A Novel Meta-Heuristic Optimization Algorithm: Current Search

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1 A Novel Meta-Heuristic Optimization Algorithm: Current Search Anusorn SAKULIN and Deacha PUANGDOWNREONG * Department of Electrical Engineering, Faculty of Engineering, South-East Asia University 9/ Petchakasem Rd., Nongkhaem, Bangkok, THAILAND * corresponding author: deachap@sau.ac.th Abstract: - Inspired by an electric current flowing through electric networks, a novel meta-heuristic optimization algorithm named the Current Search (CS) is proposed in this article. The proposed CS algorithm is an optimization algorithm based on the intelligent behavior of electric current flowing through open and short circuits. To perform its effectiveness and robustness, the proposed CS algorithm is tested against five wellknown benchmark continuous multivariable test functions collected by Ali et al. The results obtained by the proposed CS are compared with those obtained by the popular search techniques widely used to solve optimization problems, i.e., Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Tabu Search (TS). The results show that the proposed CS outperforms other algorithms. The results obtained by the proposed CS are superior within reasonable time consumed. Key-Words: - Current Search, Genetic Algorithm, Particle Swarm Optimization, Tabu Search Introduction Over five decades, many heuristic algorithms have been developed to solve combinatorial and numeric optimization problems []. By literature, several intelligent search techniques, i.e., Evolutionary Programming (EP) [2], Tabu Search [3], Simulated Annealing (SA) [4], Genetic Algorithm (GA) [5], Ant Colony Optimization (ACO) [6], Hit-and-Run (HNR) [7], Hide-and-Seek (HNS) [8], Particle Swarm Optimization (PSO) [9], Harmony Search (HS) [], Bacterial Foraging Optimization (BFO) [], Shuffled Frog Leaping Algorithm (SFLA) [2], Bee Colony Optimization (BCO) [3], Key Cutting Search (KCS) [4], and Hunting Search (HuS) [5] etc., have been proposed. These algorithms can be classified into different groups depending on their nature of criteria being considered, such as population-based (EP, GA, ACO, PSO, BFO, BCO, and HuS), neighborhoodbased (TS), iterative-based (SFLA), stochastic (KCS, HNR, and HNS), and deterministic (SA). Among them, GA, PSO, and TS are the most popular intelligent search techniques that are widely used to solve optimization and engineering problems. In this article, the current search (CS), one of the powerful and efficient meta-heuristic optimization search techniques, is proposed. The CS algorithm is inspired by the electric current flowing through electric circuits. The proposed CS algorithm is coded and tested against five benchmark continuous multi-dimensional test functions collected by Ali et al [6]. Obtained results will be compared with those obtained by GA, PSO, and TS. This article consists of five sections. The CS algorithm is described in section 2. Benchmark continuous multi-dimensional test functions used in this article are given in section 3. Performance evaluation of CS compared with GA, PSO, and TS algorithms against five benchmark multivariable test functions is illustrated in section 4, while conclusion is provided in section 5. 2 Current Search Algorithm Based on the principle of current divider in electric circuit theory [7], the electric current flows through all blanch connected in parallel form as can be seen in Fig.. Each blanch connects to a resistor R having different resistances to obstruct the current. Assume that < R < R2 < L < RN. In fundamentals of circuit theory [7], Kirchhoff s current law (KCL) stats that the algebraic sum of currents entering a node is zero. On the other hand, the sum of the currents entering a node is equal to the sum of the current leaving the node. This means that, in Fig., the sum of all currents in each blanch is equal to the total current supplied by the current source as epressed in (), where, i T is the total current and i is the current in blanch j -th. j N j j= i = i T () ISBN:

2 The behavior of electric current is like a tide that always flow to lower places. The less the resistance of blanch, the more the current flows (see Fig., the thickness of arrows representing the current quantity). Referring to Fig., in case of short circuit, the blanch resistance is zero acted as a conductor, while, in case of open circuit, the blanch resistance is infinity acted as an insulator. The Current Search (CS) algorithm is inspired by this concept. All blanches represent the feasible solutions in search space. The local entrapment is occurred when the current hits the open circuit connection. The optimum solution found is the blanch possessing the optimum resistance. Step 7. If f ( ) < f ( ), keep in set Γ k and set =, set j = and return to Step 5. Otherwise update j = j +. Step 8. If j < jma, return to Step 5. Otherwise keep in set Ξ and update k = k +. Step 9. Terminate the search process when termination criteria are satisfied. The optimum solution found is. Otherwise return to Step 4. The diagram in Fig. 2 reveals the search process of the proposed CS algorithm. Start i T i i 2 R R 2 i i 2 blanch node Initialize: - search space - k = j =, jma = - N = n =, =. - Ø i 3 R 3 i 3 Uniformly Random set of initial solutions X i, i=,,n within i N R N i N Evaluate f(x i ) and rank X i leading f(x )<f(x 2 )< <f(x N ), the store ranked X i into < R < R 2 < L < current source R N i T Let = X k be initial solution Uniformly Random set of neighborhood member i, i=,,n around within radius Fig. The behavior of electric current. The CS algorithm is described step-by-step as follows. Step. Initialize the search space Ω, iteration counter k = j =, maimum allowance of solution cycling j ma, number of initial solutions (feasible directions of currents in network) N, number of neighborhood members n, search radius ρ, and set Ψ = Γ = Ξ =. Step 2. Uniformly random initial solution X i, i =, K, N within Ω. Step 3. Evaluate the objective function f ( X i ) of X. Rank X i, i =, K, N that gives f ( X) < L < f ( X N ), then store ranked X i into Ψ. Step 4. Let = X k as selected initial solution. Step 5. Uniformly random neighborhood i, i =, K,n around within radius ρ. Step 6. Evaluate the objective function f ( i ) of. A solution giving the minimum objective function is set as. Evaluate f( i ) and let be an elite solution among i making f(.) minimum no f( )<f( ) no Update j = j+ j<jma no Store into, and update k = k+ TC met? yes Report the optimal solution Stop yes yes Store into k, then set = and set j = Fig. 2 The diagram of the proposed CS algorithm. ISBN:

3 In Step, the search space Ω is performed as the feasible boundary where the electric current can flow. The maimum allowance of solution cycling j ma implies the local entrapment occurred in the selected direction. The number of initial solutions N is set as feasible directions of the electric currents in network. The number of neighborhood members n is provided as the sub-directions of the electric currents in the selected direction, and the search radius ρ is given as the sub-search space where the electric current can flow in the selected direction. In Step 2-3, the uniformly random approach is conducted to perform the feasible directions of the electric currents. These directions will be ranked by the objective function to arrange the signification of directions from most to least. In Step 4-7, once the most significant direction of the current is selected, the search process will consecutively find the optimum solution along the most significant direction within the sub-search space where the electric current can flow in the selected direction. Each feasible solution will be evaluated via the objective function until the optimum solution is found. In Step 8-9, the local entrapment in the selected direction will be identified via the maimum allowance of solution cycling. If occurred, the second, the third, and so on, of the significant direction ranked in Step 2-3 will consecutively employed, until optimum solution will be found or the termination criteria will be met. 3 Benchmark Functions In this section, five well-known benchmark continuous multivariable test functions collected by Ali et al., [6] are described as follows. (i) Bohachevsky function (BF) is epressed as (2). The global minimum is located at = (, ) 6 with f ( ) =. Let f ma = be the maimum allowance of the global solution found. The Bohachevsky surface is depicted in Fig = + cos(2π ) cos(2π 2 ) + 2 subject to 5, 5 (3) (iii) Shekel s Fo-Holes function (SF) is epressed as (4). It is the fifth function of De Jong s test suite. The global minimum is located at = ( 32, 32) with f ( ) =. Let f ma =. 999 be the maimum allowance of the global solution found. The Shekel s Fo-Holes surface is depicted in Fig = j= 6 j + ( ) i aij i= subject to 5, where a ij = L L (4) (iv) Schwefel function (SchF) is epressed as (5). The global minimum is located at = ( , ) with f ( ) =. Let f ma = 6 be the maimum allowance of the global solution found. The Schwefel surface is depicted in Fig. 6. n = n i= subject to 5, 5 ( i sin i ), n = 2 (5) (v) Shubert function (ShuF) is epressed as (6). The global minima are located at 8 different locations with f ( ) = Let f ma = be the maimum allowance of the global solution found. The Shubert surface is depicted in Fig. 7. f ( ) = cos[ ( + ) + ] cos[ ( + ) 2 + ] i i i i i i (6) min subject to 5 i=, 5 i= 2 2 = + 2.3cos(3π ).4 cos(4π 2 ) +.7 subject to, (2) (ii) Rastrigin function (RF) is epressed as (3). The global minimum is located at = (, ) with 6 f ( ) =. Let f ma = be the maimum allowance of the global solution found. The RF s surface is depicted in Fig. 4. Fig. 3 Bohachevsky surface. ISBN:

4 Fig. 4 Rastrigin surface. Fig. 5 Shekel s Fo-Holes surface. Fig. 6 Schwefel surface. Fig. 7 Shubert surface. 4 Performance Evaluation 4. CS performance tests In this section, the CS is tested against five benchmark continuous multivariable test functions illustrated in section 3 to perform its effectiveness and robustness. The CS algorithm was coded by MATLAB running on Intel Core2 Duo 2. GHz 3 Gbytes DDR-RAM computer. The CS parameters are reasonable preset for each test function as summarized in Table, where N is number of initial solutions, n is number of neighborhood members, and ρ is search radius. Maimum search iteration = and f fma are set as termination criteria. The tests were conducted trial runs against test functions to obtain percentage of success of global minimum found (%Success). Table 2 summarizes sets of parameter values to achieve %Success of each test function obtained over trial runs. Referring to Table 2, global minima of test functions can be found with %Success. It can be noticed that the proposed CS algorithm is efficient and robust according to given search parameter values. Some movements and convergent rates of the cost function obtained by the CS over the Bohachevsky surface are depicted in Fig. 8, as an eample. The convergence curves of other test functions are omitted because they have a similar form to that of the Bohachevsky shown in Fig. 8. Results in Table 2 provide the recommendations for users to set the search parameters of the CS appropriately. However, the proposed CS is still problem-dependent, understanding the problem and selecting an appropriate parameter are essential for successful applications. Table CS parameter values for test functions. Parameters BF N, 2, 3, 4, 5, 6, 7 n, 5,, 5, 2, 25, 3 ρ.5,.75,.,.5,.2,.25,.3 Parameters RF N, 5, 2, 25, 3, 35, 4 n 5,, 5, 2, 25, 3, 35 ρ.,.25,.5,.75,.,.25,.5 Parameters SF N 5, 6, 7, 8, 9,, n 2, 3, 4, 5, 6, 7, 8 ρ.7,.9,.,.3,.5,.7,.9 Parameters SchF N, 2, 3, 4, 5, 6, 7 n, 2, 3, 4, 5, 6, 7 ρ.5,.,.25,.5,.75, 2., 2.5 Parameters ShuF N 5, 75,, 25, 5, 75, 2 n 2, 4, 6, 8,, 2, 4 ρ.,.2,.3,.4,.5,.6,.7 Table 2 Results of CS performance tests. Entry N n ρ %Success BF RF SF SchF ShuF ISBN:

5 members, ρ is search radius, and uniform random search mechanism is conducted..5 v( t + ) = ω v( + φrand(,) { pbest ( ( } (7) + φ rand(,) { g ( ( } 2 best ( t + ) = ( + v( t + ) (8) Convergent rate of objective function (a) Movements of the CS Iterations (b) Convergences of cost function. Fig. 8 Some results of the CS over BF. 4.2 Performance comparison To compare the proposed CS algorithm with GA, PSO, and TS, The CS and other algorithms were coded by MATLAB running on Intel Core2 Duo 2. GHz 3 Gbytes DDR-RAM computer. In GA, n is number of population, single point uniform crossover with the rate of.95, random selection mechanism, gaussian mutation with the rate of.. In PSO, n is number of swarm. The velocity vector v and the solution are epressed in (7) and (8), respectively, where ω is the additional inertia weight, which varies from.9 to.7 linearly with the iteration. The learning factors φ and φ 2 are set to be 2. The upper and lower bounds for v, ( v min, vma ) = ( min, ma ) are set. In TS, n is number of neighborhood members, ρ is search radius, and uniform random search mechanism is used. The aspiration criterion (backtracking mechanism) is used to escape the local entrapments. In CS, n is number of neighborhood For a fair comparison, parameter values of each algorithm are set as summarized in Table 3. Maimum search iteration = and f fma are set as termination criteria. Entry BF RF SF SchF ShuF Table 3 Parameter values of algorithms. Parameters GA PSO TS CS N N 2,5 2,5 2,5 5 ρ N n,,, 25 ρ N n 3,2 3,2 3,2 4 ρ N n 35, 35, 35, 7 ρ N n,,, 8 ρ The performance comparison tests of candidate algorithms were conducted trial runs to obtain the average cost function found, minimum solutions found, average iteration (generation or search round) used, and average search time consumed. Results are summarized in Table 4 7, respectively. It was found that the proposed CS outperforms other algorithms. Table 4 Cost function. Entry Cost GA PSO TS CS Min BF RF SF SchF ShuF Ma Ave Std Min Ma Ave Std Min Ma Ave Std Min Ma Ave Std Min Ma Ave Std ISBN:

6 Table 5 Minimum solutions. Entry GA PSO TS CS BF RF SF SchF ShuF Table 6 Average iteration. Entry GA PSO TS CS BF RF SF SchF ShuF Table 7 Average search time (sec.). Entry GA PSO TS CS BF RF, SF SchF 9, ShuF Conclusion In this article, a novel meta-heuristic optimization algorithm named the Current Search (CS) has been proposed. It is inspired by an electric current flowing through electric networks. Its effectiveness and robustness have been performed against wellknown benchmark test functions. Results obtained by the CS are compared with those obtained by GA, PSO, and TS. As results, it could be concluded that the proposed CS superior to other algorithms. Moreover, the recommendations of the CS parameter setting are appropriately given. For future trends, the CS is still needed to solve more comple and real-world problems both discrete and continuous such as engineering problems. References: [] D.T. Pham and D. Karaboga, Intelligent Optimisation Techniques, Springer, London, 2. [2] L.J. Fogel, A.J. Owens, and M.J. Walsh, Artificial Intelligence through Simulated Evolution, John Wiley, 966. [3] F. Glover and M. Laguna, Tabu Search, Kluwer Academic Publishers, 997. [4] S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, Optimization by Simulated Annealing. Science, Vol. 22, No. 4598, 983, pp [5] D.E. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning. Addison Wesley Publishers, 989. [6] M. Dorigo, Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italie, 992. [7] Z.B. Zabinsky, D.L. Graesser, M.E. Tuttle, and G.I. Kim, Global optimization of composite laminates using improving hit and run, In: Floudas C. and Pardalos P. (eds.), pp , Recent Advances in Global Optimization, Princeton University Press, 992. [8] H.E. Romeijn and R.L. Smith, Simulated annealing for constrained global optimization. Journal of Global Optimization, Vol. 5, 994, pp. 26. [9] J. Kennedy and R. Eberhart, Particle Swarm Optimization, Proceedings of IEEE International Conference on Neural Networks, Vol. 4, 995, pp [] Z.W. Geem, J.H. Kim, and G.V. Loganathan, A New Heuristic Optimization Algorithm: Harmony Search, Simulation, Vol. 76, No. 2, 2, pp [] K.M. Passino, Biomimicry of Bacterial Foraging for Distributed Optimization and Control, IEEE Control System Magazine, Vol. 22, 22, pp [2] M.M. Eusuff and K.E. Lansey, Optimization of Water Distribution Network Design using the Shuffled Frog Leaping Algorithm, Journal of Water Resource Planning and Management, Vol. 29, No. 3, 23, pp [3] D.T. Pham, A. Ghanbarzadeh, E. Koç, S. Otri, S. Rahim and M. Zaidi, The Bees Algorithm A Novel Tool for Comple Optimisation Problems, Proceedings of IPROMS 26 Conference, 26, pp [4] J. Qin, A New Optimization Algorithm and Its Application Key Cutting Algorithm, Grey Systems and Intelligent Services, 29, pp [5] R. Oftadeh, M.J. Mahjoob, and M. Shariatpanahi, A Novel Mata-Heuristic Optimization Algorithm Inspired by Group Hunting of Animals: Hunting Search, Computers and Mathematics with Applications, Vol. 6, 2, pp [6] M.M. Ali, C. Khompatraporn, and Z.B. Zabinsky, A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems, Journal of Global Optimization, Vol. 3, 25, pp [7] C.K. Aleander and M.N.O. Sadiku, Fundamentals of Electric Circuits, McGraw-Hill, 24. ISBN: