A Modified Immune Network Optimization Algorithm

Transcription

1 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 A Moifie Immune Network Optimization Algorithm Lu Hong, Joarer Kamruzzaman Abstract This stuy proposes a moifie artificial immune network algorithm for function optimization problems base on iiotypic immune network theory. A hyper-cubic mutation operator was introuce to reuce the heavy computational cost of the traitional opt-ainet algorithm. Moreover, the new symmetrical mutation can effectively improve local search. To maintain population iversity, we also evise an immune selection mechanism base on ensity an fitness. The global convergence of the algorithm was euce through the metho of pure probability an iterative formula. Simulation results of benchmark function optimization show that the moifie algorithm converges more effectively than other immune network algorithms. Inex Terms Artificial immune algorithm, Iiotypic immune network, Hyper-cubic mutation, Convergence I. INTRODUCTION IOLOGICAL immune systems has the ability of B learning, memory an recognition antigen, an the characteristics of aaptive, istribute an iversity. Inspire by biological immune mechanism, a variety of artificial immune algorithms have been evelope to solve problems in machine learning, fault iagnosis, system moeling, computer security, an other fiels of engineering [] [2]. Artificial immune algorithm is an optimization search algorithm that is bionic an intelligent. It imitates biological immunology an the mechanism of gene evolution. At present, propose immune algorithms mainly inclue clonal selection algorithm, negative selection algorithm, an immune network algorithm [3] [4]. The immune network algorithm is base on immune network theory an is generally represente by the opt-ainet algorithm, which was propose by De Castro in 22 [5]. This algorithm was inspire by iiotypic network theory an can escribe some of the ynamic characteristics of the immune system. Moreover, it possesses several unique features, incluing ynamically varie population sizes, Manuscript receive Oct 8, 23; revise Sep 26, 24. This work was supporte by the priority acaemic program evelopment of Jiangsu higher eucation institutions. Lu Hong is with the School of Electronic Engineering, Huaihai Institute of Technology, Jiangsu Province, 2225, China ( honglu92@ 63.com). Joarer Kamruzzaman is with the Gippslan School of Information Technology, Monash University, Northways Roa, Churchill, Victoria 3842, Australia ( Joarer.Kamruzzaman@monash.eu). local an global search, an the capability to maintain any number of optima. Thus, this algorithm is among the most wiely applie artificial immune optimization methos in problems of pattern recognition an multimoal optimization. However, the immune network algorithm is limite with respect to large-scale complicate optimization problems, incluing aitional parameters, heavy computational cost, sensitivity to population size, an slow convergence rate. Furthermore, the algorithm oes not fully reflect the regulation mechanism of immune networks, an it is rarely analyze in terms of mathematical theory [6 7]. In the current stuy, we present a moifie immune network algorithm (MINA) base on opt-ainet. This algorithm evaluates objective functions without negating the valuable characteristics of the original algorithm. We also introuce a hyper-cubic cloning mutation operator an a specific immune selection mechanism. The globle convergence of the algorithm is euce by aopting the pure probability an iterative formula metho. MINA is then compare with other immune network algorithms to verify its effectiveness. The paper is structure as follows. The basic principle of iiotypic immune network theory is presente in Section 2. The esign scheme of the MINA for optimization is etaile in Section 3. The globle convergence of the moifie algorithm are iscusse in Section 4. The experimental results for a set of test functions an the comparisons with other algorithms are explaine in Section 5. Finally, conclusions an recommenations for future research are provie in Section 6. II. IDIOTYPIC IMMUNE NETWORK THEORY In 974, Jerne introuce iiotypic immune network theory [9], which asserts that antiboies in the biological immune system have the uniqueness of being recognize by other antiboies. The uniqueness of antiboies can be seen as an epitope, an its role is to be recognize by other antiboies. Moreover, antiboies are ientifie by antigens through the receptors on the surface of the antiboies, which are calle paratopes. Thus, antiboies can recognize one another an form an iiotypic immune network, as shown in Fig.. (Avance online publication: 3 November 24)

2 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 Fig.. The principle of iiotype immune network. Antiboies can recognize not only external invasive antigens, but also other antiboies. However, antiboies are suppresse uring recognition, an the interaction network within the immune system reaches a ynamic equilibrium. Hence, the stimulation level of B cells (antiboies) not only epens on the affinity between the antiboy an the antigen but also on the matching egree between antiboies. If stimulation levels excee a certain threshol, the B cells prouce new antiboies through cloning an super mutation. Otherwise, these cells ie immeiately an are replace by those prouce in the marrow. Thus, the mutation rate of an antiboy is inversely proportional to its fitness uring mutation. In particular, the mutation process of antiboies involves a continuous enhancement in affinity an a graual maturation. These characteristics of immune network not only maintain the iversity of the antiboy population effectively but also facilitate self-organization an regulation in the biological immune system [] []. III. MODIFIED IMMUNE NETWORK ALGORITHM In 22, De Castro an Von Zuben propose the opt-ainet network moel [5]. Briefly, opt-ainet is an incomplete weighte connection iagram. This algorithm was originally intene for ata mining, pattern recognition, an multimoal optimization [2]; it can perform local an global searches an ajust population size ynamically. Specifically, this algorithm conucts local searches base on cloning, mutation, an antiboy selection. It performs global exploration by inserting ranom points an moifying population size. The performance levels of opt-ainet an other algorithms were compare in [3]. The results show that opt-ainet successfully etermine the global optimum of the test functions use but that the require computational cost was much higher than those of the other algorithms. The stuy conclue that this heavy computational cost may be relate to population increase. However, the possibility that the exploration of local optima causes population growth has not been stuie further [4]. Hence, the current stuy proposes a moifie MINA to reuce computational cost an to improve the convergence of opt-ainet in processing problems relate to complicate function optimization. The MINA algorithm is esigne in etail as follows: () Chaotic Initialization: An initial population of N antiboies shoul be generate using the logistic formula. (2) When the stopping criterion is unsatisfie, (2.) Fitness calculation: The fitness of each antiboy shoul be calculate. (2.2) Antiboy clone: Clones shoul be generate for each antiboy. (2.3) Antiboy mutation: Each clone mutates with the ai of the hyper-cubic mutation operator. The parent antiboy is retaine. (2.4) Clonal selection: N cells with maximum fitness from N mutation sets are selecte from each clone set base on ensity an fitness for the next generation. The ieal n antiboies are copie to the memory set. (2.5) Immune suppression: The antiboy with a poor fitness value is elete as etermine accoring to Eucliean istance an the fitness ifference between two antiboies. The antiboy with an improve fitness value is preserve in the memory set. (2.6) Immune replacement: antiboies with poor fitness values are replace with new, ranomly generate members with high fitness values (iversity introuction). (2.7) The best antiboy is etermine from the memory set before proceeing to step 2. The main operators of MINA are etaile as follows: A. Chaotic Initialization To reflect the chaotic nature of the biological immune system an to generate the initial population, we aopt a chaotic initialization operator. L chaotic variables are compute using the following logistic formulas: n n n x + n k = μxk ( xk ), x [,]. () Where k =, 2,, L, n=, 2,, N an μ = 4. k is the serial number of chaotic variables. We assume that n = an that the L chaotic initial values x k vary n ranomly. The values of L chaotic variables x k are then calculate using the logistic equation. The other N antiboies are obtaine in the same manner. B. Antiboy Clone The opt-ainet algorithm is very sensitive to the change of the number of clones. Therefore, each iniviual in the antiboy population is clone to a fixe number. Each antiboy then possesses 2n clones that are equally isplace over each imension. n enotes the number of imensions. C. Hyper-cubic Mutation To increase the probability of exceeing the local optimum, a hyper-cubic mutation operator is introuce into the algorithm [5]. The mutation operation in Step 2.3 is conucte using the following metho: Each iniviual antiboy copies two clones for each problem imension. These clones are symmetrically isplace from the original antiboy uner istance k. k =. exp f + Gauss,, (2) where f ( )( ( )) is the normalize fitness of the original antiboy, an Gauss (,) is the ranom Gaussian variable. The clones are evaluate over the objective function, an the ieal point from each imension is etermine to estimate (Avance online publication: 3 November 24)

3 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 X X estimate f ( X, X 2 ) = 5 [ X k, X 2 + k ] [ X, X 2 + k ] [ X k, X 2 ] [ X, X 2 ] f ( X, X 2 ) = 4 f ( X, X 2 ) = 3 [ X + k, X 2 ] X [ X, X 2 k ] f ( X, X 2 ) = 2 f ( X, X 2 ) = (Avance online publication: 3 November 24)

4 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 Accoring to Theorem, the convergence of sequence M M X () t, t an parameters α t an β t are very closely { } relate. To confirm the probability convergence of MINA, we must generate the following: X () t, Y() t S, σ >. We then obtain * r = min P T X t Y t > σ >, (7) t ( ( m( () ()))) * where r t is the minimum mutation rate. Theorem 2. For any initial istribution, MINA converges to global optimal solution set M in probability. X t = X, i N represent the antiboy Proof. Let () { } i population at generation t an let X N enote the best iniviual. At generation t +, we obtain X ( t+ ) = { Yi, i N} = { Y, YN, Y2}. Where Y = { Y, Y2,, YN }, YN = XN, an Y = T S = Y, Y,, Y. ( ) { } 2 g N + N + 2 N Hence, PXt+ = Y Xt= X ( ( ) ( ) ) ( ( ( ) ) ) ( 2) PTT r stm X X = Y PTg( Ω ) = Y, YN = X. (8) N =, YN XN Two cases must be consiere with regar to state transition probability: () If X M, Y M =, we erive P( X( t+ ) = Y X( t) = X) = from Equation (8). Then P( X( t+ ) M = X( t) M ). (9) = P X( t+ ) = Y X( t) = X = X M Y M= ( ) (2) If X M =, Y M, we obtain M PT ( ( Ω ) = Y2 ) g Ω PTT T X X = Y ( r s( m( ) ) ) * rt P( Ts( Z X) Y ) N Z S, Y ( Z X) (). () = = IfY ( Z X), then P T ( Z X ) = Y ( s ) D( Y ) N f ( Y ) e = e D( W ) Y Y N f ( W ) e W ( Z X ). (2) P X t M X t M. (4) + = = ( ( + ) ( ) = ) P( X( t ) Y X( t) X) ς The proof is complete with Equations (7), (9), an (4). V. SIMULATION AND RESULTS To verify the convergence performance of MINA, we test the following three complex multimoal functions. The results are compare with those of traitional opt-ainet [5] an the novel immune clonal algorithm (NICA) [7]. () Levy No. 3 f : 5 5 f ( x, x2) = icos ( i+ ) x+ i jcos ( j+ ) x2 + j i= j= < x, x2 <. This function contains approximately 76 local minima an 8 global minima with the ieal function value f = (2) Levy No. 8 f 2 : n f ( x) = sin ( πy) + ( yi ) + sin ( πyi+ ) + ( yn ). i= xi Where yi = +, x i, an i =,2,..., n. 4 When n = 3 an x = (,,) T, the global minima is Uner this conition, we can obtain approximately 25 local minimum values. (3) Levy No. f 3 : n 2 n xi xi f3 ( x) = cos 2 i= 4 + i= i. Where n = an x [ 6,6] i f =.. Using this formula, we can conuct a complex high-imensional test function an increase the number of local extreme values with the growth in problem imension. When x = (,,,), the global minimum is f ( x) =. In all of the experiments, the parameters of MINA are chosen as follows: N =, ε =., ω =.2, μ = 4, ν = 2, = 2. For opt-ainet an NICA, we follow the parameters in [5] an [7]. The experiments were run 3 times using MINA, NICA, an opt-ainet on each test function. We present the average performance in terms of mean value, computational time, an the percentage of successful convergence. Table I illustrates the performance levels of the three algorithms, an Fig. 3 compares the convergence results of the functions using these three algorithms. Base on Equations (7), (8), (), an (), we can obtain P X( t+ ) = Y X( t) = X ( ) N M σ e ς < ξ Ω N Hence, ( ). (3) (Avance online publication: 3 November 24)

5 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 - MINA NICA opt-ainet f Iterations (a) f MINA NICA opt-ainet Iterations (b) f MINA NICA opt-ainet Iterations (c) Fig. 3. Comparisons of convergence results using MINA, NICA, an opt-ainet (Avance online publication: 3 November 24)

6 IAENG International Journal of Computer Science, 4:4, IJCS_4_4_3 [7] Francisco Javier Diaz Delgaillo, Oscar Montiel-Ross, Roberto Sepulvea, an Patricia Melin, Automatic hanling of expert knowlege using artificial immune systems applie to combinatorial optimization problems, Engineering Letters, vol. 2, no., pp. 8 87, 22. [8] Vincenzo Cutello an Mario Romeo, On the Convergence of Immune Algorithms, in Proc. st IEEE Symposium on Founations of Computational Intelligence, California, 27, pp [9] N. K. Jerne. Towars a network theory of the immune system, Annual Immunology, vol. 25C, no. -2, pp , 974. [] X. Y. Huang, Z. H. Zhang, an X. B. Hu, Theory an Application of Moern Intelligent Algorithm, Beijing: Science Press, 25. [] P. A. Castro an F. J. Von Zuben, Learning ensembles of neural networks by means of a bayesian artificial immune system, IEEE Trans. Neural Networks, vol. 22, no. 2, pp , 2. [2] Y. T. Qi, F. Liu, an M. Y. Liu, Multi-objective immune algorithm with Balwinian learning. Applie Soft Computing, vol.2, no.8, pp , 22. [3] J. Timmis, C. Emons, an J. Kelsey, Assessing the performance of two immune inspire algorithms an a hybri genetic algorithm for function optimization, in Proc. IEEE Congr. Evolutionary Computation, Berlin, 24, vol., pp [4] Haktanirlar Ulutas, Berna, Kulturel-Konak, an Saan, A review of clonal selection algorithm an its applications, Artificial Intelligence Review, vol. 36, no. 2, pp.7 38, 2. [5] S. P. Gou, X. Zhuang, an Y. Y. Li, Multi-elitist immune clonal quantum clustering algorithm. Neurocomputing, vol., pp , 23. [6] M. Villalobos-Arias, C. A. C. Coello, an O. Hernanez-Lerma, Convergence analysis of a multiobjective artificial immune system algorithm, in Proc. 3 r International Conference on Artificial Immune Systems, September 3-6, 24, Catania, 24, pp [7] R. H. Shang, L. C. Jiao, an F. Liu, A novel immune clonal algorithm for MO problems. IEEE Trans. Evolutionary Computation, vol.6, no., pp.35 5, 22. Lu Hong receive his Ph.D. egree from the University of Science an Technology Beijing in 28 an is currently an associate professor in Huaihai Institute of Technology of China. His main research areas are computational intelligence an algorithm theory, as well as their applications in various omains. He has been wiely publishe in high-ranking journals an has participate in conferences relate to these research fiels. Joarer Kamruzzaman receive his Ph.D. egree in Information System Engineering from the Muroran Institute of Technology, Hokkaio, Japan in 993 an is currently the Director of Research at Gippslan School of Information Technology, Monash University. He is also the Deputy Director of the Centre for Multimeia Computing, Communications, an Applications Research, which is hoste by the Faculty of Information Technology. His research interests inclue sensor networks, wireless communications, truste computing an computational intelligence. He has publishe two eite books, nine book chapters, an more than 6 journal an conference articles. Two of these works were aware Best Paper at international conferences. His research fun excees $.5 million an is supporte by the Australian Research Council, as well as inustries such as Southern Health an Telstra. (Avance online publication: 3 November 24)