Selection Hyper-heuristics Can Provably be Helpful in Evolutionary Multi-objective Optimization

Transcription

1 Selectio Hyper-heuristics Ca Provably be Helpful i Evolutioary Multi-objective Optimizatio Chao Qia 1,2, Ke Tag 1, ad Zhi-Hua Zhou 2 1 UBRI, School of Computer Sciece ad Techology, Uiversity of Sciece ad Techology of Chia, Hefei , Chia 2 Natioal Key Laboratory for Novel Software Techology, Najig Uiversity, Najig , Chia {chaoqia,ketag}@ustc.edu.c, zhouzh@ju.edu.c Abstract Selectio hyper-heuristics are automated methodologies for selectig existig low-level heuristics to solve hard computatioal problems. They have bee foud very useful for evolutioary algorithms whe solvig both sigle ad multi-objective real-world optimizatio problems. Previous work maily focuses o empirical study, while theoretical study, particularly i multi-objective optimizatio, is largely isufficiet. I this paper, we use three mai compoets of multi-objective evolutioary algorithms (selectio mechaisms, mutatio operators, acceptace strategies) as low-level heuristics, respectively, ad prove that usig heuristic selectio (i.e., mixig low-level heuristics) ca be expoetially faster tha usig oly oe low-level heuristic. Our result provides theoretical support for multi-objective selectio hyper-heuristics, ad might be helpful for desigig efficiet heuristic selectio methods i practice. 1 Itroductio Hyper-heuristics are automated methodologies for selectig or geeratig heuristics to solve hard computatioal problems [4]. There are two mai hyper-heuristic categories: heuristic selectio ad heuristic geeratio. This paper focuses o the former type. Give a set of low-level heuristics, a heuristic selectio method chooses a appropriate oe to be applied at each decisio poit. Selectio hyper-heuristics have bee widely ad successfully applied for evolutioary algorithms (EAs) solvig sigle-objective optimizatio problems such as persoel schedulig, packig, vehicle routig, etc [3]. After that, they start to emerge i evolutioary multi-objective optimizatio. Burke et al. [5] first proposed a multiobjective hyper-heuristic approach based o tabu search for the space allocatio ad timetablig problems. McClymot ad Keedwell [17] developed a Markov chai based hyper-heuristic method for desigig water distributio etwork. By usig NSGAII, SPEA2 ad MOGA as low-level heuristics, Maashi et al. [16] desiged a choice fuctio based hyper-heuristic to solve the vehicle crashworthiess desig problem. Selec- This work was supported by the NSFC ( , ), the Fudametal Research Fuds for the Cetral Uiversities (WK ), ad the Collaborative Iovatio Ceter of Novel Software Techology ad Idustrializatio.

2 tio hyper-heuristics also achieved successes i other real-world multi-objective optimizatio problems, e.g., the 2D guillotie strip packig problem [18] ad the itegratio ad test order problem i software egieerig [10]. Most previous work focuses o empirical study. Meawhile, theoretical aalysis, particularly ruig time aalysis, is importat for ehacig our uderstadig ad desigig efficiet hyper-heuristics, as Burke et al. stated i [3]. However, the ruig time aalysis o selectio hyper-heuristics is difficult due to their complexity ad radomess, ad few results have bee reported. By usig mutatio operators as low-level heuristics, He et al. [11] first gave some coditios uder which the asymptotic hittig time of the (1+1)-EA (a simple EA) with a mixed strategy is ot larger tha that with ay pure strategy. Note that a mixed strategy (which correspods to heuristic selectio) chooses oe low-level heuristic accordig to some distributio each time, while a pure strategy uses oly oe fixed low-level heuristic. Their result was the exteded to the expected ruig time measure ad to populatio-based EAs i [12]. Lehre ad Özca [15] later gave cocrete evidece that mixig two specific mutatio operators is more efficiet tha usig oly oe operator for the (1+1)-EA solvig the GapPath fuctio. They also proved the beefit of mixig acceptace strategies for the (1+1)-EA solvig the RR k fuctio. I [19], by mixig global ad local mutatio operators, the (1+1)-EA was proved to be a polyomial time approximatio algorithm for the NP-hard sigle machie schedulig problem. I [6], the (1+1)-EA mixig two specific mutatio operators was show to be able to solve the easiest fuctio for each mutatio operator efficietly. The above studies ivestigate whether selectio hyper-heuristics ca brig a improvemet o the performace. Alaazi ad Lehre [1] also compared differet heuristic selectio methods (i.e., mixed strategies with differet distributios), ad proved their similar performace for the (1+1)-EA solvig the LeadigOes fuctio. All of the above-metioed studies cosider sigle-objective optimizatio. To the best of our kowledge, there has bee o theoretical work supportig the effectiveess of selectio hyper-heuristics i multi-objective optimizatio. I this paper, we prove that usig heuristic selectio ca speed up evolutioary multi-objective optimizatio expoetially via rigorous ruig time aalysis. The widely used multi-objective EA GSEMO i previous theoretical aalyses [8,14,21] is employed. It repeats three steps: choosig a solutio by some selectio mechaism, reproducig a ew solutio by mutatio, ad updatig the populatio by some acceptace strategy. This paper cosiders the three mai compoets of GSEMO, i.e., selectio mechaism, mutatio operator ad acceptace strategy, as the low-level heuristic, respectively. For each kid of low-level heuristic, we give a bi-objective pseudo-boolea fuctio, ad prove that the expected ruig time of GSEMO with a mixed strategy is polyomial while GSEMO with a pure strategy eeds at least expoetial ruig time. The aalysis also shows that the helpfuless of selectio hyper-heuristics is because the stregths of oe heuristic ca compesate for the weakesses of aother. For mixig acceptace strategies, we also empirically compare the ruig time of GSEMO with differet mixed strategies, ad the results imply the importace of choosig a proper heuristic selectio method. The rest of this paper is orgaized as follows. Sectio 2 itroduces some prelimiaries. The helpfuless of mixig selectio mechaisms, mutatio operators ad acceptace strategies is the theoretically aalyzed. Fially, we coclude the paper.

3 2 Prelimiaries Multi-objective optimizatio requires to simultaeously optimize two or more objective fuctios, as show i Defiitio 1. Note that maximizatio is cosidered i this paper. The objectives are usually coflicted, ad thus there is o caoical complete order o the solutio space X. The compariso betwee solutios relies o the domiatio relatioship, as preseted i Defiitio 2. A solutio is Pareto optimal if there is o other solutio i X that domiates it. The set of objective vectors of all the Pareto optimal solutios costitutes the Pareto frot. The goal of multi-objective optimizatio is to fid the Pareto frot, that is, to fid at least oe correspodig solutio for each elemet i the Pareto frot. I this paper, we cosider the Boolea space, i.e., X = {0, 1}. Defiitio 1 (Multi-Objective Optimizatio). Give a feasible solutio space X ad objective fuctios f 1,..., f m, multi-objective optimizatio ca be formulated as max x X ( f1 (x), f 2 (x),..., f m (x) ). Defiitio 2 (Domiatio). Let f = (f 1, f 2,..., f m ) : X R m be the objective vector. For two solutios x ad x X : 1. x weakly domiates x if, 1 i m, f i (x) f i (x ), deoted as x x ; 2. x domiates x if, x x ad f i (x) > f i (x ) for some i, deoted as x x. Evolutioary algorithms (EAs) have become a popular tool for multi-objective optimizatio, due to their populatio-based ature. I previous theoretical studies, GSEMO is the most widely used multi-objective EA (MOEA) [8,14,21]. As described i Algorithm 1, it first radomly selects a iitial solutio, the repeats the three steps (selectio, mutatio, acceptace) to improve the quality of the populatio. I selectio, a solutio is uiformly selected from the curret populatio; i mutatio, a ew solutio is geerated by flippig each bit of the selected solutio with probability 1 ; i acceptace, the ew solutio is compared with the solutios i the populatio, ad the oly o-domiated solutios are kept. Although simple, GSEMO explais the commo structure of various MOEAs, ad hece will be used i this paper as well. Selectio hyper-heuristics maage a set of low-level heuristics, ad select a appropriate oe to be applied at each decisio poit. Despite their practical successes, the theoretical aalysis is still i its ifacy, particularly for multi-objective optimizatio. I this paper, we take the three compoets of GSEMO, i.e., selectio, mutatio ad acceptace, as the low-level heuristic, respectively, ad compare the performace of GSEMO with a mixed strategy ad a pure strategy. For each compoet of GSEMO, we will use two cocrete low-level heuristics. A typical mixed strategy employed i our aalysis (deoted by GSEMO p ) is to use the first low-level heuristic with probability p [0, 1] i each iteratio of GSEMO, ad use the secod oe otherwise. Note that a mixed strategy correspods to usig heuristic selectio, while a pure strategy oly uses oe specific low-level heuristic ad thus implies that heuristic selectio is ot employed. The performace of the compariso algorithms is measured by their ruig time complexity. Note that ruig time aalysis has bee a leadig theoretical aspect for radomized search heuristics [2,20]. The ruig time of a MOEA is usually couted by the umber of fitess evaluatios (the most costly computatioal process) util fidig the Pareto frot [8,14,21].

4 Algorithm 1 GSEMO Give the solutio space X = {0, 1} ad the objective fuctio vector f, GSEMO cosists of the followig steps: 1: Choose x X uiformly at radom 2: P {x} 3: repeat 4: [Selectio] Choose x from P uiformly at radom 5: [Mutatio] Create x by flippig each bit of x with probability 1/ 6: [Acceptace] if z P such that z x 7: P (P {z P x z}) {x } 8: ed if 9: util some criterio is met 3 Mixig Selectio Mechaisms I this sectio, we use two fair selectio mechaisms [9,14] as low-level heuristics: fair selectio w.r.t. the decisio space: Each solutio i the curret populatio has a couter c 1, which records the umber of its offsprigs. The solutio with the smallest c 1 value will be selected for reproductio i each iteratio. That is, lie 4 of Algorithm 1 chages to be Choose x {y P c 1 (z) c 1 (y), z P } uiformly at radom. fair selectio w.r.t. the objective space: Each couter (deoted as c 2 ) is associated with a objective vector rather tha a decisio vector. Lie 4 thus chages to be Choose x {y P c 2 (f(z)) c 2 (f(y)), z P } uiformly at radom. Fairess is employed to balace the umber of offsprigs of all solutios i the curret populatio, ad thus to achieve a good spread over the Pareto frot. GSEMO with these two mechaisms are deoted by GSEMO ds ad GSEMO os, respectively. For GSEMO with the mixed strategy (deoted by GSEMO p ), it uses the fairess w.r.t the decisio space with probability p [0, 1] i each iteratio; otherwise, it uses the fairess w.r.t the objective space. Note that GSEMO ds ad GSEMO os are GSEMO p with p = 1 ad p = 0, respectively. We the compare their ruig time o the ZPLG fuctio. As show i Defiitio 3, ZPLG ca be divided ito three parts: ZeroMax, a plateau, ad a path with little gaps. It is obtaied from the PLG fuctio i [9] by replacig the secod objective value 1 i the ZeroMax part with 2. The Pareto frot is {(, 2), ( + 1, 1), ( , 0)}, ad the correspodig Pareto optimal solutios are 0, SP 1 ad 1, respectively. Defiitio 3 (ZPLG). ( x 0, 2) x / SP 1 SP 2 ZP LG(x) = ( + 1, 1) x SP 1 ( i, 0) x = 1 3/4+2i 0 /4 2i SP 2, where x 0 = j=1 (1 x j) deotes the umber of 0-bits, SP 1 = {1 i 0 i 1 i < 3/4}, SP 2 = {1 3/4+2i 0 /4 2i 0 i /8} ad = 8m, m N.

5 Theorem 1 shows that GSEMO with a pure strategy eeds expoetial ruig time with a high probability. The result of GSEMO os o ZPLG is directly from that o the PL fuctio (i.e., Theorem 1) i [9], sice ZPLG has the same structure as PL by treatig its SP 2 part as a whole. The iefficiecy is because GSEMO os allows the Pareto optimal solutio 0 to geerate ew solutios i SP 1, which stop the radom walk o the plateau SP 1 ad thus prevet from reachig SP 2. The result of GSEMO ds o ZPLG ca be directly from that o the PLG fuctio (i.e., Theorem 4) i [9], sice their proof relies o SP 1 ad SP 2, which are the same for ZPLG ad PLG. The iefficiecy is because GSEMO ds easily gets trapped i the radom walk o SP 1, which prevets from followig the path SP 2 to fid the Pareto optimal solutio 1. We the prove i Theorem 2 that by usig the mixed strategy, GSEMO p ca solve ZPLG i polyomial ruig time. The idea is that first employig GSEMO ds allows the radom walk o SP 1 to reach SP 2, ad the employig GSEMO os allows followig the path SP 2 to fid 1. Thus, we ca see that the advatage of usig heuristic selectio is that the stregths of oe heuristic ca compesate for the weakesses of aother. Theorem 1. O ZPLG, the ruig time of GSEMO ds is 2 Ω(1/2) with probability 1 2 Ω(1/2), ad that of GSEMO os is 2 Ω(1/4) with probability 1 e Ω(1/3). Theorem 2. The expected ruig time of GSEMO p with p=1 1 3 o ZPLG is O( 6 ). Proof. We divide the optimizatio process ito two phases: (1) starts after iitializatio ad fiishes util the populatio P cotais 0, a solutio from SP 1 ad a solutio from SP 2 ; (2) starts after phase (1) ad fiishes util P cotais 0, a solutio from SP 1 ad 1, i.e., the Pareto frot is foud. For the first phase, we ca follow the aalysis of GSEMO ds o PL (i.e., Theorem 2) i [9]. I their proof, the oly part relyig o the fair selectio w.r.t. the decisio space is to allow a cosecutive radom walk of legth δ 3 (δ is a costat) o the plateau SP 1, uder the coditio that the c 1 value of the maitaied solutio from SP 1 is always smaller tha that of the Pareto optimal solutio 0. Note that the fair selectio w.r.t. the decisio space is used with probability 1 1/ 3 i each iteratio of GSEMO p. Such a radom walk happes with probability (1 1 ) δ3 (2e) δ Ω(1). Thus, the 3 asymptotic ruig time is ot affected, ad the expected ruig time of this phase is the same as that of GSEMO ds o PL, i.e., O( 3 log ). For the secod phase, the populatio P always cotais three solutios, 0, a solutio from SP 1 ad a solutio from SP 2. The probability that a better solutio from SP 2 is foud uder the coditio that a solutio from SP 2 has bee selected for mutatio 1 is at least (1 1 2 ) 2 1 e, sice it is sufficiet to flip the leftmost two 0-bits. It 2 is easy to see that at most 8 such improvemets are sufficiet to fid the Pareto optimal solutio 1. The worst case is reached whe the first foud solutio from SP 2 is 1 3/4 0 /4. We cosider that the fair selectio w.r.t. the objective space is used, which happes with probability 1 i each iteratio of GSEMO 3 p. Because the c 2 values of (, 2) ad ( + 1, 1) (i.e., the objective vectors of 0 ad the solutio from SP 1 ) are ever decreased, the solutio from SP 2 is selected for reproductio at least oce i three cosecutive iteratios. Thus, the expected ruig time of this phase is at most e2 O( 6 ).

6 4 Mixig Mutatio Operators I this sectio, we use two mutatio operators [15] as low-level heuristics: oe-bit mutatio: Lie 5 of Algorithm 1 chages to be Create x by flippig oe radomly chose bit of x. Note that oe specific bit is chose with probability 1. two-bit mutatio: Lie 5 of Algorithm 1 chages to be Create x by flippig two differet ad radomly chose bits of x. Note that two specific bits are chose with probability 1/ ( ) 2 = 2 ( 1). GSEMO with these two operators are deoted by GSEMO 1b ad GSEMO 2b, respectively. GSEMO with the mixed strategy (deoted by GSEMO p ) uses oe-bit mutatio with probability p [0, 1] i each iteratio; otherwise, it uses two-bit mutatio. We the compare their ruig time o the SPG fuctio. As show i Defiitio 4, SPG has a short path SP with icreasig fitess except the solutios 1 i 0 i with i mod 3 = 1. The costructio of SPG is ispired from the GapPath fuctio i [15]. The Pareto frot is {(, 1), ( 2, 0)}, ad the correspodig Pareto optimal solutios are 0 ad 1, respectively. Defiitio 4 (SPG). ( x 0, 1) x / SP SP G(x) = ( 1, 0) x = 1 i 0 i SP, i mod 3 = 1 (i, 0) x = 1 i 0 i SP, i mod 3 = 0 or 2, where SP = {1 i 0 i 1 i } ad = 3m, m N. The followig two theorems show that the expected ruig time of GSEMO with a pure strategy is ifiite while that of GSEMO with the mixed strategy is polyomial. The proof idea is straightforward. I every three adjacet solutios o the path SP, there is a bad oe 1 i 0 i with i mod 3 = 1. Usig oe-bit ad two-bit mutatio alteratively ca jump over those bad solutios o SP ad fially reach the Pareto optimal solutio 1, while usig oly oe-bit or two-bit mutatio obviously will get stuck i some solutio 1 i 0 i with i mod 3 = 0 or 2. Theorem 3. The expected ruig time of GSEMO 1b ad GSEMO 2b o SPG is ifiite. Proof. We cosider that the iitial solutio is the Pareto optimal solutio 0, which has the objective vector (, 1). This happes with probability 1 2 due to the uiform samplig. For GSEMO 1b, oe-bit mutatio o 0 ca oly geerate solutios with the objective vector ( 1, 1) or ( 1, 0), which are domiated by 0. Thus, the populatio P will always cotai oly 0. For GSEMO 2b, two-bit mutatio o 0 geerates solutios with the objective vector ( 2, 1) or (2, 0). Thus, P cotais 0 ad after a while. Sice two-bit mutatio o caot geerate better solutios o SP, P will always keep i this state. Thus, startig from 0, either GSEMO 1b or GSEMO 2b caot fid the Pareto frot, which implies that the expected ruig time is ifiite. Theorem 4. The expected ruig time of GSEMO p with p [0, 1] beig a costat o SPG is O( 3 ).

7 Proof. The populatio P cotais at most two solutios, because the secod objective of SPG has oly two values 0 ad 1. We first aalyze the expected ruig time util the Pareto optimal solutio 0 is foud. Let x deote the solutio with the secod objective value 1 i P. Such a solutio will exist i P after at most expected ruig time. This is because a solutio from SP ca geerate a offsprig solutio ot from SP by flippig the first 1-bit, which happes with probability at least 1 by either oe-bit or two-bit mutatio. Assume that the umber of 0-bits of x is j (j 1). It is easy to see that j caot decrease, ad it ca icrease by flippig oe 1-bit (but ot the last) usig oe-bit mutatio. Because the probability of selectig x for mutatio is at least 1 2 ad oe-bit mutatio is used with probability p, the probability of icreasig j by 1 i oe iteratio is at least 1 2 p j 1 for j 2 ad 1 2 p 1 for j = 1. Thus, the expected ruig time to fid 0 is at most 2 2 j=1 p( j 1) + 2 p O( log ). Whe fidig 0, we pessimistically assume that the solutio from SP has ot bee foud. Startig from 0, the solutio ca be foud by flippig the first two 0-2 bits usig two-bit mutatio. This happes with probability (1 p) ( 1), ad thus the expected ruig time is ( 1) 2(1 p) O(2 ). Oce a solutio from SP with i mod 3 1 has bee foud, usig oe-bit ad two-bit mutatio alteratively ca follow the path SP to fid the Pareto optimal solutio 1. If i mod 3 = 2, flippig its first 0-bit by oe-bit mutatio ca geerate a better solutio. This happes with probability 1 2 p 1. If i mod 3 = 0, flippig its first two 0-bits by two-bit mutatio ca geerate a better solutio. This happes with probability 1 2 (1 p) 2 ( 1). Sice 3 such two improvemets are sufficiet to fid 1, the expected ruig time is at most 3 ( 2 p + ( 1) (1 p) ) O(3 ). Thus, the theorem holds. 5 Mixig Acceptace Strategies I this sectio, we use two acceptace strategies as low-level heuristics: elitist acceptace: As lies 6-8 of Algorithm 1, oly o-domiated solutios are kept i the populatio, ad the existig solutio i P with the same objective vector as the ewly geerated solutio will be replaced. strict elitist acceptace: It is the same as elitist acceptace, except that the old solutio with the same objective vector as the ewly geerated solutio will ot be replaced. That is, lie 6 of Algorithm 1 chages to be if z P such that z x. The differece betwee these two strategies is to accept or reject the solutio with the same fitess. This has bee theoretically show to have a sigificat effect o the performace of EAs i sigle-objective optimizatio [13]. Note that GSEMO with elitist acceptace is just GSEMO. GSEMO with the strict strategy is deoted by GSEMO strict. I each iteratio of GSEMO with the mixed strategy (deoted by GSEMO mixed ), elitist acceptace is used if the ewly geerated solutio x ad the paret solutio x have the same objective vector; otherwise, strict elitist acceptace is used. Note that the mixed strategy employed here is differet from that of GSEMO p. We the compare their ruig time o the PL fuctio. As show i Defiitio 5, PL has a short path SP {1 } with costat fitess. The Pareto frot is {(, 1), ( + 2, 0)}, ad the correspodig Pareto optimal solutios are 0 ad 1, respectively.

8 Defiitio 5 (PL). [8] ( x 0, 1) x / SP = {1 i 0 i 1 i } P L(x) = ( + 1, 0) x {1 i 0 i 1 i < } ( + 2, 0) x = 1. Theorem 5 shows that GSEMO with a pure strategy o PL eeds expoetial ruig time. The result of GSEMO was proved i [8], ad its iefficiecy is because a solutio ot from SP ca geerate a ew solutio from SP, which stops the ogoig radom walk o SP. The iefficiecy of GSEMO strict is because the first foud solutio from SP is far from the Pareto optimal solutio 1, ad strict elitist acceptace does ot allow the radom walk o SP. We the prove i Theorem 6 that GSEMO with the mixed strategy ca solve PL i polyomial ruig time. It works by allowig acceptig the solutio with the same fitess oly i the radom walk procedure. Theorem 5. O PL, the ruig time of GSEMO is 2 Ω(1/24) with probability 1 e Ω(1/24) [8], ad that of GSEMO strict is Ω( 5 ) with probability 1 2 Ω(). Proof. The iitial solutio is ot i SP with probability 1 2 due to uiform selectio, ad it has at most bits with probability 1 e Ω() by Cheroff bouds. The populatio P cotais at most two solutios, sice the secod objective of PL has oly two differet values. Note that the umber of 1-bits of the solutio ot from SP will ever icrease, sice the first objective is to maximize the umber of 0-bits. Because the probability of flippig at least 12 bits simultaeously i oe step is less tha 12, the first foud solutio from SP has at most bits with probability at least Oce a solutio from SP {1 } has bee foud, it will ever chage because SP {1 } is a plateau ad GSEMO strict will ot replace the solutio with the same fitess. Thus, P will always cotai two solutios, a solutio x from SP {1 } with x ad a solutio y ot from SP with y The probabilities of mutatig x ad y to 1 i oe step are at most 4 ad 3, respectively. Thus, after 5 steps, the Pareto optimal solutio 1 is geerated with probability at most 5 4 = 20 by the uio boud. By combiig all the above probabilities, we get that the ruig time is Ω( 5 ) with probability 1 2 Ω(). Theorem 6. The expected ruig time of GSEMO mixed o PL is O( 3 ). Proof. Sice the fuctio PL outside SP has the same structure as OeMax, the expected steps to fid 0 is O( log ) by usig the aalysis result of the (1+1)-EA o OeMax [7]. The, it eeds O() expected steps to fid a solutio from SP, as it suffices to flip the leftmost 0-bit of 0. For GSEMO mixed, if a offsprig solutio from SP is geerated by mutatio o the solutio ot from SP, it will ot replace the solutio from SP i the curret populatio; but if it is geerated by mutatio o the curret solutio from SP, the replacemet will be implemeted. Thus, the algorithm will perform the radom walk o the plateau SP ad the solutio ot from SP will ot ifluece it. Note that the solutio from SP is selected for mutatio with probability 1 2. Usig the aalysis result of the (1+1)-EA o SPC (i.e., Theorem 7) i [13], we get that the radom walk eeds O( 3 ) expected ruig time to fid 1.

9 Estimated ERT p=1/ p=0.5 mixed Problem size Figure 1. Estimated ERT of GSEMO mixed ad GSEMO p for solvig the PL problem, where a base 10 logarithmic scale is used for the y-axis. Note that the mixed strategy employed by GSEMO mixed here is differet from that by GSEMO p for mixig selectio mechaisms or mutatio operators. GSEMO p uses the first low-level heuristic with probability p [0, 1] i each iteratio ad uses the secod oe otherwise. To ivestigate the ifluece of differet mixed strategies, we coduct experimets to compare GSEMO mixed with GSEMO p for mixig elitist ad strict elitist acceptace. The parameter p is set as 1, 0.5 ad 1 1, respectively. For each compariso algorithm o each problem size {5, 10,..., 50}, we ru the algorithm 100 times idepedetly, where each ru stops whe the Pareto frot of the PL problem is foud. The average umber of fitess evaluatios is used as the estimatio of the expected ruig time (ERT). The result is plotted i Figure 1. Note that the ERT of GSEMO p for 20 is too large to estimate. We ca observe that GSEMO mixed is much more efficiet tha GSEMO p. The curves of GSEMO mixed ad GSEMO p grow i a closely logarithmic ad liear tred, respectively, which implies that their ERT is approximately polyomial ad expoetial, respectively. Thus, these empirical results suggest that choosig a proper threshold selectio method is importat. 6 Coclusio This paper presets a theoretical study o the effectiveess of selectio hyper-heuristics for multi-objective optimizatio. Rigorous ruig time aalysis showed that applyig selectio hyper-heuristics to ay of the three major compoets of a MOEA, i.e., selectio, mutatio ad acceptace, ca expoetially speed up the optimizatio. From the aalysis, we fid that selectio hyper-heuristics work by allowig the stregths of oe heuristic to compesate for the weakesses of aother. Our result provides theoretical support for multi-objective selectio hyper-heuristics. The empirical compariso o differet mixed strategies also implies the importace of choosig a proper heuristic selectio method. Refereces 1. Alaazi, F., Lehre, P.K.: Rutime aalysis of selectio hyper-heuristics with classical learig mechaisms. I: Proceedigs of CEC 14. pp Beijig, Chia (2014)

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