A Global Convergence Proof for a Class of Genetic Algorithms

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1 A Global Convergence Proof for a Class of Geneic Algorihms Richard F. Harl * Absrac: In his paper a varian of a geneic algorihm is invesigaed which combines he advanages of geneic algorihms and of simulaed annealing. In paricular, under raher mild assumpions a global convergence resul is derived. Key Words: Geneic algorihms, Simulaed annealing, Markov chains, Global convergence, Probabilisic programming, Simulaion, Randomized opimizaion. I. INTRODUCTION For many NP-had combinaorial problems and for problems wih "crazy" objecive funcion (i.e., disconinuous, nonsmooh or having many local opima) sochasic search algorihms have been successfully applied and are paricularly appreciaed by praciioners. Beside simple Mone-Carlo mehods here are wo imporan sochasic approaches o solve deerminisic problems namely "geneic (adapive )algorihms" and "simulaed annealing". Alhough he basic ideas in boh algorihms are very similar, here is no much lieraure abou he connecion of he wo approaches and abou comparisons of efficiency. I is he purpose of his paper o presen a modified geneic algorihm (GA, hereafer) which allows for all he flexibiliy of general geneic algorihms (populaion consising of more han one individual, recombinaion, ec.). On he oher hand, i has some similariies o simulaed annealing which makes i possible o prove ha under much less resricive assumpions han required for simulaed annealing he algorihm converges wih probabiliy one o he global opimal soluion. II. THE ALGORITHM Le us consider he following mahemaical model F(x) min (1) x M (2) where M is a finie se and F: M E is he objecive funcion. The assumpion of a finie number of feasible soluions may seem somewha resricive. However, he mehods can also be applied o more general problems while he finieness assumpion is required for he global convergence proof. On he oher hand, if he se of feasible soluions is bounded and if he problem is discreized on a digial compuer hen he resuling feasible se is in fac finie. Prof. Dipl.-Ing. Dr.echn. Richard F. Harl Chair of Producion and Operaions Managemen Insiue of Managemen a he Universiy of Vienna Bruennersraße 72 A-1210 Vienna. Ausria Richard.Harl@univie.ac.a

2 Richard Harl Concergence Proof 2 In order o describe he our geneic algorihm, we need he following definiions: A neighborhood srucure is a mapping N from M ino 2, i.e. for each soluion x i defines a neighborhood N(x) of x and each y N( x) is called a neighbor of x. In he case of he raveling salesman problem, he neighborhood of a given our x can e.g., be defined as he se of ours which can be generaed by he Lin (1965) 2-change or 3-change heurisic. A generaion mechanism is a rule of selecing a soluion y from he neighborhood N(x) of a given soluion x. In he conex of geneic algorihms such a generaion rule is usually called a muaion rule. The generaion mechanism can be described by a ransiion probabiliy marix R such ha (, ) = { = = } Rxy P X +1 yx x (3) where X denoes he sae of he sysem a ime (ieraion). Clearly R(x, y) > 0 if and only if y N( x). By (3) a Markov chain is defined over he se M of feasible soluions. However, in order o "solve" problem (1), (2) his Markov chain has o be modified by some accepance crierion so ha "good" soluions are seleced more ofen or wih higher probabiliy han "bad" ones. A local opimal soluion is an x M such ha ( ) Fy ( ) Fx for all y N( x) while a global opimal soluion is defined by ( ) Fy ( ) Fx for all y M. In order for he algorihm no o ge suck in a local opimum (which is no globally opimal) i is necessary o accep also deerioraions of he objecive funcion wih posiive probabiliy. A sae y is reachable from sae x if here exiss z 1,..., z m M such ha z 1 N(x), z 2 N(z 1 ),..., y N(z m ). III. GENETIC ALGORITHMS Geneic algorihms (GA, hereafer) imiae he evoluion of naure in order o opimize a sysem. In each ieraion, no a single curren soluion is considered bu here is a populaion of p soluions which can be inerpreed as he "parens". Le p { } X = x 1,..., x, i n wih x E, denoe he paren populaion in ieraion. From his a number c of new soluions is generaed which can be inerpreed as "children". Le { } X = x 1,..., x, c denoe he children populaion of ieraion. There are wo possibiliies o generae children: Muaion and Recombinaion In he conex of GA he generaion mechanism is usually called muaion rule. These c children are generaed by muaion of arbirarily seleced parens or each paren generaes a fixed number of individuals. In he laer case c is a muliple of p.

3 Richard Harl Concergence Proof 3 One of he advanages of GAs is he possibiliy of combining he informaion conained in wo paren soluions in creaing a new soluion. In allusion o he evoluion of naure his is usually called a recombinaion. In he simples case, he recombinaion can be a "crossing over" of he geneic informaion, i.e. some (arbirarily seleced) coordinaes of he newly creaed soluion are aken from one paren and he res is aken from he oher paren. Several auhors have repored significan improvemens of he convergence of GAs when recombinaions are inroduced (see, e.g., Rechenberg 1973, or Ablay 1987). Imporan references for GAs are, e.g., Fogel e al. (1966), Rechenberg (1973), Holland (1975), Schwefel (1977), and Goldberg (1988). Afer generaing he children populaion X from he paren populaion X by appropriae choice of he muaions and (possibly) recombinaions he new paren populaion X for he nex sep has o be seleced. This is usually done by one of he following selecion rules. (p+c)-rule: In his case he bes p individuals ou of all p+c parens and children 1 p 1 c 1 p x,..., x, x,..., x are seleced o form he new paren populaion x,..., x c (p,c)-rule: In his varian he bes p individuals ou of he c children x x,..., are seleced, Clearly c > p in his case. The difference is ha according o he (p+c)-rule good individuals can live infiniely long while according o he (p,c)-rule each individual has a maximum life ime of one generaion. Since he (p,c)-rule implies ha good soluions can "ge los", he convergence o a local opimum is usually a bi slower. However, o some exen i prevens he concenraion of he populaion in a small area which improves he global convergence behavior in he case of muliple local opima. Some auhors, have repored considerable improvemens of he convergence rae hrough desabilizaion phases; see, e.g., Ablay (1987) in he case of he raveling salesman problem. IV. A COMBINATION OF SA AND GA Le us consider a GA in he sense of he previous secion. The algorihm sars from an iniial X = x,..., x p. In each ieraion, from he curren paren populaion X a populaion 0 { } populaion X of c children is generaed by muaions and/or recombinaions, where a leas one child is generaed by a muaion. Then he following selecion rule is chosen: 1 p 1 (a) Selec x as he bes of all he p+c individuals x,..., x, x,... x. 2 1 c (b) Selec x +1 arbirarily among all he children x x,..., hich have no already been seleced in sep (a) and? (c) Selec x 3 p ,..., x if p > 2 by any selecion rule. In sep (c) of rule (12) he following alernaives can be hough of. (c1) Selec hose children (no already seleced) wih he bes values of he objecive funcion. (c2) Selec hose individuals (parens or children no already seleced) wih he bes value of he objecive funcion. (c3) Selec p-2 individuals arbirary among he children (no already seleced). c

4 Richard Harl Concergence Proof 4 (c4) Selec p-2 individuals arbirary among he parens and children (no already seleced). In he above selecion rules, he saemen "no already seleced" prevens he populaion from geing concenraed in a small area by having many copies of he same individual. For he following convergence resul his saemen can also be omied. By selecing one soluion arbirary in sep (b) one has he same effec of circumvening local opima as in he SA-rule (5). Furhermore, in he spiri of SA anoher possibiliy of par (c) of rule (12) can be formulaed: (c5) Selec 100r % of he p-2 individuals arbirarily and he remaining 100(1-r)% according o heir objecive funcion value. Choose r = 1 iniially and le r 0 as. Since (p-2)r < 0.5 for large enough, his implies ha afer some iniial phase his rule works like (c1) or (c2). In (c5), he parameer r plays a similar role as he "emperaure" T in he SA-algorihm. I is, however, also possible o choose he following varian: (c6) Choose he soluions according o rule (c5) wih 0 < r < 1 fixed. We can now formulae he following global convergence resul: Theorem 4.1. Le he se of feasible soluions M be finie and assume, ha for all x, y M he sae y is reachable from x by he muaions considered. Then he GA wih selecion rule (12) in any of he varians (c1) - (c6) has he following propery lim P {A leas one soluion in X is globally opimal} = 1. (13) I should be noed ha, compared o Theorem 2.1 much less resricive assumpions have o be made. Essenially he only assumpion is ha he Markovian chain described by he muaion/generaion-rule is irreducible. The proof of his resul can be found in he nex secion. V. PROOF OF THEOREM.1 Define a new Markov-chain Y as follows. The sae y M P is given by he combined vecor of p elemens of he original sae space M. However, for echnical purposes we make he following excepion for sae y 0 : he sysem is in sae y 0 if a leas one globally opimal soluion x is in he curren populaion. Now he sequence of populaions compued by he modified GA of Secion IV can be described by a Markov-chain in he above framework. The sae Y in ieraion is he combined vecor of all individuals in populaion X. On he oher hand Y = y0 if one of he elemens x 1,..., x p of X is a globally opimal soluion. According o par (a) of he selecion rule in Secion IV he sae y 0 is absorbing. This is because once a globally opimal soluion is found, i will always remain in he populaion. Furhermore, because of he assumpions of Theorem 4.1 each globally opimal soluion is reachable from any oher soluion. According o par (b) of he selecion rule his implies ha (absorbing) sae y 0 of he new Markov chain is reachable from any oher sae. This implies ha every sae y y 0 is ransien and { y} lim PY = =. 0 1

5 Richard Harl Concergence Proof 5 Thus (13) in Theorem 4.1 is proved. From he classical lieraure of Markov processes (e.g., Iosifescu, 1980) one can also obain expressions for he ime o absorpion. We menion he following: Le he ransiion marix of he new Markov process {Y } be given by P ~ = 10 RT,where R is a k 1 marix, T is a k k marix, 0 is a 1 k-zero marix, and k + 1 is he dimension of he sae space of {Y }. Then, wih P(n) denoing he vecor of probabiliy ha he ime o absorpion in sae zero saring in he ransien saes is n, we ge ( ) n 1 Pn = T ( I Te ) where e is he k-dimensional uniy vecor. Hence, he average ime o absorpion is n π nt ( I T) n e = π T e n= 1 0 n= 0 where π is he iniial disribuion of Y. However, since k is exremely large in pracical problems and herefore T canno be compued any more, his formula does no seem o be of much pracical use. For he same reason, we refrain from applying some furher sophisicaed resuls from he heory of absorbing Markov chains. VI. CONCLUDING REMARKS In his paper we have analyzed a geneic algorihm according o which in each generaion no only he bes individuals survive bu also some randomly seleced oher ones regardless of heir "finess". This is in accordance wih naure where in he long run he fies individuals survive bu in he shor run also oher individuals can survive. Furhermore, i circumvens he problem ha he populaion concenraes in a narrow neighborhood of a local opimum. As a consequence, i was possible o derive a global convergence resul in he spiri of he simulaed annealing resul by Hajek (1988) under much weaker assumpions. An advanage of he algorihm considered is ha i is robus, in he sense ha i does no require much experise o apply i whereas in he simulaed annealing case he choice of he annealing rule is always a nonrivial problem. Finally, we would like o menion ha he above GA has successfully been applied o some classical exbook examples and o he uni commimen problem of a local power company; see Harl and Perisch (1990). We should noe ha he above GA is more in he spiri of he German "Evoluionssraegien" by Rechenberg (1973) and Schwefel (1977) han of he GAs by e.g. Goldberg (1989). (14)

6 Richard Harl Concergence Proof 6 REFERENCES Aars, E., Kors, J. (1989) Simulaed annealing and Bolzman machines: a sochasic approach o combinaorial opimizaion and neural compuing. Chicheser: John Wiley & Sons. Ablay, P. (1987) Opimieren mi Evoluionssraegien, Spekrum der Wissenschafen 7, Burkard, R.E., Rendl, F. (1984) A hermodynamically moivaed simulaion procedure for combinaorial opimizaion, European Journal of Operaional Research 17, Cerny, V. (1985) Thermodynamical approach o he raveling salesman problem: an efficien simulaion algorihm, Journal of Opimizaion Theory and Applicaions 45, Fogel, L.J., Owens, A.J., Walsh, M.J. (1966) Arificial inelligence hrough simulaed evoluion, New York: Wiley. Goldberg, D.E. (1989) Geneic algorihms in search, opimizaion, and ma- chine learning. MA: Addison-Wesley. Goldberg, D.E. (1990) A noe on Bolzman ournamen selecion for geneic algorihms and populaion oriened Simulaed annealing, TCGA Repor (May 1990), Univ. of Alabama. Hajek, B. (1988) Cooling schedules for opimal annealing, Mahemaics of Operaions Research 13, Harl, R.F., Perisch, G. (1990) Krafwerkseinsazplanung miels zufallsgeseuerer Suchsraegien, in: Operaions Research Proceeding 1989, K.P. Kisner e al. (Eds.). Berlin: Springer, Holland, J.H. (1975) Adapaion in naural and arificial sysems. Ann Arbor: Univ. of Michigan Press. Iosifescu, M. (1980) Finie Markov Processes and heir Applicaions, Chicheser: Wiley. Kirkparick, S., Gela, C.D., Vecchi, M.P. (1983) Opimizaion by simulaed annealing, Science 220, Lin, S. (1965) Compuer soluions of he raveling salesman problem, Bell Sysem Technical Journal 44, Rechenberg, I. (1973) Evoluionssraegien: Opimierung echnischer Syseme nach Prinzipien der biologischen Evoluion. Sugar: Frommann-Holzboog. Schwefel, H.-P. (1977) Numerische Opimierung von Compuer-Modellen miels der Evoluionssraegie. Birkhauser.