Ant Colony Optimisation for Continuous Domains with Aggregation Pheromones Metaphor

Transcription

1 Ant Colony ptimisation for Continuous Domains wit Aggregation Peromones Metapor Sigeyosi Tsutsui Department of Management Information Sciences, Hannan University Amamiigasi, Matsubara, saka 5-5, Japan Abstract: Tis paper describes an aggregation peromone system (APS), wic is an eension of AC for continuous domains, using te collective beavior of individuals tat communicate using aggregation peromones APS is tested on several test functions Results sow APS could solve realparameter optimization problems fairly well Te sensitivity of control parameters of APS is also studied Keywords: Evolutionary Computation, Ant Colony ptimization, Aggregation Peromones, Parameter ptimization, Genetic Algoritms 1 Introduction As a bio-inspired computational paradigm, ant colony optimization (AC) as been applied wit success to a large number of computationally ard problems AC simulates te collective beavior of ants, wic communicate using peromone trails Howeve AC is mainly applicable to discrete optimization problems suc as te traveling salesman problem (TSP) [, 5, 6, 19], quadratic assignment problem [1], sceduling problem [, 7], veicle routing problem [3], as well as te routing problem in telecommunication networks [1] Altoug using peromone trail metapor is very effective in solving discrete optimization problems mentioned above, a direct application of te peromone trail metapor for solving real-parameter optimization problems is difficult In tis pape we introduce "aggregation peromones", also observed in nature, and propose an algoritm "aggregation peromone system (APS)" for solving realparameter optimization problems Wen an individual of te same species comes in contact wit peromone, it elicits a response, depending on te type of peromone In tis way, specific information is conveyed Peromones tat cause clumping or clustering beavior in a species, wic bring individuals into a closer proximity, are referred to as aggregation peromones [13] Many functions of aggregation beavior ave been observed Tese include foraging-site marking and mating [1], finding selte and defense Cockroaces produce a specific peromone wit teir excrement wen tey find safe selte wic attracts oter members of teir species [1] As a resul aggregation peromones function suc tat individuals aggregate around a "good position" wit positive feedback Peromones evaporate wit some rate Tis prevents oter individuals to aggregate to a local position Te Aggregation Peromone System (APS) proposed in tis paper uses tese aggregation peromones as te basic metapor of te model Wile AC is mainly applicable in discrete problems, APS can solve real-parameter optimization problems Altoug we need more studyesults sow tat it works fairly well in various test functions used in te evolutionary computation community Te remainder of tis paper is organized as follows We introduce te APS in Section, experimental analysis is given in Section 3, and Section concludes te paper Te Aggregation Peromone System Since te proposed algoritm is a variant of AC and uses peromone update rules similar to AC, AC is briefly described in Section 1 APS is presented in detail in Sections and 3 7 RASC

2 1 A Brief verview of AC Here AC, in tis section AC is briefly overviewed based on [5] In AC, searc activities are performed wit so-called ants A moving ant lays a peromone trail on te ground An ant encountering a previously laid trail can detect it and decide wit ig probability to follow i tus reinforcing te trail wit its own peromone In [5], te algoritm is called te ant system (AS) For a TSP, AS works as follows Let m be te total number of ants Eac ant as te following caracteristics: (i) it cooses te town to go to wit a probability tat is a function of te town distance and of te amount of trail present on te connecting edge; (ii) to force te ant to make a legal tou transitions to already visited towns are disallowed until a tour is completed; (iii) wen it completes a tou it lays a trail on eac edge (i, j) visited Let τ ij (t) be te trail intensity on edge (i, j) at iteration t At t =, initial values τ ij () for trail intensity are set on edges Wen all ants complete teir tours, te trail intensity on eac edge is updated according to te following formula: τ ( t + 1) = τ ( t) + τ (1) ij were ( < 1) is a coefficient suc tat (1 ) represents te evaporation of trail between iteration m k t and t+1, τ ij = k = 1 τ ij and τ k ij is te quantity per unit of lengt of trail laid on te edge (i,j) by te k-t ant between iteration t and t+1; it is determined to be inversely proportional to te tour lengt of k-t ant so tat larger values of τ k ij are given if te tour lengt of te k-t ant is sorter Tis process is iterated until te given termination conditions are satisfied Basic Model of te Aggregation Peromone System Tere is a big difference between AC and APS in ow te peromone functions in te searc space Peromone density in AC is defined as a trail on an edge between nodes of a given sequencing problem In APS, te aggregation peromone density is defined by a density function in searc space X in R n In te real world, aggregation peromones are used by a species to communicate wit members of teir community to sare information about te location of food, safe selte potential mates, or enemies As in AC, APS takes a cycle model In eac APS cycle, borrowed from te natural model, m individuals are attracted to positions by te aggregation peromone in searc space X Tey are attracted more to te positions were te peromone density is ige and less to te positions were te density is lower Tus, in our system, we consider individuals to coose teir positions depending on te aggregation peromone density, or more specifically, wit probability proportional to te aggregation density function in searc space X as follows Let x represent te variable in searc space X, ie x X, and τ ( be te density function of aggregation peromone at APS cycle t In initial APS cycle (t=), te aggregation peromone distributes uniformly, ie, τ (, x ) = c, were, c is a constant Te probability density function of aggregation peromone at APS cycle p τ (, is defined as τ ( pτ ( = () τ ( dx X Depending on te problem we solve, a fitness function f( is assigned for x X Eac of m individuals emits aggregation peromone around its position x in X depending on te fitness function f( Tere may be questions regarding wic approac is more similar to tat used in te natural world, to use an absolute value of f(, a relative value, or a ranking system Howeve in our system we use rank r (r = 1,,, m) of eac individual to make te fitness difference among individuals more distinguisable Te igest ranked individual as a value of m and te lowest ranked individual as a value of 1 An individual wic as rank r emits te aggregation peromone around x r wit density function represented by τ '( x r,, were x r is te position of te individual wit rank r Ten te new aggregation peromone density emitted in APS cycle t by m individuals is m τ ( = r = 1 τ '( x t, (3) Here, we assume te total aggregation peromone volume emitted by m individuals in eac cycle t is equal to C, ie, ij ij RASC

3 τ ( dx = C () X Tis assumption is important wen we consider te sampling metod in Section 3 Wen all m individuals complete coosing teir positions, te total aggregation peromone density in X is updated according to te following formula, as in Eq 1: τ ( t + 1, = τ ( + τ ( (5) After te peromone updating is performed, individuals are reset and te ne APS cycle starts Tis process is iterated Tus, te peromone density at promising points in te space increases as te APS cycle is iterated and te APS is expected to converge to a promising solution Here, we discuss te form of τ '(, of Eq 3 We assume tat (i) τ '(, is centered around position x r (position x r as te igest density), (ii) an individual wit iger rank emits more peromone tan an individual wit lower rank Furte (iii) we introduce a co-operative penomenon in wic τ '(, is affected by te distribution of oter individuals and becomes elongated in te direction of te distribution In tis researc, we use te following Gaussian functional form for τ '(, (r=1,,, m) for Eq 3 C τ '( x r, = r N( x r, β Σt ) (6) m k = 1k Here (>) is a parameter to adjust te relative importance of rank, Σ t is te covariance matrix estimated from distribution of m individuals in searc space X for cycle β (β>) is a parameter to control te widt of distribution of te peromone, N( x t, i, β Σt ) is a multivariate normal distribution Te function defined by Eq 6 satisfies all te tree conditions mentioned above It as te igest value at x r, iger rank individuals ave larger functional values, and covariance matrix β Σt reflects a distribution of individuals Wit larger values of, te ratio of te total amount of peromone produced by a iger ranked individual increases Wit larger values of β, te aggregation peromone spreads more widely in te searc space X, and wit smaller values of β, aggregation peromone spreads less Ten, τ (, te total peromone density emitted by m individuals in cycle can be obtained from Eq 3 We can see also tat τ ( satisfies Eq, ie, te total amount of aggregation peromone at eac cycle equals C 3 Sampling Tecnique In tis subsection, we discuss ow to sample individuals from aggregation peromone density function τ ( t + 1, obtained in Eq 5 As described in, we sample individuals wit probability proportional to te aggregation density τ ( t + 1, To perform te proportional sampling, we need to obtain probability density function p τ ( t + 1, from τ ( t + 1, τ ( t + 1, in Eq 5 can be rewritten as t+ 1 t τ t + 1, = τ (, + τ ( t, (7) ( = From Eqs,, and 7, p τ ( t + 1, is obtained as t+ 1 ( (, ( t, p t + τ t 1, = + t 1 k C τ τ = () + t+ 1 k C k= k= In general, if a probability density function (pdf) f( can be decomposed into sub-pdfs f k ( as f ( = p1 f1( + p f( + L + ps fs (, ten we can sample f( as follows: first we coose f s ( (s=1,,, S) according to probability (p 1, p,, p S ), ten we sample f s (, were, (p 1, p,, p S ) is a probability distribution Using tis metod, we can sample p τ ( t + 1, of Eq in te following manner In Eq, te terms τ (, x ) / C and τ ( t, / C can be pdfs from Eq Tus, first we t coose τ (, x ) / C or τ ( t, / C (=,1,, t) according to probability + 1 k / k= (=,1,, t+1) We call tis cycle sampling Nex we sample τ (, x ) / C or τ ( t, / C Sampling of τ (, x ) / C is simple because it is a uniform sampling Sampling of τ ( t, / C is similar to cycle sampling Generally, τ ( t, / C can be obtained from Eq 6 as τ ( t, m r = r = 1 N( x t, r, β Σt ), (9) C L were L = m k = 1k As wit cycle sampling, we first coose rank r according to te probability r / L (r=1,,, m) We call tis sampling rank sampling After rank sampling, we sample N(, β Σ ), using Colesky decomposition [] To perform tis sampling, based on Eq, we x t t 9 RASC

4 need a large memory to store x t, r vector values and covariance matrix β Σ t wen APS cycle t becomes large In tis case for large since < <1 Tus, we can limit te maximum number of cycles to keep data up to a constant H Ten, Eq for t H can be represented as H 1 τ ( t, pτ ( t + 1, = = (1) H 1 l 3 Experimental Study 31 Experimental Metodology C l= In te APS cycle model, we reserve te best e ( e = m Erate ) individuals at eac cycle Tey are transferred to te ne cycle Tis acts as an elitist strategy in GAs [] Parameter E rate is used to control te number of elites in eac cycle We use te following tree test functions: te Ellipsoidal function (F Ellipsoidal ), te Ridge function (F Ridge ), and te Rosenbrock function (F Rosenbrock ) F Ridge as weak linkage among variables F Rosenbrock as strong linkage among variables F Ellipsoidal as no linkage among variables Problem size n = is used for all test functions n F = = ix, ( 3 i 7), (11) Ellipsoida l i 1 i x < i ( x ), ( ), n Ridge i= 1 j= 1 j xi < F = () n F Rosenbrock = = (1( x x i ) + ( x i 1) ), ( x i < ), (13) i 1 Te default value of te parameters is as follows: m = 1, = 9, =, β = 6, E rate = 1, and H= Tese default values were cosen to tune APS to work well wen tested wit te F Rosenbrok function We evaluated APS by measuring teir #PT (number of runs in wic APS succeeded in finding te global optimum) and (mean number of function evaluations to find te global optimum in tose runs were it did find te optimum) We considered te detection of te solution successful if all parameters (x 1,,x n ) of te best individual were witin te range [(o j 1), (o j +1)] for all j, were (o 1,,o n ) is te optimal point runs are performed Eac run continued until te global optimum is found or a maximum of 5, evaluations is reaced 3 Results (1) Results wit default parameter values Table 1 sows te results of APS using te default parameter values described in Section 31 Te results are compared wit te results of SPX, a typical state-of-art crossover operator in real-coded GAs [9] APS sowed better results tan SPX on F Ellipsoidal and F Ridge, altoug it is possible to furter tune APS as described below (see Fig 3) n F Rosenbrock, APS sowed muc better results tan SPX Fig sows te convergence processes of bot APS and SPX on F Fosenbrock We can see tat APS converges more stably and rapidly tan SPX () Sensitivity of parameters To see te effect of te evaporation coefficient in APS, we varied te value of from te default value in te range [, 9] (Fig 3) Wen te value of becomes large peromone emitted in previous cycles remains for a longer period of time Wit F Ellipsoidal and F Ridge, APS wit values in [, 6] sowed #PT = and te better (smaller) values Wit F Rosenbrock, wic as strong linkage among variables, APS wit value 9 sowed smaller wit #PT = functional values Table 1 Results of APS wit default values Function F Ellipsoidal F Ridge F Rosenbrock #PT APS STD* #PT SPX STD* * STD: Standard Deviation 1E+3 1E+ 1E+1 1E+ 1E-1 1E- 1E-3 1E- 1E-5 1E-6 1E-7 1E- 1E-9 1E-1 1E-11 APS SPX 1E- 1E-13 1E-1 1E-15 1E- 1E-17 1E-1 1E-19 1E- 1E-1 1E- 1E-3 1E- 1E-5 1E-6 1E evaluations Fig Convergence process on F Rosenbrock 1 RASC

5 For tis problem te value 5 3 of is larger tan tose resulting from F Ellipsoidal #PT #PT #PT 1 and F Ridge Wit larger 5 6 values of, APS uses information of te F peromone density in te Ellipsoidal F Ridge F Rosenbrock past cycles more tan Fig 3 Effect of te coefficient of peromone evaporation wit smaller values of 15 and can find solution #PT 9 #PT efficiently on functions 6 tat ave strong linkage 3 #PT among variables To see te effect of te F Ellipsoidal F Ridge F Rosenbrock parameter, wic Fig Effect of parameter adjusts te relative importance of rank wen #PT eac individual emits #PT peromone, we tested for in te range of [, ] wit step size (Fig ) β β β F Wit larger values of, Ellipsoidal F Ridge F Rosenbrock iger ranked individuals Fig 5 Effect of parameter β emit peromone at increasing rates Wit larger values of, te performance of APS increased, as seen in te results wit F Ellipsoidal and F Ridge Howeve as seen in te results wit F Rosenbrock, wic as strong linkage among variables, larger values cause side effects in convergence #PT Parameter β controls te widt of distribution of te peromone emitted by individuals Wit increasingly larger values of β, individuals emit peromone more We tested β in te range of [5, 7] wit step size (Fig 5) Wit function F Ellipsoidal and F Ridge, aving respectively no and weak linkage among variables, smaller values of β ave better performance Wit function F Rosenbrock, wic as a strong linkage among variable, bot smaller and larger values of β ave poor performance, sowing over-exploring and over-exploiting beavior Conclusions In tis pape we ave described te aggregation peromone system (APS), wic uses aggregation peromones as te basic metapor of te model, for solving real-parameter optimization problems We studied APS wit several test problems Te results sowed tat APS could solve real-parameter optimization problems fairly well We also explored te sensitivity of te control parameters of APS Tere are many opportunities for furter researc related to APS Tese include study of te relationsip between te parameters and te resulting performance of APS, te scalability analysis of te algoritm, comparative study wit evolutionary algoritms suc as UNDX [15], SPX [9], and EDAs (estimation of distribution algoritms) [11, ] Acknowledgements Te autor would like to tank Associate Professor Gordon Wilson, Hannan Univ for is valuable comments on tis paper Tis researc is partially supported by te Ministry of Education, Culture, Sports, Science and Tecnology of Japan under Grant-in-Aid for Scientific Researc number #PT #PT #PT 11 RASC

6 References [1] Bell, W J: Cemo-orientation in Walking Insects, Cemical Ecology of Insects, Bell, W J and Carde, R T Eds, (19) [] Bullneime B, Hartl, R F, and Strauss, C: A New Rank Based Version of te Ant System: A Computational Study, Central European Journal for perations Researc and Economics, 7(1):5-3 (1999) [3] Bullneime B, Hartl, R F, and Strauss, C: Applying te Ant System to te Veicle Routing Problem, Meta-Heuristics: Advances and Trends in Local Searc Paradigms for ptimization, Voss, S, et al (Eds), Kluwer (1999) [] Costa, D and Hertz, A: Ants Can Colour Graps Journal of te perational Researc Society, :95-35 (1997) [5] Dorigo M, Maniezzo, V, and Colorni, A: Te Ant System: ptimization by a Colony of Cooperating Agents, IEEE Trans on SMC-Part B, 6(1):9-1(1996) [6] Dorigo M & LM Gambardella (1997) Ant Colony System: A Cooperative Learning Approac to te Traveling Salesman Problem IEEE TEC, 1(1):53-66 (1997) [7] Forsyt P and Wren, A: An Ant System for Bus Driver Sceduling Proc of te 7t Int Worksop on Computer-Aided Sceduling of Public Transport (1997) [] Goldberg, D E: Genetic algoritms in searc, optimization and macine learning Reading, MA: Addison-Wesley publising company (199) [9] Higuci, T, Tsutsui, S, and Yamamura, M:Teoretical analysis of simplex crossover for realcoded Genetic Algoritms, Proc of te PPSN VI, () [1] Kuntz P, Layzell, P, and Snyers, D: A Colony of Ant-like Agents for Partitioning in VLSI Tecnology Proc of te t European Conf on Artificial Life, 17-, MIT Press (1997) [11] Larranaga, P and Lozano, J A (eds): Estimation of distribution algoritms, Kluwer Academic Publisers () [] Law, J H and Regnie F E: Annual Review of Biocemistry, :533-5 (1971) [13] Lorenzo Figueiras, AN and Lazzari, C R: Aggregation beavior and interspecific responses in tree species of Triatomine, Memórias do Instituto swaldo Cruz 93(1): (199) [1] Maniezzo, V: Exact and approximate nondeterministic tree-searc procedures for te quadratic assignment problem, CSR 9-1, Scienze dell'informazione, Universitá di Bologna (199) [15] no, I and Kobayasi, S: A Real-Coded Genetic Algoritm for Function ptimization Using te Unimodal Normal Distribution Crossove Proc of te 7t ICGA, 6-53, (1997) [] Pelikan, M, Goldberg, D E, and Lobo, F: A survey of optimization by building and using probabilistic models Computational ptimization and Applications, 1(1), 5- () Also Tecnical Report IlliGAL Report 991, University of Illinois at Urbana-Campaign (1999) [17] Pena, J M, Lozano, J A, and Larranaga, P: Benefit of Data Clustering in Multimodal Function ptimization via EDAs, Estimation of Distribution Algoritms, Larranaga, P and Lozano, J A (Eds), Kluwer Academic Publisers, Cap 1:11-9 () [1] Sakuma, M and Fukami, H: Aggregation arrestant peromone of te German cockroac, Blattella germanica (L) (Dictyoptera: Blattellidae): isolation and structure elucidation of blasttellastanoside-a and -B, Journal of Cem Ecol 19:51-51 (1993) [19] Stützle, T and H Hoos, H: Te MAX-MIN Ant System and local Searc for Combinatorial ptimization Problems: Towards Adaptive Tools for Global ptimization, Proc of te nd Metaeuristics Int Conf (MIC-97) (1997) [] Scatzman, M, Taylo J, and Scatzman, M: Numerical Analysis: A Matematical Introduction, xford Univ Press () [1] Scoonderwoerd, R, Holland,, Bruten, J, and L Rotkrantz, L: Ant-based Load Balancing in Telecommunications Networks Adaptive Beavio 5():9-7 (1997) RASC