2017 8th International Conference on Information Technology (ICIT) Hybridizing cuckoo search algorithm with hill climbing for numerical optimization problems Mohammad Shehab1 Ahamad Tajudin Khader2 School of Computer Sciences Universiti Sains Malaysia Penang, Malaysia moh.shehab12@gmail.com School of Computer Sciences Universiti Sains Malaysia Penang, Malaysia Mohammed Azmi Al-Betar3 Department of Information Technology, Al-Huson University College, Al-Balqa Applied University, P.O. Box 50, Al-Huson, Irbid, Jordan Laith Mohammad Abualigah4 Abstract— The Cuckoo Search Algorithm (CSA) is a promising metaheuristic algorithm. It is applied to solve many problems in different fields. This paper proposes a new cuckoo search algorithm by combining the cuckoo search algorithm with the Hill Climbing method for solving the integer and minimax optimization problems. The proposed algorithm is named hybrid cuckoo search and hill climbing (CSAHC). CSAHC starts the search by applying the standard cuckoo search for the number of iterations then the best-obtained solution is passed to the hill climbing algorithm as an intensification process to accelerate the search and overcome the slow convergence of the standard cuckoo search algorithm. The proposed algorithm balances between the global exploration of the cuckoo search algorithm and the deep exploitation of the hill climbing method. The validation of the performance is determined by applying 13 benchmarks. The results of experimental simulations indicated that the CSAHC performs better than standard CSA. Key words: Metaheuristic algorithms, Cuckoo search algorithm, Hill climbing, exploration and exploitation. I. INTRODUCTION Optimization resides in many domains, such as engineering, energy, economics, medical, and computer science [1]. It is mainly concerned with finding the optimal values for several decision variables to form a solution to problem optimization. This solution is optimally considered when the decision maker is satisfied with it [2]. An optimization problem is the minimization or maximization of a suitable decision-making algorithm normally adapted to the approximation methods. The principle of decision making entails choosing between several alternatives. The result of this choice is the selection of the best decision from all choices. School of Computer Sciences Universiti Sains Malaysia Penang, Malaysia Optimization algorithms developed based on natureinspired ideas deal with selecting the best alternative in the sense of the given objective function. The optimization algorithm can be either a heuristic or a metaheuristic approach. Heuristic approaches are problem-designed approaches where each optimization problem has its own heuristic methods that are not applicable for other kinds of optimization problems. The metaheuristic-based algorithm is also a general solver template that can be adapted for various kinds of optimization problems by properly tweaking its operators and configuring its parameters. To elaborate, each optimization algorithm can be categorized into three classes: evolutionary algorithms (EAs), swarm-based algorithms, and trajectory-based algorithms. Examples of EAs include genetic algorithms (GAs) [3], genetic programming (GP) [4], and differential evolution (DE) [5]. Examples of swarm-based algorithms include artificial bee colony (ABC) [6], particle swarm optimization (PSO)[7], and cuckoo search algorithm (CSA) [8]. Examples of trajectorybased algorithms includes tabu search (TS) [9], simulated annealing (SA) [10], β-hill climbing [11]. The performance of the population-based algorithms is measured through checking its ability to establish a proper trade-off between exploration and exploitation. Where, the algorithm which has a weak balance between exploration and exploitation be more likely to the trapping in local optima, premature convergence, and stagnation [12]. Population-based search algorithm is normally very powerful in exploring several regions of the problem search space. However, it has difficulty in determining the local optima in each region. By contrast, deep searching of the local searchbased algorithm is very efficient in a single search space region 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 36 2017 8th International Conference on Information Technology (ICIT) but not for several search space regions [13]. Thus, sometimes, it is very beneficial to hybridize a local and a population search-based method to complement their advantages in a single optimization framework. Based on the above suggestion and through hybridization, the search can strike a balance between the wide range of exploration and nearby exploitation of the problem search space. In this context, CSA has been hybridized with other population-based algorithms to improve its performance in tackling complex optimization problems [14]. Quadratic assignment problems (QAPs) are considered to be NP-hard problems, which cannot be easily solved by exact methods. Therefore, Dejam et al. [15] proposed a hybrid algorithm combined with the CSA of TS (i.e., CSA-TS) to solve QAPs. In their research, the QAPs were initially tackled using CSA. Thereafter, these were combined with TS, which focused on the local search to increase the optimization precision. The experimental results indicated that the proposed algorithm performs better than ABC and GA. Ali and Tawhid [16] proposed a hybrid algorithm, which includes the CSA and Nelder–Mead method (HCSANM), to solve the integer and minimax optimization problems. HCSANM started with a basic CSA for a number of iterations, chose the best results achieved, and then passed it to Nelder– Mead, similar to intensification for increasing the search and eliminating the slow convergence struggle of the basic CSA. Therefore, HCSANM controls the balance using the basic CSA for global exploration, and NM method is utilized for deep exploitation [17]. The validation of the performance was determined by applying the HCSANM on 7 integer programming and 10 minimax problems. The results showed the efficiency of HCSANM and its ability to deal with minimax optimization problems in a reasonable time. The linear least squares problem solved by hybridization algorithm between Newton method (NM) and CSA is called CSANM [18]. The authors benefited from CSA for fast convergence and global search as well as from NM for the ability of strong local search. The experimental results showed the convergence efficiency and computational accuracy of the CSANM in comparison with the basic CSA and HS based on NM (HSNM). In this research, a new hybrid optimization approach is developed by hybridizing the cuckoo search algorithm with hill climbing to solve optimization problems. The proposed approach is evaluated on thirteen benchmark functions. Experimental results demonstrate that the proposed algorithm performs better than the basic cuckoo search algorithm. The structure of this paper is organized as follows. Sections II and III describes the CSA and HC in brief, respectively. The CSAHC approach is presented in Section IV. Subsequently, our method is evaluated through fourteen benchmarks in Section V by comparing with the original CSA. Finally, the conclusion and future works are given in Section VI. II. CUCKOO SEARCH ALGORITHM The use of CSA in the optimization context was proposed by Yang and Deb in 2009 [8]. To date, work on this algorithm has significantly increased, and the CSA has succeeded in having its rightful place among other optimization methodologies [19]. This algorithm is based on the obligate brood parasitic behaviour found in some cuckoo species, in combination with the Levy flight behaviour discovered in some birds and fruit flies. The CSA is an efficient metaheuristic swarm based algorithm that efficiently strikes a balance between local nearby exploitation and global-wide exploration in the search space problem [20]. The cuckoo has a specific way of laying its eggs to distinguish it from the rest of the birds [21]. The following three idealized rules clarify and describe the standard cuckoo search: x Each cuckoo lays one egg at a time and dumps it in a randomly chosen nest. x The best nests with high-quality eggs will be carried over to the next generations. x The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability PD 0,1 . In this case, the host bird can either get rid of the egg or simply abandon the nest and build a completely new nest. In addition, probability PD can be used by the n host nest to replace the new nests. Figure 1 shows the representation of the CSA search process. Similar to other swarm-based algorithms, the CSA starts with an initial population of n host nests. These initial host nests will be randomly attracted by the cuckoos with eggs and also by random Levy flights to lay the eggs. Thereafter, nest quality will be evaluated and compared with another random host nest. In case the host nest is better, it will replace the old host nests. This new solution has the egg laid by a cuckoo. If the host bird discovers the egg with a probability PD 0,1 , the host either throws out the eggs, or abandons it and builds a new nest. This step is done by replacing the abundant solutions with the new random solutions. 1: Objective function f(X), X= (x1, x2, ..., xd)T 2: Generate initial population of n host nests Xi (i=1, 2, …, n) 3: While t Max_iterations do 4: Get a cuckoo randomly by Levy flights 5: Evaluate its quality/ fitness Fi 6: Choose a nest among n (say, j) randomly 7: If Fi ! Fj then 8: replace j by the new solution; 9: End If 10: A fraction (Pa) of worse nests are abandoned and new ones are built; 11: Keep the best solutions 12: Rank the solutions and find the current best 13: End While 14: Postprocess results and visualization 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 37 2017 8th International Conference on Information Technology (ICIT) Fig. 1. Algorithm 1, Pseudo code of the cuckoo search algorithm Yang and Deb used a certain and simple representation for the implementation, with each egg representing a solution. As the cuckoo lays only one egg, it also represents one solution. The purpose is to increase the diversity of new, and probably better, cuckoos (solutions) and replace them instead with the worst solutions. By contrast, the CSA can be more complicated by using multiple eggs in each nest to represent a set of solutions. The CSA, as a bat algorithm [22] and an FA [23], uses a balance between exploration and exploitation. The CSA is equiponderant to the integration of a local random walk. The local random walk can be written as xit 1 Where x t j xit Ds H ( PD H ) ( xtj xkt ) and x t k (1) are two different solutions selected randomly by random permutation, H (u) is a Heaviside function, generates a random number drawn from a uniform distribution between 0 and 1 and s is the step size. Furthermore, global explorative random walk by using Levy flights can be expressed as follows: xit 1 And DL ( s , O ) xit DL( s, O ), (2) O*(O ) sin( SO / 2) 1 , s !! s 0 ! 0 S s 1O (3) Where L is the characteristic scale of the problem of interest, and D > 0 is a factor of size scaling. The value can be calculated by D O(L / 10) ; for more affectivity, D O(L / 100) can be used. In addition, xit in the above equation represents the current location, which is the only way t 1 to determine the next location x i . This is called the random walk and the Markov chain. In the second term, DL( s, O ) represents the transition probability. However, to avoid premature convergence and increase diversity (not only confined in a local optimum), the new solutions should be generated on a remote sufficient distance from the current best solution and some random elements should be included. possible. Then, loop until a solution is found or until there are no new operators left to be applied in the current state. Also, inside the loop there are two steps. The first step, select an operator that has not yet been applied to the current state and apply it to produce the new state. The second step, evaluate the new state. Algorithm 2 shows the pseudo-code of the HC algorithm, which proves the simplicity of hill climbing. Based on the above, in HC the basic idea is to always head towards a state which is better than the current one. So, it always improves the quality of a solution [25]. 1: i = initial solution 2: While f(s) d f (i) for all s Neighbours (i) do 3: Generates an s Neighbours (i); 4: If fitness (s) ! fitness (i) then 5: Replace s with the i; 6: End If 7: End While Fig. 2. Algorithm 2, Pseudo code of the hill climbing algorithm IV. THE PROPOSED METHODOLOGY: CSA-HILL CLIMBING Based on the introduction of CSA and HC in the previous sections, this section provides a detailed description of the proposed cuckoo search algorithm with hill climbing (CSAHC). CSA is inspired from the “brood parasitic” cuckoo species in combination with certain birds and fruit flies [6]. Levy flight has been applied for global exploration and proved its efficiency through achieving good results [26][27]. This resulted in CSA being balanced between exploration and exploitation [28][29]. However, sometimes it exploits solutions poorly with slow convergence. For that reason, the proposed algorithm balances between the global exploration of the CSA and the deep exploitation of the HC. CSAHC starts the search by applying the standard cuckoo search for the number of iterations. The best-obtained solution is then passed to the HC to accelerate the search and overcome the slow convergence of the standard cuckoo search algorithm. As shown in Figure 3, the HC is an iterative algorithm that starts with an arbitrary solution to a problem and subsequently attempts to determine a better solution by incrementally changing a single element of the solution. When the change produces a better solution, incremental change is performed on the new solution, which is repeated until no further improvements can be found. It then returns the solution to the CSA to check it through the fraction probability PD . III. HILL CLIMBING Hill Climbing (HC) is a mathematical optimization technique which belongs to the family of local search [24]. It searches for a better solution in the neighborhood through evaluating the current state. If it is also goal state, then return to it and quit. Otherwise, continue updating the current state, if V. THE EXPERIMENTAL RESULTS ANALYSIS In this section, the proposed CSAHC was tested through an array of experiments. For testing purposes, we implemented the original version of CSA. We compared results of CSAHC 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 38 2017 8th International Conference on Information Technology (ICIT) with the original results. This comparison is shown in the tables within this section. A. Benchmark Functions To test the performance of a CSAHC, 13 well-known benchmark functions are used for comparison. Table I describes these benchmark functions in terms of the optimum solution after a predefined number of iterations and the rate of convergence to the optimum solution. Further information about all the benchmark functions can be found in [30][31][32]. A. Experimental results and algorithms settings The two parameters of the CSA are the number of host nests (population size n) and the probability of discovery PD . In this section, the different settings have been used such as n (5, 10, 15, 20, 50, 100, 150, 250, 500) and for PD (0, 0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.4, 0.5). Fig. 3. Flowchart of the CSAHC Algorithm The simulations illustrated the best value for n=25 and PD =0.25 which used for most optimisation problems leaving no need to perform parameter tuning. The tests have been run on 10, 25, 50, and 100 dimensions for a maximum of 100000 function evaluations. All tests have been run 100 times. Table II lists the parameter settings. Mean, standard deviation and the best results for basic CSA and hybrid CSAHC are shown in Table III. TABLE I. BENCHMARK FUNCTIONS 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 39 2017 8th International Conference on Information Technology (ICIT) TABLE II. PARAMETER SETTING Parameter Runtime Max Cycle n D Value generation until the twenty-seventh generation) showed that the CSA achieved better solutions than CSAHC. 100 100000 25 5/10/50/100 PD 0.25 All the experiments are conducted using a computer with processor Intel(R) Core (TM) i7-6700K CPU 4.00 GHz with 16 GB of RAM and 64-bit for Microsoft Windows 10 Pro. The source code is implemented using MATLAB (R2015a). Experimental results with 5, 10, 50 and 100 dimensions are shown in Table III. The table is divided into four parts based on dimension. Each part has two columns with results for side-by-side comparison. In the first column, we show results of basic CSA, and in the second, results of the hybrid CSAHC. As shown in Table III, the CSAHC achieved the better solutions compared with basic CSA in all 13 benchmark functions. The results illustrated the preference of CSAHC especially in tests with five dimensions. Regarding the other dimensions, the difference between the results was stable based on the results of five dimensions. Fig. 4. Performance comparison for the F1 Ackley func. Fig. 5. Performance comparison for the F2 Griewank func. The most representative convergent curves are illustrated in Figs. 4 to 13. The values in the figures are the mean function optimum, which are the true values. Figure 4 exhibits that CSBA is more capable of finding better solutions compared with the CSA. While, both Figures 5 and 7 illustrated that CSAHC and CSA are very close in the initial analysis, but after that, the CSAHC shows clear superiority until the end. Fig. 6. Performance comparison for the F3 Penalty 1 func. The results in Figures 6, 10 and 11 show that the CSA initially has better solutions compared with CSAHC. However, in Figure 6, CSAHC achieved better solutions after the seventh generation. Nevertheless, the results of CSAHC and CS are very close until the end, with a preference for CSAHC. Figures 10 and 11 show that the CSAHC started to investigate better solutions after the fourth generation. Figures 8 and 12 recorded similar results with the results for both CSAHC and CSA being very close from the beginning through to the end. Figure 9 is divided into three parts. The initial and end parts show that CSAHC has better results compared with CSA. While the middle part (from twelfth Fig. 7. Performance comparison for the the F4 Penalty 1 func. 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 40 2017 8th International Conference on Information Technology (ICIT) Fig. 8. Performance comparison for the F6 Rastrigin func. Fig. 11. Performance comparison for the the F10 Schwefel 2.22 func. Fig. 12. Performance comparison for the the F12 Sphere func. Fig. 9. Performance comparison for the F7 Rosenbrock func. VI. CONCLUSION AND FUTURE WORK Fig. 10. Performance comparison for the F9 Schwefel 1.2 func. In the present work, a novel metaheuristic CSAHC method is proposed for solving global optimisation tasks. We improved the CSA by combining it with HC and evaluated the performance of the both CSAHC and CSA on the 13 benchmark functions. The hybridization enhanced the exploration of basic CSA by using the HC which is capable of dealing with local searches. The experimental results present that the CSAHC achieved better results compared with the basic CSA. Future work will investigate the hybrid CSA performance in other benchmarks and real-life problems. REFERENCES [1] [2] [3] [4] S. M. Z. Mohammed, A. T. Khader, and M. A. AlBetar, “3-SAT Using Island-based Genetic Algorithm,” IEEJ Trans. Electron. Inf. Syst., vol. 136, no. 12, pp. 1694–1698, 2016. A. L. BOLAJI, A. T. Khader, M. A. Al-Betar, and M. A. Awadallah, “Artificial bee colony algorithm, its variants and applications: A survey.,” J. Theor. Appl. Inf. Technol., vol. 47, no. 2, 2013. J. H. Holland, Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. U Michigan Press, 1975. J. R. Koza, “Genetic programming II: Automatic [5] [6] [7] discovery of reusable subprograms,” Cambridge, MA, USA, 1994. R. Storn and K. V Price, “Minimizing the Real Functions of the ICEC’96 Contest by Differential Evolution.,” in International Conference on Evolutionary Computation, 1996, pp. 842–844. D. Karaboga, “An idea based on honey bee swarm for numerical optimization,” 2005. K. James and E. Russell, “Particle swarm optimization,” in Proceedings of 1995 IEEE International Conference on Neural Networks, 978-1-5090-6332-1/17/$31.00 ©2017 IEEE 41 2017 8th International Conference on Information Technology (ICIT) TABLE III. MEAN, STANDARD DEVIATION AND BEST OPTIMA FOR CSA AND CSAHC WITH 5, 10, 50 AND 100 DIMENSIONS D=5 Function Ackley Griewank Mean Std. Best Mean Std. Best Penalty #1 Mean Std. Best Penalty #2 Mean Std. Best Quartic with noise Mean Std. Best Rastrigin Mean Std. Best Rosenbroc k Mean Std. Best Schwefel 2.26 Mean Std. Best Schwefel 1.2 Mean Std. Best D=10 D=50 D=100 CSA CSAHC CSA CSAHC CSA CSAHC CSA CSAHC 6.63E-13 5.08E-21 4.61E-14 5.08E-23 1.25E-06 4.75E-10 0.01 2.35E-03 1.48E-12 2.14E-19 1.02E-12 2.14E-23 5.68E-06 1.06E-07 0.63 5.24E-02 8.88E-16 1.18E-27 2.84E-18 2.51E-29 5.18E-08 2.98E-13 1.91E-02 4.54E-05 2.52E-14 5.20E-18 2.34E-14 2.48E-19 2.22E-04 5.90E-09 2.14 7.42E-02 5.64E-14 7.37E-18 5.22E-14 9.50E-18 4.97E-04 8.25E-09 2.96 9.59E-02 4.27E-15 6.57E-21 5.25E-16 7.25E-23 8.47E-06 2.91E-14 7.39E-02 9.99E-07 3.88 5.65 8.65E-04 1.45E-04 7.21E-03 7.14E-08 1.03 9.05 3.56E-04 2.17E-05 4.37E-05 1.02E-08 3.38 7.56 4.85E-02 2.70E-02 0.28 7.42E-05 6.87 7.50 2.99 5.01E-02 1.14 2.05E-02 1.65E-30 4.01E-38 1.61E-30 2.21E-39 1.60E-18 3.57E-22 3.26E-08 3.46E-11 3.70E-30 8.25E-36 3.60E-30 7.87E-39 3.58E-18 9.60E-22 7.28E-08 7.74E-10 1.35E-32 2.19E-42 1.05E-32 3.25E-44 1.35E-20 5.62E-27 2.09E-09 1.34E-13 2.761 8.67E-03 1.14E-02 5.86E-03 16.58 9.93E-03 414.98 33.68 7.479 1.85E-02 8.55E-02 7.58E-03 37.11 2.69E-02 927.70 42.30 8.14E-02 9.22 E-05 7.86E-04 2.27E-05 14.81 3.05E-03 19.28 5.61E-02 3.44E-12 3.68E-16 2.88E-14 4.29E-18 7.85E-03 8.24E-08 3.21 1.68E-02 6.07E-12 5.99E-16 6.43E-12 2.27E-16 1.75E-02 3.51E-06 4.08 7.02E-02 1.65E-14 4.41E-19 8.21E-17 1.87E-20 3.97E-05 1.02E-10 0.01 2.23E-08 1.13E-28 8.29E-33 1.90E-28 1.51E-34 1.82E-14 2.22E-15 2.78E-04 4.62E-09 2.52E-28 5.36E-31 4.26E-28 7.94E-34 4.08E-14 5.64E-15 6.10E-02 7.12E-09 8.17E-30 2.01E-40 2.87E-32 6.79E-41 5.87E-15 3.92E-19 6.42E-05 3.93E-12 1.038 7.25 E-03 2.40E-02 7.19E-04 0.14 1.49E-02 1.02 8.56 E-02 5.84 2.22E-02 0.49 1.60E-03 5.84 3.27E-02 3.03 0.35 9.75E-07 6.39E-13 2.76E-09 1.15E-012 1.15E-02 8.91E-07 0.74 1.96E-04 8.84E-310 8.65E-335 4.13E-316 4.77E-339 8.80E-36 9.88E-58 3.61E-02 1.15E-22 5.63E-309 7.82E-333 2.58E-314 9.07E-339 1.78E-35 2.78E-56 8.75E-02 4.55E-22 8.37E-321 3.55E-346 9.38E-322 1.20E-347 2.46E-41 7.35E-73 2.91E-07 1.83E-28 1.08E-315 2.58E-330 4.11E-318 2.65E-331 1.77E-28 9.93E-55 5.24E-08 6.43E-25 2.01E-313 9.15E-330 3.21E-315 4.99E-330 5.90E-27 7.59E-54 1.07E-06 9.29E-23 1.36E-321 3.66E-339 6.91E-322 4.76E-344 1.84E-33 5.79E-76 3.86E-10 4.21E-28 6.28E-315 5.28E-322 1.19E-318 1.28E-328 1.33E-33 2.30E-51 2.28E-10 9.88E-24 3.47E-312 6.77E-319 6.39E-316 7.94E-328 1.38E-30 9.84E-51 8.86E-08 3.74E-22 1.54E-321 8.62E-329 7.36E-322 1.49E-430 7.11E-38 9.24E-71 8.44E-13 8.39E-25 Schwefel 2.22 Mean Std. 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