Benchmarking a Hybrid Multi Level Single Linkage Algorithm on the BBOB Noiseless Testbed

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1 Benchmarking a Hyrid ulti Level Single Linkage Algorithm on the BBOB Noiseless Tested László Pál Sapientia - Hungarian University of Transylvania 00 iercurea-ciuc, Piata Liertatii, Nr., Romania pallaszlo@sapientia.siculorum.ro ABSTRACT ulti Level Single Linkage () is a well known stochastic gloal optimization method. In this paper, a new hyrid variant (H) of the algorithm is presented. The most important improvements are related to the sampling phase: the sample is generated from a Sool quasi-random sequence and a few percent of the population is further improved y using crossover and mutation operators like in a traditional differential evolution () method. The aim of this study is to evaluate the performance of the new H algorithm on the tested of noiseless functions. The new algorithm is also compared against a simple and a traditional in order to identify the enefits of the applied improvements. The results confirm that the H outperforms the and methods. The new method has a larger proaility of success and usually is faster especially in the final stage of the optimization than the other two algorithms. Categories and Suject Descriptors G.. [Numerical Analysis]: Optimization gloal optimization, unconstrained optimization; F.. [Analysis of Algorithms and Prolem Complexity]: Numerical Algorithms and Prolems General Terms Algorithms Keywords Benchmarking, Black-ox optimization, ulti level methods, Differential evolution. INTRODUCTION The ulti Level Single Linkage () [] method has een derived from clustering [] methods which enale the exploration of the whole feasile region through random Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distriuted for profit or commercial advantage and that copies ear this notice and the full citation on the first page. To copy otherwise, to repulish, to post on servers or to redistriute to lists, requires prior specific permission and/or a fee. GECCO Companion, July 0, 0, Amsterdam, The Netherlands. Copyright 0 AC //07...$.00. sampling followed y local search methods. It is considered one of the est known and efficient stochastic algorithm for gloal optimization prolems of moderate size of dimensions. A similar clustering type algorithm [] achieved good results [] on the BBOB-009 functions with moderate numer of local minima using a small udget of function evaluations. In this paper, we introduce a new hyrid variant of the method denoted y H. The most important improvements are related two the sampling phase: the sample is generated from a Sool quasi-random sequence [7] and a few percent of the sample is further improved using crossover and mutation operators like in a traditional differential evolution () [] method. The purpose of this paper is to evaluate the performance of the H algorithm using the COCO framework [] and to assess the enefits of the introduced improvements. We also compare the H method against a simple and a traditional method. The rest of this article is organized as follows. Section reviews the algorithm, and also presents the new hyrid version of the and methods. In Section, we descrie the experiment design together with the algorithms parameter settings. The results are presented in Section and discussed in Section. Section concludes the paper and points out some directions for future work.. ALGORITH PRESENTATION Similarly to the clustering methods, has two phases: a gloal and a local one. The gloal phase consists of sampling, while the local phase is ased on local searches. The local minimizer points are found y means of a local search procedure (LS), starting from appropriately chosen points from the sample drawn uniformly within the set of feasiility. The local search procedure is applied to every sample point from the reduced sample, except if there is another sample point within some critical distance r k, which has a lower function value (see Algorithm ). The reduced sample consists of the γkn est points (0<γ ) from the cumulated sample x,..., x kn. The critical distance will e chosen to depend on kn only so as to minimize the proailities of two possile failures of the method: the proaility that a local search is started, although the resulting minimum is known already, and the proaility that no local search is started in a level set which contains reduced sample points. The critical distance is given y the following formula r k (x) = (Γ( + n /n ζ ln(kn) ) m(x) π kn ),

2 Algorithm : The algorithm X ; k 0 repeat k k + Generate N points x (k )N+,..., x kn with uniform distriution on X. Determine the reduced sample (X r) consisting of the γkn est points from the cumulated sample x,..., x kn. for i to length(x r ) do 7 if NOT (there is such a j that f(x j ) < f(x i ) and x j x i < r k ) then 8 Start a local search method (LS) from x i. 9 x LS(x i ) 0 X X {x } until Some gloal stopping rule is satisfied. return The smallest local minimum value found. where Γ is the gamma function, n is the numer of variales of the prolem, m(x) is the Leesgue measure of the domain X, kn is the total numer of sampled points, k is the iteration counter and ζ is some positive constant. The algorithm continues repeating the gloal and local phases until some stopping rule is satisfied. It has een proved that the algorithm has good asymptotic properties (depending on the ζ value): the asymptotic proailistic correctness and proailistic guarantee of finding all local minimizers. In our calculations the parameter ζ was taken to e. Based on the presented method, we introduced some improvements which are mainly related to the gloal step of the algorithm. Low-discrepancy sequences have een used instead of purely random samples. We use sample points from Sool quasi-random sequences [7] which fill the space more uniformly. Sool low-discrepancy sequences are superior to pseudorandom sampling especially for low and moderately dimensional prolems [8]. Furthermore a few percent of the sample points are improved y using crossover and mutation operators similar to those used in the method. In other words, a few iterations are applied to the est points of the actual sample. This last step is executed in each iteration efore the local phase of the optimization. The aim of these improvements are to help the method to overcome the difficulties arising in prolems with a large numer of local optima or in the cases when the local search method cannot make further improvements.. EXPERIENTAL PROCEDURE The main purpose of the experiment is to assess the enefits of the improvements applied to the method. Thus we compare the three algorithms on the noiseless function tested. Each of the algorithms was run on instances of all the functions in dimensions,,, 0, and 0. The evaluations udget was set to 0 D for each run. The applied udget is enough to capture all relevant features of the three algorithms. has four parameters to set: the numer of sample points in an iteration, the size of the reduced sample, the maximum numer of function evaluations for local search, and the used local search procedure. The sample size in each iteration was set to 0D, while the size of the reduced sample to D. This latter setting is also motivated y the same population size of the method (see elow). On the whole tested we use the ATLAB s fmincon local search method in all dimensions. fmincon is an interior-point algorithm for constrained nonlinear prolems which approximates the gradient using the finite difference method and ased on a recent study [9], it performed well on most of the test functions. The maximum numer of function evaluations for local search was set to 0% of the total udget, while the termination tolerance parameter value was set to 0. The H method posses the same parameter settings as the algorithm. Additionally we apply D iterations to the reduced sample. The iterations numer was selected after a small systematic study and provides a good alance etween the two methods. The population size for was set to D while the crossover and mutation rates to 0.. Similar population size was also applied in [0]. The crossover strategy is the exponential one and the mutation operator comines the est memer with other two randomly chosen individuals.. RESULTS Results from experiments according to [] on the enchmark functions given in [, ] are presented in Figures, and and in Tales and. The expected running time (ERT), used in the figures and tale, depends on a given target function value, f t = f opt + f, and is computed over all relevant trials as the numer of function evaluations executed during each trial while the est function value did not reach f t, summed over all trials and divided y the numer of trials that actually reached f t [, ]. Statistical significance is tested with the rank-sum test for a given target f t (0 8 as in Figure ) using, for each trial, either the numer of needed function evaluations to reach f t (inverted and multiplied y ), or, if the target was not reached, the est f-value achieved, measured only up to the smallest numer of overall function evaluations for any unsuccessful trial under consideration.. CPU Timing Experiments The three algorithms were run on the test function f 8, and restarted until at least 0 seconds had passed. These experiments were carried out on a machine with Intel Dual- Core processor,. Ghz, with GB RA, on Windows 7 it in ATLAB R0 it. The average time per function evaluation in,,, 0, 0, 0 dimensions was aout, 9.,.7,.0,.9,.7 0 s for H, aout, 8.9,.9,.,.9,.7 0 s for, and aout.,.,.,.,.,. 0 s for.. DISCUSSION As a result of the hyridization the HLS method is usually etter than the algorithm in terms of the ERT needed to find the f = 0 8. oreover H is significantly faster than on the f, f, f 7, f 0, f, f, f, f, f 7, and f 8 functions (see Figure ). Compared to the method, H is significantly faster on the f, f 8, f 9, f, and f functions. Considering the proportion of solved instances we can state that the new H method inherits the speed of

3 the simple algorithm on the initial phase of the optimization, while the use of the method inside the provides a etter performance in the final stage. In -D (see Figure ), the general aspect is that the H method is as fast as the algorithm in the initial stage (#FEs < 00D) of the optimization, while in the middle and final phases (#FEs > 00D) it is usually faster than the and methods. These properties can e nicely followed in the figure with all functions aggregated and on the multi-modal functions sugroup. On the weakly structured multi-modal functions the is slightly faster than H in the initial stage (00D<#FEs < 700D) of the optimization. After 700D evaluations the H method ecomes the leader and up to the final udget solves around 78% of the prolems. As a result of the hyridization, the H method is significantly etter then the on the separale, moderate and multi-modal function sugroups. This increase is caused y solving the f, f, f, f 7, f 8, and f 9 functions, where was ale to solve only the prolems with loose target levels. In the 0-D space, similar aspects can e oserved as in -D (see Figure ). Considering all functions aggregated for larger udgets than 0 D, H is the est algorithm, solving almost 70% of the prolems, followed y, and solving aout 8%, and % of the prolems, respectively. Significant improvements can e oserved on the moderate functions sugroup where H solved 00% of the prolems, followed y and (7% and 70%, respectively). This is due to the one solved instance of the f 7 function y H. The lowest percentage (aout %) of the solved prolems y H can e oserved on the multi-modal functions sugroup. This is due to the difficulties of the and methods on these functions. On the ill-conditioned and weakly structured functions the method is slightly faster in the middle stage of the optimization, while on the moderate and ill-conditioned function sugroups the H method (with ) is even faster than the est algorithm from BBOB-009 on the initial phase (D < #FEs < 00D) of the optimization.. CONCLUSIONS We enchmarked the H algorithm, a hyrid version of the classic and methods. The new hyrid algorithm differs from in that it applies a few iterations in the gloal phase. The new algorithm was extensively compared with the and methods on the tested of noiseless functions in order to reveal the enefits of the new improvements. The results show that the H outperforms the and methods. The new method has a larger success proaility and is as fast as the method in the initial and middle phase while in the final stage of the optimization it is usually faster than the other two algorithms. Further improvements y using adaptive remains to e investigated as a future work. 7. REFERENCES [] C. G. E. Boender, A. H. G. Rinnooy Kan, G. T. Timmer, and L. Stougie. A stochastic method for gloal optimization. athematical Programming, : 0, 98. [] T. Csendes, L. Pál, J.-O. H. Sendín, and J. R. Banga. The GLOBAL optimization method revisited. Optimization Letters, ():, 008. [] S. Finck, N. Hansen, R. Ros, and A. Auger. Real-parameter lack-ox optimization enchmarking 009: Presentation of the noiseless functions. Technical Report 009/0, Research Center PPE, 009. Updated Feruary 00. [] N. Hansen, A. Auger, S. Finck, and R. Ros. Real-parameter lack-ox optimization enchmarking 0: Experimental setup. Technical report, INRIA, 0. [] N. Hansen, A. Auger, R. Ros, S. Finck, and P. Pošík. Comparing results of algorithms from the lack-ox optimization enchmarking BBOB-009. In GECCO 0: Proceedings of the th annual conference comp on Genetic and evolutionary computation, pages 89 9, New York, NY, USA, 00. AC. [] N. Hansen, S. Finck, R. Ros, and A. Auger. Real-parameter lack-ox optimization enchmarking 009: Noiseless functions definitions. Technical Report RR-89, INRIA, 009. Updated Feruary 00. [7] H. S. Hong and F. J. Hickernell. Algorithm 8: Implementing Scramled Digital Sequences. AC Transactions on athematical Software, 9:9 09, 00. [8] S. Kucherenko and Y. Sytsko. Application of Deterministic Low-Discrepancy Sequences in Gloal Optimization. Computational Optimization and Applications, 0:97 8, 00. [9] L. Pál, T. Csendes,. C. arkót, and A. Neumaier. Black-ox optimization enchmarking of the GLOBAL method. Evolutionary Computation, 0:09 9, 0. [0] P. Pošík and V. Klemš. JA, an Adaptive Differential Evolution Algorithm, Benchmarked on the BBOB Noiseless Tested. In GECCO 0: Genetic and Evolutionary Computation Conference Companion, pages 97 0, New York, NY, USA, 0. AC. [] K. Price. Differential evolution vs. the functions of the second ICEO. In Proceedings of the IEEE International Congress on Evolutionary Computation, pages 7, 997. [] A. H. G. Rinnooy Kan and G. T. Timmer. Stochastic gloal optimization methods part II: ulti level methods. athematical Programming, 9:7 78, 987. [] R. Storn and K. Price. Differential evolution a simple and efficient heuristic for gloal optimization over continuous spaces. Journal of Gloal Optimization, : 9, 997. Acknowledgements This work was supported y the Sapientia Foundation - Institute for Scientific Research with the grant No. 0/9/0.

4 Sphere Ellipsoid separale Rastrigin separale Skew Rastrigin-Bueche separ H 0ftarget=e ftarget=e ftarget=e ftarget=e Linear slope Attractive sector 7 Step-ellipsoid 8 Rosenrock original 0ftarget=e ftarget=e ftarget=e ftarget=e Rosenrock rotated 0 Ellipsoid Discus Bent cigar 0ftarget=e Sharp ridge 0ftarget=e Sum of different powers 0ftarget=e Rastrigin 0ftarget=e Weierstrass 0ftarget=e ftarget=e ftarget=e ftarget=e Schaffer F7, condition 0 8 Schaffer F7, condition Griewank-Rosenrock F8F 0 Schwefel x*sin(x) 0ftarget=e ftarget=e ftarget=e ftarget=e Gallagher 0 peaks Gallagher peaks Katsuuras 7 Lunacek i-rastrigin H 0ftarget=e ftarget=e ftarget=e ftarget=e Figure : Expected running time (ERT in numer of f-evaluations) divided y dimension for target function value 0 8 as log 0 values versus dimension. Different symols correspond to different algorithms given in the legend of f and f. Light symols give the maximum numer of function evaluations from the longest trial divided y dimension. Horizontal lines give linear scaling, slanted dotted lines give quadratic scaling. Black stars indicate statistically etter result compared to all other algorithms with p < and Bonferroni correction numer of dimensions (six). Legend: :, :, :H

5 .0 f-,-d separale fcts est 009 HH D log0 of (# f-evals / dimension) ill-conditioned fcts.0 f0-,-d est 009 D HH log0 of (# f-evals / dimension) weakly structured multi-modal fcts.0 f0-,-d D log0 of (# f-evals / dimension) est 009 HH.0 f-9,-d moderate fcts est 009 D HH log0 of (# f-evals / dimension) multi-modal fcts.0 f-9,-d est 009 HH D log0 of (# f-evals / dimension) all functions.0 f-,-d est est H H log0 of (# f-evals / dimension) Figure : Bootstrapped empirical cumulative distriution of the numer of ojective function evaluations divided y dimension (FEvals/D) for 0 targets in 0 [ 8..] for all functions and sugroups in -D. The est 009 line corresponds to the est ERT oserved during BBOB 009 for each single target.

6 .0 f-,0-d separale fcts est 009 HH D log0 of (# f-evals / dimension) ill-conditioned fcts.0 f0-,0-d D log0 of (# f-evals / dimension) weakly structured multi-modal fcts.0 f0-,0-d D log0 of (# f-evals / dimension) est 009 HH est 009 HH.0 f-9,0-d moderate fcts D log0 of (# f-evals / dimension) multi-modal fcts.0 f-9,0-d est 009 HH est 009 HH D log0 of (# f-evals / dimension) all functions.0 f-,0-d log0 of (# f-evals / dimension) est est H H Figure : Bootstrapped empirical cumulative distriution of the numer of ojective function evaluations divided y dimension (FEvals/D) for 0 targets in 0 [ 8..] for all functions and sugroups in 0-D. The est 009 line corresponds to the est ERT oserved during BBOB 009 for each single target.

7 f opt e e0 e- e- e- e-7 #succ f / 0.7(0.).(0.).0(0).(0.).(0.).9(0.) /.9() 0(7) 9() 79(7) 0(9) 7(8) / H0.7(0.).(0.).0(0).(0.).(0.).9(0.) / f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.9().().().().7().() / (0.) 0.(0.) 0.(0.) 0.7(0.) 0.7(0.) 79(909) 0/ () () () 0() () 0() /.().() () (7) () (9) / H.9().().().().9().(7) / H(0.) 0.(0.) 0.(0.) 0.7(0.) 0.7(0.) 7() / f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.() 79() e 0/.(0.).7(0.).9(0.).(0.).(0.) 0(7) e 0/.8(0.) /.().().9() 7.9(7) 7.7() 7.(8) 8/ H.(0.).().(.0).().().() / H.().() 0() 0(0) 0() 0(0) / f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f / () e 0/.(0.) ().7() 70() 9e 0/ (08) (9) 9(8) 7(8) /.() (8) 8(8) e 0/ H.(0.7) () 9(9) 87() 8(0) 8(0) / H.() (0) 0(9) e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.(0).(0).(0).(0).7(0) 8() / () (7) e 0/ 9() 08(7) 9() 89() 7(7) 77() /.().().(0.9).7(0.).(0.9).(0.9) / H.(0).(0).(0).(0).7(0) 87(9) / f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.(0.).(0.).(0.) (0.) 0.7(0.) 0.7(0.) / 0() 0() e 0/.().() 7.() 8.9() 8.() 8.9() / H.(0.).(0.).(0.) (0.) 0.7(0.) 0.7(0.) /.9(.0).9().().(0.).().() / f opt e e0 e- e- e- e-7 #succ f f / opt e e0 e- e- e- e-7 #succ () 097() e 0/ f9.e.e.e / 9.0(9).8().7().8().8().0() / (0) (0) 0.7() e 0/ H().8() 8.7() 8.() 8.() 8.8() / (8) 0(88) 087(0) e 0/ f opt e e0 e- e- e- e-7 #succ f / f opt e e0 e- e- e- e-7 #succ.0(0.).0(.0) 0.99() 0.9(0.7) 0.97(0.) 0.9(0.) / f / 8.0() 7.() 7() () () () /.(0) 7.(0) () (8) () () / H.0(0.).().().().().() / 7.7().().().7().7().7() 8/ f opt e e0 e- e- e- e-7 #succ f / f opt e e0 e- e- e- e-7 #succ 0.9(0) 0.(0.) 0.(0.) 0.7() 0.() 0.() / f / (8) (9) 0() 8(0) 9() 7() /.8() 0.7(0.7).().().().() / H 0.9(0) 0.(0.) 0.(0.) 0.7() 0.() 0.() /.().9() () () () () / f opt e e0 e- e- e- e-7 #succ f f / opt e e0 e- e- e- e-7 #succ (0.) 0.(0.) 0.(0.) (0.) () 9(7) / f / (9) () 9() () () () /.7().8().().().().() / H (0.) 0.(0.) 0.(0.) (0.) 8(7) 0() 8/.() 9() 9(08) 9(0) 9() 9(9) 8/ f opt e e0 e- e- e- e-7 #succ f / 0.(0.) 0.(0.) 7().() () 88(98) 0/ () () () 8() 0() () / H 0.(0.) 0.(0.) 7().0() 7(0) (7) / f opt e e0 e- e- e- e-7 #succ f / 0.7(0.) 0.7(0.) 0(0.).(0.) 0() 7e 0/ () (0) () 0(7) (7) 8() / H 0.7(0.) 0.7(0.) 0(0.).0() () (7) 0/ H() (8) 8.(8).().().9() / H0(8) 0(7).().().9() () / H (0) (0) 0.7() () () () / H.(0).().().9().8().8() / H.() 8(0.7).(0.).(0.).(0.).(0.) / H.().().().(0.7).(0.7).() / f opt e e0 e- e- e- e-7 #succ f / 9.().9().0() () e 0/.9() (7) e 0/ H 9.().().7() () e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f.e.e 9.e.e7.e7 /.(0.) 0.9(0.) 9(0.7) 0.9().() 8() /.().() e 0/ (8) 0(0) 70(7) 9(8) 8() () / 9.(8) e 0/ H.(0.) 0.9(0.) 9(0.7) 0.9().() 0() 8/ H.8() e 0/ Tale : Expected running time (ERT in numer of function evaluations) divided y the respective est ERT measured during BBOB-009 (given in the respective first row) for different f values in dimension. The central 80% range divided y two is given in races. The median numer of conducted function evaluations is additionally given in italics, if ERT(0 7 ) =. #succ is the numer of trials that reached the final target f opt Best results are printed in old.

8 f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f / 0.77(0.).7(0.).9(0.).8(0.).7(0.).7(0.) /.(0.) (0.) 0.97(.0) 0.9(0.7) e 0/ () 0() 0() 77(9) 0(0) 87() / 0(9) 0(0) (99) e 0/ H 0.77(0.).7(0.).9(0.).8(0.).7(0.).7(0.) / H.(0.) (0.) () () e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.().7().() 7.() 0() (9) / 0.7(0.) 0.9(0.) 0.(0.) 0.(0.) 0.7(0.) e 0/ () () () 8() 8() 0() / (7) 9() () e 0/ H.().7().() 7.() 0() () / H 0.7(0.) 0.9(0.) 0.(0.) 0.(0.) 0.7(0.) e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f 078.e.e.e.e.e / e 0/ e 0/ e 0/ e 0/ H9(08) e 0/ H e 0/ f opt e e0 e- e- e- e-7 #succ f e 9/ e 0/ 0(9) e 0/ H () e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f /.(0).(0).7(0).7(0) 7.(0) 8(9) / (0) e 0/ 98() 8(0) 98() 987(7) 70(97) 7() / 8.() 7() () 0(7) 0(7) e 0/ H.(0).(0).7(0).7(0) 7.(0) () / H() (7) 8() (9) e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f e.e /.8().(0.9).(0.9).8(0.7).(0.).7(0.) / e 0/ 9(7) (8) (8) (8) e 0/ () 9() 9() e 0/ H.8().(0.9).(0.9).8(0.7).(0.).(0.) / H 8() () 0() e 0/ f opt e e0 e- e- e- e-7 #succ f opt e e0 e- e- e- e-7 #succ f / f9.e.e.7e.7e / e 0/ (0) (0) 7.e- e 0/ () 77(7) 7() 8() 8() (9) / (0) H() 7() (9) 7(8) 7(7) (9) / 7() e 0/ f opt e e0 e- e- e- e-7 #succ f / (0.) 0.78(0.) 0.79(0.) 0.79(0.) 0.79(0.) 0.78(0.) / () 7() 7(0) 9(8) 0() () / H (0.).0().0().0().0() 0.99() / f opt e e0 e- e- e- e-7 #succ f / 0.7() 0.() () () () () / e 0/ H0.7() 0.() () () () () / f opt e e0 e- e- e- e-7 #succ f / 0.() 0.() 0.() 0.() (0) e 0/ e 0/ H0.() 0.() 0.() 0.() (0) e 0/ f opt e e0 e- e- e- e-7 #succ f / 0.7() () (9e-) () e 0/ e 0/ H0.7() () (9e-).() e 0/ f opt e e0 e- e- e- e-7 #succ f e.0e.e / 7() e 0/ e 0/ H 78(97) e 0/ H (0) (0) 7.e- (0) e 0/ f opt e e0 e- e- e- e-7 #succ f0 8 0.e.e.e.e /.(0) (0) e 0/ ().8() e 0/ H.(0).(0.).9() e 0/ f opt e e0 e- e- e- e-7 #succ f /.().0(0.9).00() 0.98() 0.9() 0.9() 8/ (7) 9() 8(7) (8) () 8(7) / H.(9) () () () 0() 9.() 7/ f opt e e0 e- e- e- e-7 #succ f e /.().().().().8() () / 8(89) 00() e 0/ H0(9) (08) (8) 0(8) 97() () / f opt e e0 e- e- e- e-7 #succ f. 77.9e 8.e 8.e / ().() (0) e 0/ f opt e e0 e- e- e- e-7 #succ.9() e 0/ f / H().() 8(9) e 0/ (0.) 9(0.) (0.) 0.7(0.) (0.) () / f opt e e0 e- e- e- e-7 #succ 0() (0) 08() e 0/ f.e 7.e.e7.e7.e7.e7 / H (0.) 9(0.) (0.) 0.7(0.) 8.8() 7(7) / e 0/ e 0/ H e 0/ Tale : Expected running time (ERT in numer of function evaluations) divided y the respective est ERT measured during BBOB-009 (given in the respective first row) for different f values in dimension 0. The central 80% range divided y two is given in races. The median numer of conducted function evaluations is additionally given in italics, if ERT(0 7 ) =. #succ is the numer of trials that reached the final target f opt Best results are printed in old.