13	ICSA	0.001451912	0.003586402	0.005875586	0.001166245	2.5906	0
	CSA	0.26864036	3.341504939	9.317091409	2.459242632	0.9788	0
	DA	1.948973033	11.59901345	33.92239689	7.271545365	45.0891	0
	BA	8.957216473	29.24617611	52.71212578	9.954942266	2.2538	0
	FA	3.94959E-06	0.005987502	0.089547671	0.022712683	5.7776	0
	PSO	0.090094886	3.660322993	12.99833848	2.759457708	4.9417	0
	GA	0.125465159	0.307863338	0.483111962	0.086673689	6.7406	0
F14	ICSA	0	0	0	0	1.9884	30
	CSA	9.950081492	32.23710354	56.71356871	10.42409791	0.7587	0
	DA	76.46318256	169.9386745	248.0069654	40.28814333	43.2988	0
	BA	9.950081492	32.23710354	56.71356871	10.42409791	0.8396	0
	FA	21.88953151	47.7585934	73.62750789	13.83846297	3.8477	0
	PSO	29.84952283	55.91931126	84.57132938	13.15124479	3.1208	0
	GA	71.89845292	126.128554	196.1517561	30.66954968	5.5992	0
F15	ICSA	1.92947E-17	3.67644E-05	0.001102933	0.000201367	2.0178	28
	CSA	0.599873346	0.791152554	1.005516934	0.104714425	0.7369	0
	DA	2.799873346	4.413206679	6.399873346	0.861327497	45.4200	0
	BA	17.19987335	21.46654001	28.89987335	2.823587023	1.0786	0
	FA	0.299873346	0.403206679	0.599873346	0.07648905	4.1207	0
	PSO	0.799873346	1.439873346	2.699873346	0.375637389	3.7470	0
	GA	0.299873346	0.299873346	0.299873346	4.92747E-16	5.6497	0

Table 8

Average error rates offered by all the algorithms on 15 benchmark problems

F(x)	ICSA	CSA	DA	BA	FA	PSO	GA
F1	2.07649E-24	584.7759207	697649238.6	40167366979	3672.857654	26202.86437	5761532.421
F2	1.54778E-25	1551.884228	54004.06484	290740.1174	22704.25684	1.653068306	754.6156638
F3	1.35422E-90	8.17011E-09	5.34062E-05	3.80555E-06	3.314E-21	1.237E-07	0.218968714
F4	2.19084E-15	4.408719534	7.545816548	19.8554813	0.008647631	2.049234212	3.44136285
F5	0	0.023683373	7.756107591	421.4736255	0.004618614	0.114288924	0.265509465
F6	2.59851E-30	0.00066698	719.3233919	41355.03441	0.001231892	0.078541282	5.890235216
F7	0	11.8	905.4666667	43586.4	0	18.43333333	7.933333333
F8	6.10908E-15	1.959627583	18.89253945	972445.8112	0.04031065	0.368761657	11.61257964
F9	5.03138E-27	38.71011662	320.4699812	395.396481	0.282054262	0.019158737	7369593.027
F10	7.94153E-16	1.72211973	19.55243011	16.13887222	0.027548365	0.081545804	6.679953109
F11	1.416004564	33.11608896	1784.94971	200742.6436	42.36573681	45.82788865	13.23259029
F12	5.33282E-29	0.008131471	1.741527818	0.007082975	0.079617562	0.001672578	62.314256
F13	0.003586402	3.341504939	11.59901345	29.24617611	0.005987502	3.660322993	0.307863338
F14	0	32.23710354	169.9386745	32.23710354	47.7585934	55.91931126	126.128554
F15	6.03169E-13	0.791152554	4.413206679	21.46654001	0.403206679	1.439873346	0.299873346

Table 9

Rank of algorithms for all functions with 30 dimension using MAE

Algorithm	MAE	Rank
ICSA	0.0946	1
CSA	1.5099E+02	2
PSO	1.7555E+03	3
FA	1.7645E+03	4
GA	8.7547E+05	5
DA	4.6514E+07	6
BA	2.6779E+09	7

Table 10

Results of Wilcoxon’s test for ICSA against the other six algorithms for 15 functions

ICSA vs.	P-value	α = 0.05
CSA	5.80E-07	+
DA	5.15738E-08	+
BA	9.02542E-08	+
FA	0.000268558	+
PSO	1.25066E-06	+
GA	1.54721E-07	+

Table 11

Comparison of modified algorithms

Function	Algorithm	Best	Mean	Worst	SD
F1	GA	4631588.228	5761532.421	7235807.917	567790.5835
	MGA	1306.265	5.34198E+03	1.22538E+04	3.19199E+03
	FA	800.7938888	3672.857654	9008.248203	2301.733428
	MFA	6.33050E+03	2.60467E+04	6.72997E+04	1.54320E+04
F2	GA	6.727647356	754.6156638	7358.354288	1609.365942
	MGA	0.0062193407	1.47678E+02	1.20191E+03	3.52936E+02
	FA	13537.67164	22704.25684	40029.80303	6208.112837
	MFA	0.272718082	1.110880578	4.435545843	0.793479520
F3	GA	0.018155541	0.218968714	0.546296402	0.155445323
	MGA	3.38021E-20	4.26535E-08	8.13999E-07	1.50497E-07
	FA	3.97299E-23	3.314E-21	7.14931E-20	1.29007E-20
	MFA	1.75632E-06	4.867E-05	2.19503E-04	4.29226E-05
F4	GA	2.751206131	3.44136285	3.706600974	0.176737866
	MGA	6.2723823622	12.4249641	17.12691929	2.544621689
	FA	0.006784197	0.008647631	0.011013161	0.000906503
	MFA	0.154115274	15.36198667	19.96676885	8.489618506

5.3 Convergence rate analysis

To investigate the performance of proposed ICSA at run time for high-dimensional benchmark functions, a graphical comparison analysis is performed by plotting the convergence curves of ICSA and other employed algorithms (Figs. 6 and 7). It is observed that ICSA achieves accurate global optimal results with significantly accelerated convergence rate for all benchmark functions (except F11 and F13) while standard CSA and other algorithms trap in local optimal regions with poor convergence rate. Only ICSA and FA successfully find global optimum for function F7 but ICSA depicts better convergence rate (Fig. 6 (g)). In case of F11 all the algorithms stuck in near optimal region but ICSA minimizes the test function with fast convergence rate. ICSA traps in local minima on function F13 while FA is still exploring the search space and achieves only near global optimal region (Fig. 7 (e)). The analysis of convergence characteristics proves that the proposed technique offers quick convergence and achieves optimal results by continuous exploration of search space whereas standard CSA and other state-of-art algorithms trap in local optimums. The fast convergence of ICSA allows less number of iterations for optimization. Thus overall time taken by ICSA for optimization will be less and increase in computational time is compensated by fast convergence. It is obvious that the improved performance is achieved with slightly increased computational effort.

The improved balance between exploration and exploitation phases results in enhanced performance of the proposed algorithm. During exploration phase population diversity is achieved by unique combination of Lévy flight operator and adaptive flight adjustment operator. This combination redirects the search in the unexplored regions of the search space. Thus new candidate solutions are generated which increased population diversity and premature convergence is prevented. Further experience factor enhances the accuracy by improving the exploitation phase and thus accelerated convergence is achieved. The viability of proposed modifications is also evaluated on other evolutionary methods as discussed in the next section.

Fig.6

The convergence curves of ICSA and other methods (a) F1, (b) F2, (c) F3, (d) F4, (e) F5, (f) F6, (g) F7 and (h) F8.

Fig.7

The convergence curves of ICSA and other methods (a) F9, (b) F10, (c) F11, (d) F12, (e) F13, (f) F14 and (g) F15.

5.4 Experimental test 3

As discussed previously there is no thumb rule to achieve a proper balance between exploration and exploitation while designing or modifying any evolutionary method. The algorithms involving random probability distributions may be modified using Lévy distribution. Further the updating mechanism of parameters in an algorithm may be improved using the concept of experience. In this work, these modifications are attempted on GA and FA and their modified versions are named as MGA and MFA respectively. The mutation operation of GA uses uniform distribution which is replaced by Lévy distribution in MGA. On the other hand, in FA [33] the movement of i^th firefly is determined by following Equation $x_{i} = x_{i} + β_{0} e^{- γ r_{ij}^{2}} (x_{j} - x_{i}) + α ɛ_{i}$ (12) where in simplest form ɛ_i = (rand - 1/2) and rand is a random number considered from uniform distribution. This equation is modified by introducing ef, Lévy flight and adaptive adjustment operator to achieve MFA as follows:

$\begin{matrix} x_{i} & = & ef \times x_{i} + β_{0} e^{- γ r_{ij}^{2}} (x_{j} - x_{i}) \\ + α (L \overset{´}{e} vy (n) - 0.5) + α_{1} (1 - I / I_{m}) \end{matrix}$ (13) where I is current iteration and I_m is maximum iterations. The modified algorithms are compared statistically to their corresponding variants over 30 trial runs and few recorded results are presented in Table 11. It is observed from the results that performance of MGA is slightly better than GA on 10 test cases out of 15. However, significant improvement in results is not obtained using the modification. On contrary, the performance of MFA is degraded on 10 test cases in comparison to FA. Thus it is revealed from experimental test 3 that achieving a proper balance between exploration and exploitation phases is always a challenging problem and hence motivates the researchers to design new optimizers and improve the existing one continuously.

The present work focuses on the improved version of single-objective CSA. However the implementation, testing and viability of multi-objective CSA for high dimensional problems with the proposed modifications is still to be investigated and may be considered as a new research direction. In future ICSA can be extended for solving high dimensional multi-objective optimization problems with Pareto front solution. The improved version can be compared with its standard multi-objective variant as well as other existing multi-objective optimization algorithms.

6 Conclusion

In this paper, a novel metaheuristic search technique, ICSA is proposed for high-dimensional global optimization problems. The global search capability of standard CSA is improved by including Lévy flight. Additionally, the convergence efficiency is enhanced by incorporating experience factor and adaptive adjustment operator in the position updating mechanism of CSA. A test bed of 15 benchmark functions each with 50, 100 and 500 dimensions is used to evaluate the performance of proposed ICSA. The simulation results show that ICSA is better than CSA in terms of stability, search ability and convergence speed. Further ICSA shows superior performance on almost every benchmark function in comparison to other existing optimizers like DA, BA, FA, PSO and GA. Hence it is concluded from the results that ICSA provides better and robust optimization of high-dimensional problems while retaining simplicity and ease of implementation features of CSA. In future the proposed technique will also be tested on more challenging large scale optimization problems provided in CEC2008 and CEC2010.

Footnotes

Appendix A. Benchmark functions

Table 12

Benchmark functions

Function	Range	F (x^*)
$F 1 = x_{1}^{2} + 10^{6} \sum_{i = 2}^{n} x_{i}^{2}$	[-100, 100]	0
$F 2 = 10^{6} x_{1}^{2} + \sum_{i = 2}^{n} x_{i}^{2}$	[-100, 100]	0
$\begin{matrix} F 3 = \sum_{i = 1}^{n} x_{i}^{6} (2 + \sin \frac{1}{x_{i}}) \end{matrix}$	[-1, 1]	0
$\begin{matrix} [t] F 4 = - 20 \exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) \\ - \exp (\frac{1}{n} \sum_{i = 1}^{n} \cos (2 π x_{i})) + 20 + e \end{matrix}$	[-32, 32]	0
$\begin{matrix} [t] F 5 = 1 + \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} \\ - \prod_{i = 1}^{n} \cos (\frac{x_{i}}{\sqrt{i}}) \end{matrix}$	[-600, 600]	0
$F 6 = \sum_{i = 1}^{n} x_{i}^{2}$	[-100, 100]	0
$F 7 = \sum_{i = 1}^{n} (⌊ x_{i} + 0.5 ⌋)^{2}$	[-100, 100]	0
$F 8 = \sum_{i = 1}^{n} \| x_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	[-10, 10]	0
$\begin{matrix} [t] F 9 = \sum_{i = 1}^{n} x_{i}^{2} + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{2} \\ + {(\sum_{i = 1}^{n} 0.5 {ix}_{i})}^{4} \end{matrix}$	[-5, 10]	0
$\begin{matrix} [t] F 10 = \sum_{i = 1}^{n} \| x_{i} \sin (x_{i}) + 0.1 x_{i} \| \end{matrix}$	[-10, 10]	0
$\begin{matrix} [t] F 11 = \sum_{i = 1}^{n - 1} (100 (x_{i}^{2} - x_{i + 1})^{2} \\ + (1 - x_{i})^{2}) \end{matrix}$	[-10, 10]	0
$\begin{matrix} [t] F 12 = - \frac{1}{n} \sum_{i = 1}^{n} \sin^{6} (5 π x_{i}) \end{matrix}$	[-1, 1]	0
$\begin{matrix} [t] F 13 = \sin^{2} (π w_{1}) + \sum_{i = 1}^{n - 1} (w_{i} - 1)^{2} \\ \times [1 + 10 \sin^{2} (π w_{i} + 1)] \\ + (w_{n} - 1)^{2} [1 + \sin^{2} (2 π w_{n})] \end{matrix}$	[-10, 10]	0
$\begin{matrix} [t] F 14 = 10 n + \sum_{i = 1}^{n} (x_{i}^{2} - 10 \cos (2 π x_{i})) \end{matrix}$	[-5.12, 5.12]	0
$\begin{matrix} [t] F 15 = 1 - \cos (2 π \| \| x \| \|) + 0.1 \| \| x \| \|, \\ where \| \| x \| \| = \sqrt{\sum_{i = 1}^{n} x_{i}^{2}} \end{matrix}$	[-100, 100]	0

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