Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 555{560 c International Academic Publishers Vol. 38, No. 5, November 15, 2002 Capability Analysis

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1 Cmmun. Ther. Phys. (Beijing, China) 38 (2002) pp. 555{560 c Internatinal Academic Publishers Vl. 38, N. 5, Nvember 15, 2002 Capability Analysis f Chatic Mutatin and Its Self-Adaptin YANG Li-Jiang and CHEN Tian-Lun Department f Physics, Nankai University, Tianjin , China (Received May 17, 2002) Abstract Thrugh studying several kinds f chatic mappings' distributins f rbital pints, we analyze the capability f the chatic mutatins based n these mappings. Numerical experiments supprt ur cnclusins very well. The capability analysis als led t a self-adaptive mechanism f chatic mutatin. The intrducing f the self-adaptive chatic mutatin can imprve the perfrmance f genetic algrithm very prminently. PACS numbers: Gg Key wrds: genetic algrithms, chatic mutatin, functin ptimizatin, self-adaptin 1 Intrductin Genetic algrithms play a signicant rle, as search techniques fr handling cmplex ptimizatin prblems. The basic idea f genetic algrithms is t maintain a ppulatin f chrmsmes, which represent candidate slutins t cncrete prblem and prcess the ppulatin with three peratins: selectin, crssver and mutatin. Generally, the candidate slutins in genetic algrithms are cded using the binary alphabet. Hwever, the gd prperties related with these algrithms d nt stem frm the use f this alphabet ther cding methds have been cnsidered fr the representatin issue, such as real cding methd which wuld seem particularly natural when tackling ptimizatin prblems f parameters with variables in cntinuus dmains. Accrding t this type f cding methd, crssver and mutatin peratrs must make sme changes. In Ref. [1], a chatic mutatin based n Lgistic mapping is intrduced t real-cded genetic algrithm (GA-LM). And in Ref. [2], we intrduced a chatic mapping t cnstruct mutatin peratrs (GA- NM). When the tw genetic algrithms with chatic mutatins GA-LM and GA-NM were applied t functin ptimizatin prblems, [3] the results shwed that the perfrmance f GA-NM is better than that f GA-LM. Hwever, there is an imprtant questin: why is the perfrmance f GA-NM better? In this article, we will try t nd the answer thrugh analyzing the relatinship between chatic mappings' distributins f rbital pints and the perfrmances f chatic mutatins. The rest f this paper is rganized as fllws. Sectin 2 describes chatic mutatin, genetic algrithm with chatic mutatin and their implementatins. Sectin 3 investigates the cmputatin f chatic mapping's distributin f rbital pints. Then six types f chatic mappings are given and their distributins f rbital pints are studied. Based n the abve analyses f chatic mutatins, a self-adaptive selectin mechanism f chatic mutatin and an imprved genetic algrithm with chatic mutatin are prpsed and tested in Sec. 4. Finally, sectin 5 cncludes with sme remarks and future research directins. 2 Real-Cded Genetic Algrithm and Chatic Mutatin A glbal minimizatin prblem can be frmalized as a pair (S f), where S R n is a bunded set n R n and f : S! R is an n-dimensinal real-valued functin. The prblem is t nd a pint X min 2 S such that f(x min ) is a glbal minimum n S. Mre specically, it is required t nd an X min 2 S such that 8X 2 S : f(x min ) f(x), where f des nt need t be cntinuus but it must be bunded. This paper nly cnsiders uncnstrained functin ptimizatin prblems. It wuld seem particularly natural t represent the genes directly as real numbers fr functin ptimizatin prblems whse variables are dened in cntinuus dmains. Then a chrmsme is a vectr f ating pint numbers. The size f the chrmsmes is kept the same as the length f the vectr which is the slutin t the prblem in this way, each gene represents a variable f the prblem. The values f the chrmsme genes are frced t remain in the interval established by the variables which they represent. Accrding t this cdingmethd, the realcded genetic algrithms with chatic mutatins is implemented as fllws in this study: i) Generate the initial ppulatin P (t) f chrmsmes and set t = 0. Each chrmsme is taken as real-valued vectr X(j) = (x 1 x 2 ::: x n ), j = 1 2 :::, where x i is bjective variable. ii) Evaluate the tness scre fr each chrmsme X(j) 8j 2 f1 2 ::: g f the ppulatin based n the bjective functin f(x(j)). iii) t = t+1, select P (t) frm P (t;1) thrugh rulette wheel selectin. iv) Select a pair f parents randmly, apply arithmatical crssver t parents accrding t a prbability in rder t generate tw springs. v) Apply chatic mutatin X 0 i(j) = X i (j) + i i = 1 ::: G max j = 1 ::: (1) The prject supprted by Natinal Natural Science Fundatin f China under Grant N

2 556 YANG Li-Jiang and CHEN Tian-Lun Vl. 38 vi) t the springs generated by crssver accrding t a prbability, where G max is the largest evlutinary generatin, is the size f the ppulatin, i is the current generatin, i is a rbital pint f chatic mapping, and is the scale f mutatin. Stp if the halting criterin is satised therwise g t step 3. Equatin (1) is the general frm f chatic mutatin. The dierence between chatic mutatin f GA-LM and that f GA-NM is that in GA-LM, i is the rbital pint f chatic mapping r n+1 = r n (1 ; r n ) (2) while in GA-NM i is the rbital pint f chatic neurn mdel x n+1 = kx n ; 2 tanh(x n )exp(;3x 2 n) : (3) Althugh a single value f r n in ne mutatin can be similar t that f x n, the dierent distributins f r n and x n may result in widely dierent search strategies in lng time cmputatin. These dierent search strategies f mutatins wuld determine the evlutin directins f ppulatins s that the perfrmances f the tw kinds f genetic algrithms are dierent. In the latter parts f this article, we will analyze the capability f several types f chatic mutatins thrugh studying the distributins f their chatic mappings' rbital pints. 3 Capability Analysis f Chatic Mutatins 3.1 Chatic Mapping and the Distributin f Orbital Pints In this sectin, we will study several new chatic mutatins which are based n fur kinds f nnlinear mappings respectively i) The parablic mapping ii) x n+1 = 1 ; 2x 2 n (4) The dissymmetrical cubic mapping x n+1 = ax 3 n + (1 ; a)x n (5) iii) The symmetrical quartic mapping iv) The Hennn mapping x n+1 = ;ax 4 n + ax 2 n ; 1 (6) x n+1 = y n + 1 ; ax 2 n y n+1 = bx n (7) and the chatic mutatins f GA-LM and GA-NM, which are based n nnlinear mappings (2) and (3). In rder t make sure that these mappings are chatic nes, the parameters in these nnlinear mappings are taken as a = 4:0 in Eq. (5), a = 8:0 in Eq. (6), a = 1:4, b = 0:3 in Eq. (7), = 4:0 in Eq. (2), and k = 0:9, = 2:3 r 5.0 in Eq. (3). Please nte the mappings: Eqs. (2), (4), and (7). In fact the mappings (4) and (2) are dierent frms f the parablic mapping. They have the similar dynamical behavirs. Hwever, in GA-LM the denitin dmain f the mapping (2), x 2 [0 1] is transfrmed t x 2 [;1 1] in rder that the mutatin has the ability f increasing r decreasing the parent. S the distributin f rbital pints in the mapping (2) is dierent frm that f the mapping (4). Frm Eq. (7) the Hennn mapping is a 2-dimensinal ne which has tw variables: x and y. Thus when we cnstruct chatic mutatin using the Hennn mapping, we will get tw types f chatic mutatins, ne is based n the variable x f the Hennn mapping and the ther is based n the variable y. Usually, chatic mappings' distributins f rbital pints are calculated by Perrn{Frbenius equatin X (x i ) (y) = jf 0 (x i )j (8) fx i =f;1 (y)g where f is the functin f mapping and is the distributin f rbital pints. In numerical cmputatin Perrn{ Frbenius equatin (8) takes the iterative frm n (y) = X fx i =f;1 (y)g n;1 (x i ) jf 0 (x i )j : (9) Taking a reasnable initial distributin (x 0 ), the invariable distributin (y) will be btained thrugh iterating equatin (9) in sme steps. Hwever, in the implementatin f this methd, the inverse functin f functin f in many kinds f mappings is nt easy t be calculated. S we intrduce a simple statistical methd t calculate the distributins f rbital pints. Given chatic mapping, x n+1 = f(x n ) x n 2 [x l x h ] (10) i) Take an initial pint x 0 arbitrarily, then substitute x 0 t the mapping (10) and iterate mapping (10) times t get rbital pints. ii) Let the interval f the mapping x 2 [x l x h ] be divided int 100 subintervals x i i = 1 2 ::: 100 and cunt the numbers f rbital pints in each subinterval N i = fn 1 N 2 ::: N 100 g. iii) Let N 1 N 2 ::: N 100 be nrmalized, that is, N 1 =(1000 ) N 2 =(1000 ) ::: N 100 =(1000 ) where is the length f the denitin dmain ( = jx h ; x l j). The plt f N i with regard t x i wuld illustrate the distributin f rbital pints apprximately. 3.2 Capability Analysis f Chatic Mutatins Based n Studying f Distributin f Orbital Pints Applying the statistical methd cnsidered abve, we calculate distributins f chatic mapping equatins (2) (7), which are shwn in Figs The mapping (2) in GA-LM and the parablic mapping (4) Frm Fig. 1(b), we can see that the parablic mapping's distributin f rbital pints is ppsite t the

3 N. 5 Capability Analysis f Chatic Mutatin and Its Self-Adaptin 557 distributin f Gaussian randm numbers. Gaussian randm numbers are mainly distributed near the pint x = 0, while the rbital pints f the parablic mapping cncentrate at the edge f the interval. That is t say: unlike Gaussian mutatin generating new chrmsme near its parent, the chatic mutatin based n the parablic mapping which may intrduce greater changes prduces a new chrmsme being further away frm its parent. Thus we can cnclude that this type f chatic mutatin can imprve the genetic algrithm's ability f escaping frm lcal minimum. But it is t many large jumps in search space that results in the lcal ptimizatin ability f the algrithm being bad. Frm Fig. 1(a), just like what we discussed abve, the distributin f the mapping (2) in GA-LM is dierent frm that f the parablic mapping. Since the transfrmatin f interval, the mapping (2)'s rbital pints mainly distribute nt nly at the edge f the interval: x = ;1 and x = 1 but als near the middle pint x = 0. As a result, the chatic mutatin f GA-LM will have a higher prbability f generating new chrmsme near its parent, s the perfrmance f the chatic mutatin in GA-LM will be better than that f the chatic mutatin based n the parablic mapping. mutatin based n parablic mapping presented abve can be easily extended t the chatic mappings based n the dissymmetrical cubic mapping and the symmetrical quartic mapping. The perfrmance f the chatic mutatins based n these tw mappings have n great dierence t that f the chatic mutatin using the parablic mapping. Fig. 2 The distributin f rbital pints (a) the dissymmetrical cubic mapping (b) the symmetrical quartic mapping. The Hennn mapping Fig. 1 The distributin f rbital (a) the mapping in GA-LM (b) the parablic mapping. The dissymmetrical cubic mapping and the symmetrical quartic mapping The shapes f curves in Figs. 2(a), 2(b), and 1(b) are very alike. S the dissymmetrical cubic mapping, the symmetrical quartic mapping, and the parablic mapping have similar distributin f rbital pints. The capability analysis f the chatic Fig. 3 The distributin f the Hennn mapping's rbital pints (a) (x) vs. variable x (b) (y) vs. variable y.

4 558 YANG Li-Jiang and CHEN Tian-Lun Vl. 38 We can cnclude frm Fig. 3 that the perfrmance f the chatic mutatin based n the Hennn mapping is nt gd. Either the distributin f variable x r that f the variable y des nt have symmetry, s chatic mutatins based n them will alter parent with a single directin. Thus, the glbal minimum cannt be btained by algrithms using this kind f chatic mutatin. The chatic mapping in GA-NM Fig. 4 The distributin f the mapping (3)'s rbital pints under dierent values f (a) = 2:3 (b) = 5:0. Figure 4 clearly shws the reasn why GA-NM can btain the glbal minimum eciently. Due t the ideal distributin f rbital pints, the chatic mutatin f GA-NM cmbines glbal search and lcal search harmniusly. S the mapping (3) is the mst suitable chatic mapping t cnstruct mutatin peratr. Hwever, just as presented in Ref. [2], the distributin f the mapping (3) is nt the same fr = 2:3 and = 5:0. We can see that when = 2:3, rbital pints have the highest frequency near zer. That is t say, under this circumstance, chatic mutatin will generate new chrmsme near its parent just like what Gaussian mutatin des. Hwever, unlike Gaussian mutatin which has lng at tails, ur chatic mutatin has higher prbability t generate springs far away frm their parents due t the tw peaks at the edge in Fig. 4(a). Cnsequently the capability f escaping frm lcal ptimum is imprved. When = 5:0, rbital pints distribute mainly away frm zer. Therefre the cnvergent prperty will be wrse than that when = 2:3 because f the mre frequent large jumps during searching, hwever the quality f slutin may be imprved. 3.3 Numerical Simulatin In rder t prve the qualitative cnclusins abut chatic mutatins, which are btained thrugh analyzing the distributin f chatic mapping's rbital pints, we cnstruct several kinds f genetic algrithms with these chatic mutatins. They are called GA-PM (using the chatic mutatin based n the parablic mapping), GA- CM (using the chatic mutatin based n the dissymmetrical cubic mapping), GA-QM (using the chatic mutatin based n the symmetrical quartic mapping), GA- HXM and GA-HYM (using the chatic mutatin based n the Hennn mapping). Then all these ve algrithms tgether with GA-LM and GA-NM are applied t functin ptimizatin prblem. In numerical simulatin, the multimdal Shubert functin f(x) = i cs[(i + 1)x 1 + i ] i cs[(i + 1)x 2 + i ] + 0:5[(x 1 + 1:42513) 2 + (x 2 + 0:80032) 2 ] x 1 x 2 2 [;10 10] which has 760 lcal minimum in interval [;10 10] is selected t examine the capability f all these algrithms. We have run each algrithm ver Shubert functin 30 times. Fr the sake f the results being cmparative, the parameters in algrithm such as ppulatin size, prbability f mutatin P m, prbability f crssver P c, the largest generatin G max, and the scale f mutatin are taken the same value in all algrithms, = 50, P m = 0:1, P c = 0:9, G max = 500, and = 1:0. Averaged results are shwn in Table 1. (In Table 1 \f min " is the knwn glbal minimum,\best" is the best functin values fund by the algrithms, \Mean Best" indicates the mean best functin values fund by the algrithms, \Std Dev" stands fr the standard deviatin, and \N min " dentes the numbers f glbal minimums fund in cmputatin.) Table 1 Cmparisn between 7 kinds f genetic algrithms with chatic mutatins n Shubert functin. Algrithms f min Best Mean Best Std Dev N min GA-HXM ;185:812 ;164:393 19:788 0 GA-HYM ;184:347 ;158:442 19:952 0 GA-PM ;186:73 ;182:071 8: GA-CM ;186:73 ;186:73 ;182:388 8: GA-QM ;186:73 ;182:005 9: GA-LM ;186:73 ;183:618 8: GA-NM ;186:73 ;186:173 0: Apparently, the capability f the chatic mutatins based n the Hennn mapping (crrespnding t algrithms GA-HXM and GA-HYM) is the wrst. The mean

5 N. 5 Capability Analysis f Chatic Mutatin and Its Self-Adaptin 559 best slutins fund by GA-HXM and GA-HYM have greater dierences frm the knwn best. Furthermre, there is nt even ne glbal minimum being reached during 30 times cmputatin using GA-HXM and GA-HYM. The results btained by GA-PM, GA-CM and GA-QM have n great dierence. While the results f GA-LM are better than thse f GA-PM, GA-CM and GA-QM, especially the values f Mean Best and Std Dev. That is t say that algrithm GA-LM imprves the cnvergent prperty f GA-PM, etc. GA-NM is the algrithm with the highest perfrmance. The dierence between the Mean Best btained by GA-NM and the knwn Best is very little and the results f the algrithm GA-NM are very stable, which have smallest Std Dev. All these numerical results supprt the capability analysis f chatic mutatins thrugh analyzing the distributins f rbital pints very well. 4 Self-Adaptin f Chatic Mutatin The cnclusins f previus sectins shw that the chatic mutatin f GA-NM has the best perfrmance in ptimizatin prblems. Mrever, Ref. [2] and previus analyses in this paper indicate that in GA-NM the prperties f chatic mutatin are very dierent fr = 2:3 and = 5:0. When = 2:3, the chatic mutatin is easy t generate new individuals near parent, thus it has better lcal searching ability. While, when = 5:0, the changes t parent prduced by the chatic mutatin are larger, s the mutatin has better glbal searching ability under this situatin. It wuld be ideal if the chatic mutatin with = 5:0 is used when search pints are far away frm the glbal ptimum and the chatic mutatin with = 2:3 is adpted when search pints are in the neighbrhd f the glbal ptimum. Unfrtunately, the glbal ptimum is usually unknwn in practice, making the ideal switch frm = 5:0 t = 2:3 very diculty. This paper prpses a self-adaptin mechanism f chatic mutatin. We cnstruct an algrithm with this self-adaptive chatic mutatin, calling it \Imprved Real- Cded Genetic Algrithm with Chatic Neurn Mutatin", r IGA-NM fr shrt. The idea is t mix dierent search biases f the chatic mutatin when = 2:3 and that when = 5:0. The implementatin f IGA-NM is very simple. It diers frm GA-NM nly in Step 5 f the algrithm described in Sec. 2. Instead f using Eq. (3) with = 2:3 r = 5:0 alne, IGA-NM generates tw springs frm each parent, ne by the chatic mutatin with = 2:3 and the ther by the chatic mutatin with = 5:0. The better ne is then chsen as the spring. The rest f the algrithm is exactly the same as GA-NM. IGA-NM is tested by applying t 4 functin minimizatin prblems: i) Rsenbrck Functin f 1 (x) = 100(x 2 1 ; x 2 ) 2 + (1 ; x 1 ) 2 x 1 x 2 2 [;2:048 2:048] : ii) Shubert Functin f 2 (x) = i cs[(i + 1)x 1 + i ] i cs[(i + 1)x 2 + i ] + 0:5[(x 1 + 1:42513) 2 + (x 2 + 0:80032) 2 ] x 1 x 2 2 [;10 10] : iii) Sum f Square Functin with Tw Variables vi) f 3 (x) = nx x 2 i n = 2 x i 2 [;5:12 5:12] : Sum f Square Functin with 30 Variables f 4 (x) = nx x 2 i n = 30 x i 2 [;5:12 5:12] : The reasns why we chse these functins are that they have the characteristics as unimdal/multimdal, lw-dimensinal/hige-dimensinal, therefre the examining thrugh applying algrithm t these functins is believable. IGA-NM and GA-NM have been run ver each functin 30 times. In rder t cmpare with the results f the tw algrithms, we take the same parameters, = 50, P c = 0:9, P m = 0:1, and = 1:0. The average minimal value f bjective functin btained by the tw algrithms is shwn in Table 2. It is very clear frm Table 2 that IGA-NM has imprved GA-NM's perfrmance signicantly fr all test functins. Especially, IGA-NM has better perfrmance than GA-NM in functin f 3 and f 4. Obtained minimal values f these tw functins using IGA-NM are ne rder f magnitude smaller than thse btained by GA-NM. Table 2 Cmparisn between IGA-NM and GA-NM n functins f1 f2 f3, and f4. \Mean Best" dentes the averaged result f 30 times cmputatin. Functin Generatin Best Mean Best f IGA-NM Mean Best f GA-NM f : ;4 4: ;4 f ;186:73 ;186:575 ;186:173 f : ;6 1: ;5 f : ;3 1: ;2 During the evlutin f the IGA-NM, we can recrd the numbers f the chatic mutatins ( = 2:3 r 5.0) which are perated practically in every generatin. At this time the prbability f crssver P c must be set 0 t eliminate the inuence intrduced by crssver peratin. And we als take larger prbability f mutatin P m = 0:9 and ppulatin size = 100 in rder that mutatin peratins can be executed mre times. Thus the self-adaptive prcess f the chatic mutatin can be shwn prminently.

6 560 YANG Li-Jiang and CHEN Tian-Lun Vl. 38 The alteratins f the numbers f successful chatic mutatins crrespnding t generatins in functin f 4 are shwn in Fig. 5. Fig. 5 The numbers f successful chatic mutatins. (a) = 5:0 (b) = 2:3. Thrugh Fig. 5(a), the numbers f successful chatic mutatins with = 5:0 are larger at the beginning f evlutin (befre 200 generatin). This circumstance can be easily understd. Because the chatic mutatin with = 5:0 can generate larger jump in search space, the algrithm can apprach the glbal ptimum quickly using this kind f mutatin at the beginning. Hwever, with mving clsely t the glbal ptimum, larger jump is nt benecial, s the numbers f mutatins with = 5:0 decrease gradually. Accrdingly, gure 5(b) shws an ppsite prcess t Fig. 5(a). Since the chatic mutatin with = 2:3 has better lcal searching ability, the numbers f this kind f mutatins which are perated successfully are larger at the later stage f evlutin. The self-adaptive prcess shwn in Fig. 5 indicates that the tw kinds f chatic mutatins can be selected autmatically at suitable stage. Therefre the self-adaptin mechanism f chatic mutatin is reasnable. 5 Cnclusin During the prcedure f bilgical evlutin, chas, chance, temprality and nnlinear interactivity are presented typically. They inspire us with the idea f intrducing chas in genetic algrithm. But the cmbinatin f genetic algrithm and chas is in explring stage. Recently, sme kinds f genetic algrithms with chatic mutatin have been presented. Althugh the results f applying these algrithms t ptimizatin prblems shw their eectiveness, further analyses abut these algrithms is scarce. This paper prpses a general methd, which utilizes the analysis f the distributin f chatic mapping's rbital pints, t study the perfrmance f all kinds f chatic mutatins. The results f numerical experiments indicate that ur methd is feasible. The abve analyses als led t an imprved GA-NM (IGA-NM) and a self-adaptin mechanism f chatic mutatin which is very simple yet eective. IGA-NM uses the idea f mixing search biases f dierent chatic mutatins. IGA-NM is rbust, assumes n prir knwledge f the prblem t be slved, and intrduces n parameters. Future wrk n IGA-NM includes the applicatin t mre practical ptimizatin prblems and imprving the perfrmance f IGA-NM thrugh designing the algrithm delicately. At present, the capability f chatic mutatin is the fcus f studying. But there are many details, such as suitable types f selectin strategies and crssver peratrs etc., being wrth which cnsidering carefully in the future. References [1] LUO Chen-Zhng and SHAO Hui-He, Evlutinary Algrithms with Chatic Mutatins, Cntrl and Decisin, Vl. 15, N. 5 (2000) pp. 557{560. [2] YANG Li-Jiang and CHEN Tian-Lun, Cmmun. Ther. Phys. (Beijing, China) 38 (2002) 168. [3] YANG Li-Jiang, CHEN Tian-Lun, and HUANG Wu- Qun, Cmmun. Ther. Phys. (Beijing, China) 35 (2001) 22.

CSE 331, Spring 2000

YOUR NAME: SECTION NUMBER: CSE 331, Spring 2000 Class Exercise 15 Algrithm Design Techniques-Greedy Algrithms March 13, 2000 Fllwing are five f the cmmn types f algrithms. Fr many prblems it is quite likely