Multi-agent system based on self-adaptive differential evolution for solving dynamic optimization problems

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1 Mult-agent system based on self-adaptve dfferental evoluton for solvng dynamc optmzaton problems Aleš Čep Faculty of Electrcal Engneerng and Computer Scence Unversty of Marbor ABSTRACT Ths artcle presents the mult-agent system based on a selfadaptve dfferental evoluton algorthm for solvng dynamc problems. Characterstcs of dynamc problems are that objectve functon, wth whch the qualty of solutons s evaluated, changes over tme. These changes can occur ether after some predefned number of generatons or, more commonly, n each generaton, where the onlne responses are desred by the evolutonary algorthms. In ths study, the former knd of problems were taken nto consderaton and were solved usng mult-agent systems. Each agent n the systems s mplemented as a self-adapted dfferental evoluton wth constant populaton sze. In our comparatve study, multagent systems consstng of varous populaton szes n combnaton wth varous number of agents were compared, by solvng the benchmark functons provded for CEC 09 Competton on Dynamc Optmzaton wth respect to the same number of the ftness functon evaluaton. The man obstacle when usng smaller populatons s to prevent the stagnaton of the populaton. That was partally overcome by usng addtonal mechansms such as: dversty measurement and nformaton sharng between agents. As a result, the most approprate mult-agent system was searched for, and the results of the best found mult-agent system were compared wth the state-of-the-art algorthms. Keywords Dfferental evoluton, self-adaptve dfferental evoluton, mult-agent system, dynamc problems 1. INTRODUCTION Many of real-world problems are dynamc n ther nature. Dynamc optmzaton problems (DOP) [4] can be descrbed as problems, where decson maker has to make multple decsons over tme and the overall performance depends on all decsons made n that tme. Decsons are made sequentally over tme, but t depends on the type of problems, f ths decsons are made by tme trggered (perodc) or event Iztok Fster Faculty of Electrcal Engneerng and Computer Scence Unversty of Marbor ztok.fster@um.s trggered events. Consequently, the value of a ftness functon that s optmal at some tme, s not necessary optmal at another and vce versa. Dfferental evoluton (DE) was proposed by Storn and Prce n [11] and s a powerful evolutonary algorthm (EA) frequently used for solvng global optmzaton problems. Its smplcty and effectveness have been proven n many realworld applcatons. DE emulates natural evoluton process usng three operators: mutaton, crossover and selecton. Brest et al [1] proposed jde algorthm whch extends the orgnal DE algorthm wth self-adaptaton of DE control parameters,.e., the scale rate and the frequency of crossover rate. Ths self-adaptaton s performed before generatng the tral vector and therefore t nfluences creaton of new tral vectors. These controls parameters are added to a representaton of ndvduals and undergo operaton of varaton operators (.e., mutaton and crossover) durng the evolutonary cycle. Morrson [8] presented new EA archtecture for solvng DOP where he uses so-called sentnels,.e., ndvduals n a populaton that are spread across search space whose locaton does not change. In ths way they can be used for detectng changes of envronment. Brest et al [2] presented multpopulaton jde (jde*) algorthm wth the ageng mechansm and the use of archve where they stored the current best ndvduals after the envronment change was detected. There was no nformaton sharng between ndvduals. Yang et al [13] proposed an algorthm for mprovng DE wth populaton adaptaton approach, where t calculates the standard devaton of ndvduals n j -th dmenson at generaton G. Halder et al [5] presented an algorthm CDDE Ar that uses a mult-populaton method where clusters (sub-populatons) are parttoned accordng to the spatal locatons of tral solutons. Durng the evoluton process, populaton s perodcally dvded n dfferent number of clusters whch causes certan nformaton sharng durng the optmzaton process. Lepagnot et al [6] presented MLSDO algorthm, whch s based on several coordnated local searches and on the archvng of the found local optma, n order to track them after a change n the objectve functon. Novoa et al [9] proposed an algorthm msqde, a mult-populaton algorthm wth self-adaptve strategy for controllng the populaton dversty and an nteracton mechansm between ndvduals. Ths paper proposes mult-agent system based on self-ada- StuCoSReC Proceedngs of the th Student Computer Scence Research Conference DOI: Ljubljana, Slovena, 11 October 35

2 ptve dfferental evoluton (MAS-jDE) for solvng DOP. These are a class of problems, where the envronment changes over tme. Autonomous agents that are capable of detectng these changes on ther own, try to solve DOP by usng jde and mantanng varous szed populatons. An evolutonary progress can typcally stagnate when usng agents wth smaller populatons. Therefore, we tred to preval ths problem by mplementng two mechansm: detecton of stagnaton by calculatng populaton s dversty and mgraton of ndvduals between agents. Experments wth varous confguratons of mult-agent systems were conducted, where the number of agents and the populaton szes were vared. Indeed, these both parameters were selected such that the algorthms n each confguratons spent the same number of the ftness functon evaluatons. In ths way, far comparson between dfferent MAS-jDE confguratons can be expected. Addtonally, the performance of the proposed algorthm MAS-jDE was also compared to the others state-of-the-art algorthms for solvng DOP. The remander of the paper s organzed n the followng way. Secton 2 descrbes the DOPs. Secton 3 descrbes the orgnal DE, ts extenson jde and the proposed MAS-jDE algorthm. Secton 4 llustrates the results of experments that were conducted and provdes dscusson on them. Fnally, Secton 5 concludes ths paper wth summarzng the performed work and outlnng drectons for the future work. 2. PROBLEM DEFINITION Ths paper s devoted to solvng DOPs. L et al. n [7] defned these knd of problems as follows: DOP = f(x, φ, t), (1) where DOP s the dynamc optmzaton problem, f s an evaluaton functon, x s a feasble soluton n the soluton set X, φ (so called change type) s a system control parameter determnng the soluton s dstrbuton n the ftness landscape and t s real-world tme. Changes n envronment can be results of a change of system control parameters descrbng envronment state or a change n evaluaton functon. These changes can occur ether after some predefned number of generatons or, more commonly, n each generaton, where the onlne responses are desred by the evolutonary algorthms. The goal of the algorthm for solvng DOP s to detect the change of envronment and fnd the new optmum. Problem frequently occurrng n evolutonary algorthms for DOP s a stagnaton of populaton. Evolutonary process moves populaton to or very near to the local or global optmum, whch causes a very small dversty of the populaton. When t comes to envronmental change, the populaton wth small dversty cannot explore search space effcently enough. Usually ths could mean, that the algorthm wll not be able to fnd the next global optmum. 3. THE PROPOSED ALGORITHM The proposed MAS-jDE algorthm combnes a knowledge from two domans: evolutonary algorthms (EAs) [3] and mult-agent systems (MAS) [12]. In lne wth ths, three algorthm are descrbed at frst,.e., dfferental evoluton (DE), followed by self-adaptaton dfferental evoluton (jde) that s an extenson of DE. Fnally, the proposed MAS-jDE s presented from the mult-agent aspect. 3.1 Dfferental evoluton DE [11] s a well establshed EA, developed by Storn and Prce [11]. It s a populaton-based algorthm, whose populaton s defned as a set of real-valued vectors representng dfferent solutons of the problem: x = x,1, x,2,..., x,d}, (2) for = 1,..., NP, where NP represents the populaton sze and D s the dmensonalty of the problem. Lke n other EAs, the frst step n DE s a random ntalzaton of the populaton. Untl the termnaton condton s satsfed, the populaton undergoes operaton of three evolutonary operators by creatng tral vectors: mutaton, crossover and selecton. Mutaton creates mutant vector v (G+1) from parent s populaton. Dfferent mutaton DE strateges have been used n lterature [10, 11], but n orgnal artcle [11] authors used the so-called DE/rand/1/bn : v (G+1) = x (G) r0 + F (x(g) r1 x(g) r2 ), (3) where r0, r1 and r2 represent random nteger numbers n range [1, NP], where t holds r0 r1 r2, and the scale factor s a real number n range F [0, 2]. Crossover operator ntroduces a lttle dversty n the populaton of solutons. It creates a tral vector u (G+1),j from a combnaton of elements from ether the mutaton vector or the correspondng parent vector as: v (G+1) u (G+1),j =,j f rand(0, 1) CR or j = j rand, otherwse, x (G),j where the crossover rate s defned n range CR [0, 1) and determnes a probablty of creatng a tral vector from the mutaton vector. Index j rand s randomly selected nteger defned n range [1, D] that s used to ensure that the algorthm makes at least one change n tral vector regardng prevous generaton. Selecton compares the results of ftness functon from newly created tral vector u (G+1),j and parent vector x (G),j. If the problem s mnmzaton problem, t s defned as follows: u (G+1) f f(u (G+1) ) f(x (G) ) x (G+1) = x (G) otherwse. The ftness functon depends on the problem to be solved. 3.2 Self-adaptve DE Brest et al. [1] presented self-adaptve dfferental evoluton algorthm (jde), where control parameters F and CR are added to representaton of ndvduals and adapted durng the evoluton process usng followng equatons: F l + rand 1 F u f rand 2 < τ 1, F (G+1) = F (G) CR (G+1) = otherwse, rand 3, f rand 4 < τ 2, CR (G) otherwse, (4) (5) (6) (7) StuCoSReC Proceedngs of the th Student Computer Scence Research Conference Ljubljana, Slovena, 11 October 36

3 where rand j, j [1 4] represents random number n range [0, 1] and F (G+1) s n range [0.1, 1]. Constants τ 1 and τ 2 are so-called learnng rates determnng the frequency of updatng control parameters F and CR, and were set to 0.1. Adaptaton of control parameters accordng to Eq. (6)-(7) s performed after 10 generatons n average, and therefore has a bg nfluence on mutaton and crossover operators. 3.3 Mult-agent system based on jde for solvng DOPs Mult-agent system based on jde for solvng DOPs (MASjDE) mplements system of multple autonomous agents, that run autonomously based on jde algorthm usng separated populatons of the same sze. Mult-agent systems are systems composed of multple nterfacng ntellgent agents. Each agent s a computatonally entty lke an algorthm that s placed n an envronment n order to acheve ts objectves [12]. In our case, each agent s devoted to search for a best soluton n own regon of the whole search space. Usng the nteracton between agents, they are able to enrch ther lmted knowledge wth nformatons obtaned from other agents n the MAS. There are four man ssues that must be solved when developng the algorthm: agent s populaton sze determnaton, envronment change detecton, mantanng the populaton dversty and nformaton exchange between agents. Accordng to our assumpton of farness, t holds that f system contans more agents, the populaton sze of each agent s decreased. Ths fact can cause a stagnaton problem, where algorthm gets stuck n a local optmum and can not get out, due to the lower populaton dversty. To avod ths problem, we ntroduce mechansm for measurng the populaton dversty that uses the followng prncple: When the dversty of populaton s nsgnfcant, the MAS-jDE begns mgraton of ndvduals between agents. The pseudo-code of the algorthm s llustrated n Algorthm 1. Algorthm 1 Algorthm MAS-jDE NP MAS: sum of populatons szes NP: sze of agent s populaton N: number of agents 1: NP = NP MAS/N 2: Generate and evaluate all agent s populatons 3: Update global best 4: whle the termnaton condton s not meet do 5: for each agent s populaton do 6: Update parameters F and CR (Eq. (6)-(7)) 7: Create tral populaton j wth DE operators mutaton and crossover 8: Evaluate j 9: f envronment change detected then 10: Generate and evaluate new populaton 11: else 12: Selecton between and j 13: Update global best 14: Check dversty and exchange ndvduals In the remander of the paper, the exposed ssues of the proposed MAS-jDE algorthm are dscussed n detals Agent s populaton sze determnaton In order to ensure that the comparson between dfferent MAS s as far as possble, we ntroduce the total populaton sze NP MAS, that determnes the number of vectors n the so-called grand populaton. Obvously, the MAS-jDE wth grand populaton s equvalent to the orgnal jde usng a sngle populaton. In MAS-jDE, the members of ths grand populaton are dvded among agents evenly. The hgher the number of agents n the MAS, the smaller s ther populaton sze. In lne wth ths, the agent s populaton sze NP s obtaned as: NP MAS NP = N, (8) where N represents the number of agents. Because all three mentoned varables have to be ntegers, not all combnatons of NP MAS and N are feasble Envronment change detecton An evolutonary cycle starts after a random ntalzaton and evaluaton of agent s populaton. The DE mutaton and crossover operate on NP 1 tral vectors only, because the last vector s used as a sentnel [8] that never changes. Thus, the sentnel s ftness functon value s changed wth the change of the envronment. Therefore, we frst check after the newly generated populaton s evaluated, f the ftness value of sentnel dstngushes from the old value. In ths case, we declare that the envronment has changed, consequently rentalze agent populaton randomly and evaluate the new ftness values for the whole populaton Mantanng the populaton dversty The stagnaton problem can be evdent at the populaton as whole or at the component level [13]. In MAS-jDE, we detect stagnaton by calculatng agent s populaton dversty usng the followng equatons [8, 13]: a (G) j = j=1 NP 1 dversty = D =1 x (G),j, (9) NP 1 NP 1 =1 (x (G),j a(g) j ) 2, (10) where a (G) j stores the mean of populaton n j th dmenson. j s n range [1, D], where D s a problem dmenson. x s an ndvdual n populaton. Dversty n populaton s checked after each generaton to detect stagnaton as quckly as possble. If t falls bellow threshold level, mgraton between agents starts. Threshold level must be set hgh enough, so that agent has enough dversty n populaton to contnue explorng search-space and low enough, so t can close-n to the optmum Informaton exchange between agents When dversty n populaton falls bellow threshold, mgraton between agents starts (Algorthm 2). Idea s as follows. Each agent begns exploraton n dfferent part of search space and therefore t s search s lmted to the specfc regon. When new vectors are mgrated from another agent, t s expected that the dversty of populaton wll ncrease. Thereby, the agent wll be attracted to explore the new promsng regons n the search space. Procedure StuCoSReC Proceedngs of the th Student Computer Scence Research Conference Ljubljana, Slovena, 11 October 37

4 of sharng nformaton between agents s descrbed n three steps, as follows: 1. step: The best vectors from all agents are found. 2. step: From these vectors, fnd the one that s the most dverse from the best vector n current agent based on Eucldan dstance (Eq. (11)). 3. step: Usng Eucldan dstance (Eq. (11)), fnd two vectors that are closest to each other n the current agent (Algorthm 3) and replace the worse based on ftness functon result, wth the vector found s step 2. Dstance between vectors s calculated as Eucldean dstance: d(p, q) = D (p j q j) 2, (11) j=1 where p and q represent D dmensonal vector. Algorthm 2 Calculate dversty and exchange vectors 1: for each agent a do 2: Calculate populaton dversty for a (Eq. (9, 10)) 3: f a.dversty < threshold then 4: bestvectors[] =BestVectorForEachAgent() 5: Fnd the most dverse vector from the best vector n a 6: r = IndexOfMostSmllar(a) 7: Replace vector on ndex r wth the most remote vector Algorthm 3 Algorthm that fnds vector to replace NP: sze of agent s populaton Agent s populaton s ordered by ftness value 1: functon IndexOfMostSmllar(a) 2: for = 0 to (NP - 2) do 3: for j = + 1 to (NP - 1) do 4: dstance = EucldanDstance(a[], a[j]) 5: f dstance < smallestdstance then 6: smallestdstance = dstance 7: nd = j 8: return a[nd] 4. EXPERIMENT AND RESULTS The goal of our expermental work was twofold. Frstly to test the MAS-jDE wth dfferent parameters settngs, and thus fnd the best settngs for parameters: the populaton sze NP and the number of agents N. Secondly to compare the results obtaned by the MAS-jDE wth the results of the other state-of-the-art algorthms. In both experments, all algorthms solved CEC 09 benchmark test functons for DOP. The algorthm MAS-jDE used the followng parameters durng experments: the total populaton sze was vared as NP MAS 50, 100, 200, 400}, the number of agents was set to values N 1, 5, 8, 10, 16, 20, 25}, from whch only the feasble (Table 1) were consdered, the scale sze was ntally drawn randomly from unform dstrbuton n nterval F (0) [0, 1], the crossover probablty was ntally drawn randomly from unform dstrbuton n nterval CR (0) [0, 1], the dversty threshold n Algorthm 2 was set to threshold = Dynamc optmzaton benchmark Algorthms n the study were tested on benchmark test sute provded by CEC 09 organzers for Competton on Dynamc Optmzaton [7], whch uses the generalzed dynamc benchmark generator. Benchmark defnes seven change types: small-step change (T 1), large-step change (T 2), random change (T 3), chaotc change (T 4), recurrent change (T 5), recurrent change wth nose (T 6), and random change wth changed dmenson (T 7). There are sx test functons defned n the real-space: F 1: Rotaton peak functon (wth 10 and 50 peaks) F 2: Composton of Sphere s functon F 3: Composton of Rastrgn s functon F 4: Composton of Grewank s functon F 5: Composton of Ackley s functon F 6: Hybrd Composton functon In summary, there are 49 specfc test cases and each test case contans 60 changes. Test cases run ndependently 20 tmes to obtan the mark on specfc case. Overall performance perf s the sum of the performance measures obtaned for each functon and each change type as follows: perf = 49 =1 mark 100, (12) where the performance measure mark s defned as follows: mark = w R C R l=1 j=1 C r lj, (13) where w denotes weght of the problem (see [7] for more detals), R the number of runs and C s the number of changes (n our case R = 20 and C = 60). The varable r lj s expressed as: r lj = r last lj /(1 + S (1 rlj)/s), s (14) s=1 where rlj last denotes the relatve rato between ftness value of best ndvdual and global optmum of the j-th envronment, rlj s the relatve rato between ftness value of best ndvdual and global optmum at the s-th samplng durng one change, and S s total number of samples for each envronment. StuCoSReC Proceedngs of the th Student Computer Scence Research Conference Ljubljana, Slovena, 11 October 38

5 4.2 Computer confguraton The computer used for expermental work had the followng confguraton: System: Wndows 10 CPU: Intel Core RAM: 16 GB Programmng language: C Results As mentoned n Secton 3.3.1, not all combnatons of parameters NP MAS and N are feasble. Therefore, n the frst experment, the followng feasble combnatons as presented n Table 1 were used n our study. Table 1: Feasble combnatons of agent s number N. NP MAS N Values n the table present agent s populaton sze NP and must be nteger. When the results of dvson of NP MAS by N s not an nteger value, ths combnaton of parameters s declared as nfeasble. The nfeasble values are presented bold n the table. Value n row NP MAS = 50 and column N = 25 s also nfeasble, because mutaton strategy DE/rand/1/bn n DE (Eq. (3)) requres at least 4 ndvduals n populaton. Table 2 shows the overall performance perf (Eq. (12)) obtaned n the frst experment by the MAS-jDE algorthm for all combnatons of NP MAS and N. Hgher values n table present the better results. Non-vald combnatons of parameters are marked wth n/a n the table. Thus, the last row depcts the average overall performance values accordng to the number of agents (N), where the nfeasble values of populaton szes were gnored. The hghest performance value perf = was acheved, when the grand populaton sze was NP MAS = 200 and the MAS conssted of N = 10 agents. The last row wth average performance results shows, that the MAS-jDE consstng of 8 agents returned the best results. However, ths result was based only on two nstances. Obvously, when all four overall performance values are taken nto consderaton, the MASjde usng 5 agents has shown to be the best. The results n Table 2 also show the stagnaton effect n algorthm, although both features,.e., the dversty measurements and mgraton of ndvduals were actvated. Consequently, the worst results were acheved, when the grand populaton was dvded between large number of agents. Performance also worsened, when the number of agents n the MAS wth the sngle populaton ncreased. Results from Table 2 are llustrated n Fgure 1, comparng the mult-agent systems of dfferent total populaton szes Overall performance NP MAS =050 NP MAS =100 NP MAS =200 NP MAS = Number of agents Fgure 1: Comparson of varous populaton szes and dfferent number of agents accordng to overall performance results of CEC 09 benchmark. It can be observed that the best result of MAS-jDE was obtaned by the MAS wth the total populaton sze of NP MAS = 200. Thus, t s evdent that the total populaton sze of NP MAS = 50 s napproprate for buldng MAS, because of nsuffcent populaton sze n agent. In general, the MASs wth hgher total populaton szes are more convenent for achevng the best results. Table 3 presents more detaled results for the best soluton of MAS-jDE, wth N = 10 agents and the total populaton sze of NP MAS = 200. The best results were obtaned by solvng the Ackley s functon (F 5) for all 7 change types. Good results were acheved for all functons at small-step (T 1) and chaotc (T 4) change. Algorthm performed very well for functons F 2 and F 4 for recurrent change wth nose (T 6) and for functon F 6 wth random change (T 3), recurrent change (T 5) and recurrent change wth nose (T 6). In the second experment, the soluton wth hghest overall performance perf (Eq. (12)) found n the frst experment, was compared to the DOP DE, DOP jde (.e., equvalent to sngle agent MAS-jDE) and the followng stateof-the-art algorthms: jde* [2], CDDE Ar [5], MLSDO [6] and msqde- [9]. Algorthms DOP DE and DOP jde are orgnal DE [11] and jde [1] algorthms that were adopted to solve DOP, by randomly rentalzng populaton when change n envronment s detected. The results for algorthms DOP DE and DOP jde were contrbuted usng our own mplementatons of the algorthms, whle the results for others were obtaned from ther respectve lterature. The comparatve study of mentoned algorthms was made accordng to the overall performance values perf for 49 test cases of CEC 09 benchmark functons. The results are depcted n Table 4 and show that the MAS-jDE algorthm acheved better results than the DOP DE, DOP jde, jde* and CDDE Ar algorthms, whle the msqde- and MLSDO StuCoSReC Proceedngs of the th Student Computer Scence Research Conference Ljubljana, Slovena, 11 October 39

6 Table 2: Overall performance by total populaton szes and number of agents NP MAS N n/a n/a n/a n/a n/a n/a n/a Avg Table 3: Algorthm MAS-jDE overall performance for the best soluton (NP MAS = 200 and N = 10) F 1(10) F 1(50) F 2 F 3 F 4 F 5 F 6 T T T T T T T Mark Performance (summed the mark obtaned for each case and multpled by 100): algorthms outperformed t. Table 4: Overall performance comparson Algorthm Performance MAS-jDE DOP DE DOP jde jde* CDDE Ar MLSDO msqde CONCLUSIONS In ths paper we proposed MAS-jDE algorthm that s a MAS, where agents ndependently of each other explore the search space usng the jde. To prevent stagnaton of the agent s populatons, thers populaton dversty s measured. If dversty falls bellow threshold, mgraton mechansm between agents takes place. Algorthm was tested on CEC 09 benchmark functons for DOP. In lne wth ths, two experments were conducted. In the frst, we compared mult-agent systems wth dfferent confguratons, where the populaton szes and the number of agents were vared. The results were compared accordng to the benchmark s overall performance measure. The best soluton was found when the total populaton sze was 200 and the number of agents was 10. From the results of the experments we can observe, that although we used dversty measurements and mgraton of ndvduals, stagnaton of populaton appears n combnatons, where agent s had smaller populaton szes. The results of the MAS-jDE wth the best parameter settng from the frst experment were also compared wth other state-of-the-art optmzaton algorthms. The MAS-jDE showed much better results then DOP DE and DOP jde. Smlar, but better results were acheved aganst jde* and CDDE Ar. Smlarty between MAS-jDE and JDE* results can be contrbuted to the fact that both algorthms use the same self-adaptve algorthm to search for a global optmum. Our algorthm was outperformed by MLSDO and msqde-. Future plan s to study the results of dversty measurements and consequences of mgratons of ndvduals more closely, and add mechansms lke k-means clusterng algorthm to spread ntal populaton more equally over search-space. Intal tests show promsng results. 6. REFERENCES [1] J. Brest, S. Grener, B. Boškovć, M. Mernk, and V. Žumer. Self-adaptng control parameters n dfferental evoluton: A comparatve study on numercal benchmark problems. IEEE Transactons on Evolutonary Computaton, 10(6): , [2] J. Brest, A. Zamuda, B. Boškovć, M. S. Maučec, and V. Žumer. Dynamc optmzaton usng self-adaptve dfferental evoluton. In Evolutonary Computaton, CEC 09. IEEE Congress on., pages IEEE, May [3] A. Eben and J. E. Smth. Introducton to Evolutonary Computng. Natural Computng Seres. Sprnger-Verlag Berln Hedelberg, 1 edton, [4] H. Fu, P. R. Lews, B. Sendhoff, K. Tang, and X. Yao. What are dynamc optmzaton problems? In Evolutonary Computaton (CEC), 2014 IEEE Congress on, [5] U. Halder, S. Das, and D. Maty. A cluster-based dfferental evoluton algorthm wth external archve for optmzaton n dynamc envronments. IEEE transactons on cybernetcs, 43(3): , June [6] J. Lepagnot, A. Nakb, H. Oulhadj, and P. Sarry. A multple local search algorthm for contnuous dynamc optmzaton. Journal of Heurstcs, 19(1):35 76, Feb StuCoSReC Proceedngs of the th Student Computer Scence Research Conference Ljubljana, Slovena, 11 October 40

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