A New Mechanism for Maintaining Diversity of Pareto Archive in Multiobjective Optimization

A New Mec hanism for Main taining Div ersit y of P areto Arc hiv e in Multiob jectiv e Optimization Jarosla v H´ ajek 1 , Andr´ as Sz¨ oll¨ os 1 , Jakub ˇ S ´ ıstek 2 † 1 Aeronautical Researc h and T est Institute, Berano v ´ yc h 130, Praha, CZ-199 05, Czech Republic 2 Institute of Mathematics of the AS CR, ˇ Zitn´ a 25, Praha, CZ-115 67, Czec h Republic † Corresp onding author hajek@vzlu.cz, szollos@vzlu.cz, sistek@math.cas.cz Abstract The article in troduces a new mec hanism for selecting individuals to a P areto arc hive. It w as com bined with a micro-genetic algorithm and tested on sev eral problems. The abilit y of this approach to produce indi- viduals uniformly distributed along the Pareto set without negative im- pact on conv ergence is demonstrated on presented results. The new con- cept was confronted with NSGA-I I, SPEA2, and IBEA algorithms from the PISA pack age. Another studied effect is the size of population v ersus n umber of generations for small p opulations. Keyw ords: m ulti-ob jectiv e optimization; micro-genetic algorithms; di- v ersity preserving; P areto archiv e; selection to archiv e 1 1 In tro duction The field of m ultiob jectiv e design optimization has evolv ed very fast during last y ears, reflecting the need of solving tasks with several conflicting criteria, which is common in practical problems. F rom the mathematical p oin t of view, this corresp onds to minimization/maximization of a vector-v alued function, which rarely leads to a single solution. Consequen tly , a whole hyperplane of trade-off solutions, called Pareto-optimal set, is exp ected as the result instead of a single optim um. A num b er of algorithms hav e b een presented that generate a set of solutions appro ximating this hyperplane. The quality of the approximation is usually considered from tw o p oints of view: (i) the closeness to the exact trade-off surface and (ii) its distribution. The former is related to con vergence prop erties of an algorithm while the latter describ es its abilit y to main tain diversit y . An ideal algorithm should pro duce well conv erged solutions perfectly distributed along the Pareto front. Ho wev er, these requirements are conflicting, and many curren t approaches concentrates on one of them finding reasonable compromise in the other. In this study , our atten tion is focused on the second aspect of div ersity of the P areto-optimal set, namely we presen t a new strategy for maintaining v ariety of mem b ers of a P areto archiv e. The problem of maintaining uniform distribution at an affordable cost has b een addressed b y many algorithms. It is known that the notion of cro wding distance prop osed by Deb et al. for algorithm NSGA-I I [5, 6] is not sufficien t to maintain diversit y of the evolution for more than tw o ob jectiv es (e.g. [7, 8]). On the other hand, SPEA2 by Zitzler et al. [18] is usually able to pro duce well spread solutions even for three or more ob jectiv es. The concept of arc hiving promising design vectors was first introduced for SPEA by Zitzler and Thiele 2 [19]. Knowles and Corne presented the Pareto Archiv ed Evolutionary Strategy (P AES) [11] and prop osed the adaptive grid algorithm ([12]) to main tain diver- sit y . How ever, it is difficult to keep the efficiency of this approach in cases with more than three ob jectiv es. The new mec hanism presented in this paper w as implemented in micro- genetic algorithm µ ARMOGA prop osed by Sz¨ oll¨ os et al. [15], and results for three standard three-ob jectiv e b enc hmark problems are presented. Our second aim is to in vestigate the effect of population size for small (some- times called micro) p opulations on the p erformance of µ ARMOGA. It w as re- p orted by Krishnakumar [13] for single-ob jectiv e optimization and b y Co ello and Pulido [3] for m ultiple ob jectiv es, that very small populations can lead to fast conv ergence to the Pareto front. In this context, most exp eriments were p erformed using p opulations of 4, 10 and 20 individuals. Results got b y µ ARMOGA equipp ed with the new arc hiving mechanism are compared with those obtained b y tw o leading metho ds in the field, namely NSGA-I I by Deb et al. [6] and SPEA2 by Zitzler et al. [18], and a recent inter- esting algorithm IBEA by Zitzler and K ¨ unzli [17]. All these are implemen ted in the Platform and Programming Language Independent In terface for Search Algorithms (PISA) 1 [2]. PISA is an interesting op en source pack age developed b y the team of Prof. E. Zitzler at ETH Z¨ uric h. The soft ware implemen ts v arious selection, crossov er, and mutation op erators and ob jective function ev aluations. An imp ortan t idea of the pro ject is to separate the selection of promising candi- dates from ob jective function ev aluation, crosso v er and m utation and implemen t these in t wo separate programs, in terchanging information via formatted files. These programs are called sele ctors and variators in the PISA context. There is an increasing num b er of ready-to-use v ariators and selectors that can b e down- loaded from the web page of the PISA pro ject. Therefore, the system offers 1 http://www.tik.ee.ethz.ch/sop/pisa 3 a simple wa y to pro duce fair comparisons of v arious selection sc hemes with the same v ariator. While the describ ed scheme of splitting an evolutionary algo- rithm in to t wo separate programs is very useful for some techniques, in our opinion, it do es not fit to algorithms with strong coupling b etw een b oth stages via the use of an archiving pro cedure. That is the reason wh y our implemen- tation of µ ARMOGA w as used, instead of integrating the prop osed archiving tec hnique into the PISA framework. Three metrics, measuring b oth conv ergence to the exact front and diversit y of the appro ximate set, are used for the comparison. It is observed, that the new algorithm pro duces very goo d distribution of individuals outp erforming in this respect the other algorithms in many cases. The archiving strategy do es not seem to affect its conv ergence. Moreo ver, div ersit y is maintained in an affordable w ay as suggested by presen ted numerical exp eriments. The rest of the pap er is organised as follows. In Section 2, µ ARMOGA is recalled with an emphasis on its main asp ects. Section 3 con tains the main con tribution, which is the prop osition of a new archiving mec hanism. T ests and comparisons with the other ev olutionary techniques can be found in Section 4, where w e describ e the test problems (4.1), metrics used for ev aluating the p er- formance (4.2), detailed setting of particular algorithms (4.3), and organization of the exp eriments (4.4), resp ectively . Our findings are discussed in detail in Section 5, while Section 6 contains summary of the work and concluding re- marks. 4 2 µ ARMOGA – Multiob jectiv e micro-genetic al- gorithm with range adaptation: A Brief In tro- duction T o minimize the costly ev aluation of individuals, it is straightforw ard to see that one wa y to go is to minimize their num ber. It is well known for evolutionary practitioners, that using smaller p opulations and applying the evolutionary op- erators many times is often more fav ourable than vice versa (e.g. [14]). This idea can b e brought to an extreme b y using a micro-p opulation (e.g. 4, 5, 10 individuals), what we really did when we utilized some ideas of Krishnakumar [13] and of Coello and Pulido [3]. Krishnakumar came with the concept of micro-genetic approach first, and used it for single-ob jective optimization. His algorithm contained only selection and crossov er op erators, and no m utation op erator. Instead, the author introduced a reinitialization technique, which w as in vok ed once in a few generations to ensure diversit y for the evolution. The latter t wo researchers prop osed a micro-genetic algorithm enabling to tackle m ulti-ob jective problems. Their concept was similar to that of Krishnakumar, i.e. it con tained selection, crosso ver, and reinitialization op erators supplemen ted b y a m utation op erator. Both algorithms w ere v erified on v arious test problems. In both the cases, the micro-genetic v ariants conv erged to the optimum (Pareto- fron t) muc h faster than their macrogenetic counterparts used for comparison. In the approach by Sz¨ oll¨ os [15], microgenetic algorithm is supplemented by range adaptation and “knowledge-based” reinitializion pro cedure exploiting the P areto-archiv e to generate better individuals. The concept of range adaptation was originally introduced by Arak aw a and Hagiw ara [1], who used it with binary coding of the design v ariables. Its essence lies in ability to promote the evolution to wards promising regions of the design 5 space via sophisticated manipulation with the p opulation statistics. Due to the co ding, it contained some artificial parameters which were hard to guess in general. Oyama [14] used range adaptation in real domain and w as successful in a voiding this dra wback via enco ding a design v ariable to a real num b er r i ∈ (0 , 1) defined b y integration of the Gaussian distribution N (0 , 1) r i = Z e p i −∞ N (0 , 1)( z )d z , (1) where e p i is link ed to the original design v ariable p i b y p i = σ i · e p i + µ i . (2) W e are using this encoding scheme too, with one important difference: Oyama originally calculated the av erage µ i and the standard deviation σ i b y sampling the upp er half of the p opulation, which is justified as long as macro-p opulations are used (e.g. with more than fift y individuals). But such approac h would b e to o restrictive in the case of micro evolution, since the upp er half of the microp- opulation contains to o little information to keep the diversit y . Consequen tly , the evolution quickly ends up in premature conv ergence. Thus, we calculate b oth by taking into account the whole p opulation. “Kno wledge based” reinitialization resulted from an attempt to use the mem- b ers of the Pareto-arc hive to get new mem b ers, sup erseding the old ones b y putting several of them in to the reinitialized p opulation. Moreov er, only a sub- set of the arc hiv e is considered. F or instance, t wo archiv e mem b ers with extreme v alues of tw o different ob jectiv es chosen randomly are usually exploited. In this w ay , it is possible to further impro ve the whole arc hive b y improving its subsets. The functioning of µ ARMOGA can b e seen in Figure 15. After initialization of the p opulation by Latin hypercub e sampling (LHS) and ev aluation depicted as 6 arc hive up date, the ev olution go es through selection, mating and m utation to ev aluation of the new p opulation. Eac h n -th generation the p opulation statistics is up dated, range-adaptation takes place, follo wed by knowledge based (elitist- random) reinitialization. A thorough description of the algorithm is to b e found in [15]. Our approach con tains tw o new system parameters: adaptation factor δ and minimal standard deviation σ min . In short, µ ARMOGA strives to k eep the evo- lution in a p ermanen tly “excited” state via forced modification of the p opulation statistics. It practically means that the standard deviation is not allow ed to fall under certain minimal v alue of σ min for any design v ariable. This helps to pre- v ent the micro-genetic algorithm from getting stuc k in premature conv ergence. The role of the adaptation factor δ lies in controlling the frequency of range adaptation: if reinitialization is necessary , and the new standard deviation of a design v ariable is changed b y more than δ · σ old , where σ old is the standard deviation when the last reinitialization to ok place, then the range of that design v ariable is adapted. 3 The arc hiving algorithm Par eto ar chive is a key comp onent of many evolutionary algorithms. It acts as a collector of goo d individuals during the evolution, and is often used to give the resulting Pareto fron t appro ximation at the end of the evolution. After new individuals are ev aluated, the archiv e is improv ed if these individuals dominate or are non-dominated with resp ect to the existing individuals of the archiv e. During reinitialization, the micro-genetic algorithm retriev es information from the arc hive, using it to explore the promising regions of the search space. Ob viously , in any real setup we must limit the n umber of individuals stored in an arc hive. This is necessary not just to keep the amount of information pro- 7 cessed feasible, but also to get a go o d div ersity of the resulting approximation. In our strategy , we use a fixed upp er limit on the num b er of individuals stored in the archiv e. Ideally , we wan t to end up with a full archiv e of Pareto-optimal solutions that is “well spread” o ver the true Pareto front of the problem. Our approac h is an archiv e dealing with a single new individual at a time. This is particularly suitable for micro-evolutionary approac hes, where we only hav e a few new individuals from each generation. When a new individual arrives, it is first chec k ed for Pareto dominance with all existing members of the archiv e. Now, we distinguish among three cases: • The new individual is dominated by one or more mem b ers of the archiv e. In this case, the new individual is discarded. • The new individual dominates one or more mem b ers of the archiv e. The dominated ones are remo ved, and the new individual is added to the arc hive and the in ternal information of the archiv e is updated (see b e- lo w). • The new individual is non-dominated and non-dominating. If the n umber of members of the archiv e has not y et reached the upp er limit, the new individual is added as in the previous case. In the opp osite case, we need to discard at least one individual (either the newcomer or one from the arc hive), but w e can not decide this by P areto dominance. In this case, w e pro ceed to the secondary decision pro cedure describ ed b elo w: If we arrive at the case that can not b e resolved by P areto dominance, our secondary goal is to maximize the distance b etw een neighbouring individuals, based on some distance-measure in the ob jective space. In this pap er, w e use the standard Euclidean distance, whic h is meaningful for any dimension of the ob jective space. 8 First, w e consider the minimum pairwise distance, i.e., min i,j ∈ P i 6 = j k f i − f j k , (3) where P denotes the set of arc hived individuals and f i stands for the vector of ob jective v alues of individual i . W e take the pair of individuals that achiev es the minimum in the ab o ve expression. If there are m ultiple pairs, we take any of them. Without the loss of generality , we assume that the minim um pair is f 1 , f 2 . F urther, we denote the vector of ob jective v alues of the new individual as h . If min k ∈ P k 6 =1 k f k − h k > k f 1 − f 2 k , (4) w e can replace f 1 b y h . Alternativ ely , if min k ∈ P k 6 =2 k f k − h k > k f 1 − f 2 k , (5) w e can replace f 2 b y h . If either of the ab o ve conditions is satisfied, the ov erall minim um pairwise distance will b e improv ed b y the substitution or, if there w ere multiple minimal pairs, it will sta y the same but the num b er of minimal pairs will reduce. W e call this as the glob al impr ovement chec k. If neither of these conditions is satisfied, w e consider the closest archiv ed individual to h , say , f c instead. If min k ∈ P k 6 = c k f k − h k > min k ∈ P k 6 = c k f k − f c k , (6) w e replace f c b y h . If this condition holds, there is a certain subset of the arc hived individuals whose pairwise minim um will improv e. This is the lo c al impr ovement chec k. 9 If neither chec k is successful, w e discard the new individual. Searc hing for the minimum-distance pair of the archiv e afresh each time an individual is considered w ould be too costly . T o mak e the pro cedure efficien t, we main tain for eac h archiv ed individual a p oin ter to its closest neighbour (or any of them). Therefore, searc hing for the pairwise minimum in (3) requires only one pass through the arc hive. Similarly , the right-hand side of equation (6) is simply the distance of f c to its closest neighbour. Hence, these tw o chec ks only require computing the distances of the new individual to all archiv ed individuals, and computing the minima on left-hand sides of the equations (4), (5), and (6). Th us, de ciding whether to add a new individual has linear complexity in terms of n umber of archiv ed individuals (ev aluating mutual pairwise dominance also has linear complexity). If the new individual is to b e added, the existing closest-neigh b our links need to b e updated. Eac h resulting archiv e member is considered in turn. If the link is v alid (i.e. the closest neighbour in the archiv e w as not discarded), we simply c heck if the newcomer is closer, and p ossibly up date the link. This takes only constan t time. How ever, if the link b ecame in v alid (the former closest neigh b our w as discarded), we need to compute the closest neighbour afresh by computing ob jective distances of the up dated individual to all others. It can b e prov en b y a simple argumen t based on k -dimensional ball v olumes that the maximum n um b er of points in k -dimensional space ha ving a single common closest neigh b our is b ounded from ab ov e by a constan t depending on k . Since the Pareto archiv e consists of mutually non-dominating vectors, whic h can not b e arranged arbitrarily , in our case the constant is ev en smaller. F or instance, for a tw o-ob jective optimization, i.e. k = 2, a single arc hive member can be the closest neighbour to at most t wo other members at the same time. Using this argument, it can b e easily seen that the complexity of a single 10 arc hive up date has complexity O ( N + M N ), where N is the size of the Pareto arc hive and M is the num b er of archiv e members dominated b y the new indi- vidual. As was already said, merely de ciding whether the newcomer is to b e added costs O ( N ). If the decision is p ositive, there are tw o cases: either the new comer dominates some M existing archiv e members, or it was added based on the secondary decision pro cedure. In the former case, M mem b ers will b e discarded, so at most c M nearest-neighbour links will need to b e up dated, c b eing the upp er b ound constant discussed in the previous paragraph. In the latter case, one existing mem b er is discarded, so at most c existing links m ust b e up dated. Given that up dating a single link costs O ( N ), together we hav e the cost O ( N + c (1 + M ) N ) which can b e simplified to O ( N + M N ), given that c is a constant indep endent of M , N . While in principle M can b e as high as N , in practice it drops to M  N v ery quic kly as the conv ergence pro ceeds and new dominating individuals b ecome increasingly rare. It should also b e noted that if M is high at one step, the ev olution contin ues with an archiv e of N − M which will be significantly smaller than N . Numerical exp eriments confirm that in real evolutionary runs, the a verage n umber of in v alid links p er archiv e up date is v ery small, even muc h smaller than the theoretical b ounds suggested ab ov e. This might b e observ ed from T ables 19 and 20. Therefore, w e can conclude that the pro cedure of adding new individual to our archiv e is essen tially of linear complexity . 4 Comparison of results The abilities of the new archiving mechanism are first demonstrated on test functions DTLZ1, DTLZ2 and DTLZ4, suggested by Deb et al. [8]. T o exam- ine the influence of p opulation size, our algorithm w as run separately with 4, 10 and 20 individuals. Obviously , it is preferable to maintain the num b er of 11 function ev aluations as low as p ossible, therefore w e study the b eha viour of the aforemen tioned approaches for three fixed n umbers of ev aluations, 4 000, 20 000 and 40 000. F or test problem DTLZ1, the n um b er of function ev aluations is extended to 100 000 and 200 000, since the algorithms w ere unable to conv erge to the global Pareto front with just 40 000 computations. T o further inv estigate the behaviour of the prop osed metho d, we p erformed an exp eriment with test problem WFG1 suggested by Huband et al. [10]. F or this difficult problem, it w as necessary to run the ev olution to as man y as 2 000 000 ev aluations to obtain reasonable con vergence to the Pareto front. 4.1 T est problems The algorithms are compared on three benchmark problems introduced in [8]. The follo wing form of them is considered: • DTLZ1 Minimize f 1 , f 2 , f 3 , where f 1 ( x ) = 1 2 x 1 x 2 (1 + g ( x M )) , (7) f 2 ( x ) = 1 2 x 1 (1 − x 2 )(1 + g ( x M )) , (8) f 3 ( x ) = 1 2 (1 − x 1 )(1 + g ( x M )) , (9) g ( x M ) = 100 5 + X x i ∈ x M  ( x i − 0 . 5) 2 − cos(20 π ( x i − 0 . 5))  ! , (10) sub ject to 0 ≤ x i ≤ 1, for i = 1 , 2 , . . . , 7. Here x = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) and x M = ( x 3 , x 4 , x 5 , x 6 , x 7 ). • DTLZ2 12 Minimize f 1 , f 2 , f 3 , where f 1 ( x ) = (1 + g ( x M )) cos( x 1 π / 2) cos( x 2 π / 2) , (11) f 2 ( x ) = (1 + g ( x M )) cos( x 1 π / 2) sin( x 2 π / 2) , (12) f 3 ( x ) = (1 + g ( x M )) sin( x 1 π / 2) , (13) g ( x M ) = X x i ∈ x M ( x i − 0 . 5) 2 , (14) sub ject to 0 ≤ x i ≤ 1, for i = 1 , 2 , . . . , 12. Here x = ( x 1 , x 2 , . . . , x 12 ) and x M = ( x 3 , x 4 , . . . , x 12 ). • DTLZ4 Minimize f 1 , f 2 , f 3 , where f 1 ( x ) = (1 + g ( x M )) cos( x 100 1 π / 2) cos( x 100 2 π / 2) , (15) f 2 ( x ) = (1 + g ( x M )) cos( x 100 1 π / 2) sin( x 100 2 π / 2) , (16) f 3 ( x ) = (1 + g ( x M )) sin( x 100 1 π / 2) , (17) g ( x M ) = X x i ∈ x M ( x i − 0 . 5) 2 , (18) sub ject to 0 ≤ x i ≤ 1, for i = 1 , 2 , . . . , 12. Here x = ( x 1 , x 2 , . . . , x 12 ) and x M = ( x 3 , x 4 , . . . , x 12 ). Problem WF G1 is a b enchmark problems introduced in [10]. The following form is considered: • WF G1 13 Minimize f 1 , f 2 , f 3 , where f 1 ( x ) = 2 [(1 − cos( z 1 π / 2))(1 − cos( z 2 π / 2))] , (19) f 2 ( x ) = 4 [(1 − cos( z 1 π / 2))(1 − sin( z 2 π / 2))] , (20) f 3 ( x ) = 6  1 − z 1 − cos(10 π z 1 + π / 2) 10 π  , (21) where z 1 = z 1 ( x ), and z 2 = z 2 ( x ) are auxiliary v ariables obtained from design v ariables x = ( x 1 , x 2 , . . . , x 24 ) by a series of nonlinear transforma- tions (see [10] for their definition). How ever, there is a slight difference in our application of transformation b poly , which w e use with exp onent 0.2 instead of 0.02 suggested in [10] due to numerical issues in floating p oin t arithmetic. Design v ariables ha ve range 0 ≤ x i ≤ 2 i , for i = 1 , 2 , . . . , 24. The exact front of this problem is visualized in Figure 16. 4.2 Metrics The results are ev aluated according to three measures. The distance of members of the Pareto archiv e to the true Pareto front is measured using the genera- tional distance (GD) [16], which is defined as GD = v u u t 1 n n X i =1 d 2 i , (22) where n is the num ber of nondominated solutions found by an algorithm, and d i is the Euclidean distance of the i -th solution to the exact fron t. In order to ev aluate the distance accurately , we implemented an approach, that is able to iterativ ely find the closest p oint of the exact fron t for each approximate solution, pro vided the analytic expression of the exact front is known. This p oint is then used for measuring the distance. Zero v alue of GD corresp onds to all members 14 of the archiv e on the exact fron t. W e ev aluate also another measure of conv ergence, denoted as TOL5 . It is defined as the low est v alue, suc h that d i > TOL5 holds for at most 5 % of individuals. In statistics, it is called the 95-th p ercentile with resp ect to distance. Again, the low er v alue of TOL5 the b etter con vergence. Zero v alue indicates, that at least 95% of arc hive members are on the exact fron t. This metric is less sensitive to remote individuals than the GD v alue. The uniformity of distribution of arc hive members is measured b y spacing defined in [4]. It is based on the distance to the nearest neigh b our for each mem b er of the arc hive, which is defined as dn i = min j ∈ P j 6 = i k f i − f j k . (23) No w spacing is the ratio of standard deviation of v alues of these squared dis- tances and their av erage, i.e. spacing = 1 dn v u u t 1 n − 1 n X i =1 ( dn i − dn ) 2 , (24) where dn stands for the mean v alue dn = 1 n n X i =1 dn i . (25) Consequen tly , zero spacing corresponds to uniform distribution of distances to the nearest neigh b our. Although this do es not assure global uniformit y of dis- tribution (e.g. for pairs of individuals), our exp erience with this metric is satis- factory . The cov erage of the Pareto fron t is not ev aluated b y means of a metric, but rather compared qualitatively at presented plots. 15 4.3 Setting of algorithms All the algorithms from PISA pac k age [2] (in the PISA context called selectors), i.e. NSGA-I I, SPEA2, and IBEA, are used with the following setting of v ariator DTLZ: • individual mutation probabilit y . . . 1, • individual recombination probabilit y . . . 1, • v ariable mutation probability . . . 0.01 , • v ariable swap probability . . . 0.5, • v ariable recombination probability . . . 1, • η mutation . . . 20, • η recombination . . . 15, • use symmetric recom bination . . . 1, F or v ariator WFG, these v alues are the same except the v alue of v ariable mu- tation probabilit y preset to 1 (default). The simulations with PISA are p erformed with population of 100 individu- als. All of them are selected for mating, pro ducing 100 new individuals in eac h generation. The tournament of 2 individuals is used in these selectors. Ex- p erimen ts with IBEA are p erformed using the additive  -indicator with scaling factor κ equal to 0.05. The µ ARMOGA is run with 4, 10 and 20 individuals in p opulation, mark ed as µ ARMOGA(4), µ ARMOGA(10) and µ ARMOGA(20), respectively (for WF G1, only 4 members of p opulation are considered). The archiv e size is alwa ys set to 100 to pro duce results comparable with those of PISA algorithms. Simple one- p oin t crossov er scheme without mutation is used. It was rep orted by Oyama 16 [14], that this sc heme derived for binary co ded algorithms [9] w orks reasonably w ell also for real-domain. The v ersion with 4 members uses reinitialization in eac h generation (DTLZ1, DTLZ2, DTLZ4) or once in four generations (WFG1), while for larger p opulations, the reinitialization is performed once p er 3 gener- ations for all problems. After reinitialization, several existing arc hive mem b ers are put into the new p opulation. Their num b er is 2, 4 and 6 for the p opula- tion of 4, 10 and 20 mem b ers, resp ectively . Random selection of individuals for mating is then p erformed with this modified p opulation. The other imp ortant parameters of µ ARMOGA are set to the following v alues: • adaptation factor δ . . . 1.4, • minimal standard deviation σ min . . . 0.8 (DTLZ1, WF G1), 0.005 (DTLZ2, DTLZ4), • recom bination probability . . . 1. Larger v alue of σ min helps to attain the global Pareto optimal fron t of mul- timo dal problems such as DTLZ1 and leads to faster con vergence also in the case of WF G1. In general, its large v alues emphasizes global exploration of the design space while small v alues lead to refined search. 4.4 Exp erimen ts The results for problems DTLZ1, DTLZ2 and DTLZ4 are summarized in T a- bles 1 – 11, and visualized in Figures 1 – 11. F or problem WF G1, results are summarized in T ables 12–18 and Figures 12 – 14. The v alues in tables are ob- tained as av erages for 20 differen t seeds and where it makes sense, the b est v alue is emphasized b y b old font. The approximation with the b est distribution is selected out of the tw ent y runs of eac h algorithm for visualization. The exact P areto fron t is mark ed b y grid of small dots in presen ted figures. How ev er, 17 not all tw ent y seeds lead to a successful approximation of the P areto set in some instances, esp ecially for problem DTLZ4. Most of the algorithms suffer from problems with robustness with resp ect to initial population and pro duce de gener ate d fron ts for some seeds. W e consider a front as degenerated, if all in- dividuals hav e almost identical v alue of an ob jective, and thus cov er just a line on the three dimensional surface of the exact front. Numbers of degenerated fron ts for all problems and metho ds are summarized in T able 21. The efficiency of the prop osed arc hiving tec hnique is further demonstrated in comparison with the same approac h, i.e. µ ARMOGA, using cro wding distance [5, 6]. That algorithm w as describ ed in [15]. Outputs of these exp eriments are summarized for DTLZ1 in T ables 22 and 23, for DTLZ2 in T ables 24 and 25, for DTLZ4 in T ables 26 and 27, and for WFG1 in T ables 28 and 29. Obtained P areto fronts are plotted in Figures 17–28. 5 Discussion of results 5.1 DTLZ1 This problem with three ob jectives has a linear Pareto optimal fron t that inter- sects the axes of the ob jective space at v alue 0.5. Apart of the exact front, there exist a n umber of other parallel planes corresp onding to local Pareto fronts. As these also attract an ev olution, problem DTLZ1 tests the ability of a genetic algorithm to cop e with multi-modality . As can b e seen in Figure 1, none of the algorithms is able to reach the global P areto front in 4 000 ev aluations for any seed, and metrics in T able 1 do not pro vide muc h v aluable information. How ev er, we can remark that IBEA and µ ARMOGA(4) pro vide one order b etter conv ergence than the other algorithms and µ ARMOGA for all sizes of p opulation pro vides reasonable spacing. 18 Ho wev er, all algorithms except NSGA-I I are able to reac h the global fron t for some seeds in 20 000 ev aluations (Figure 2). Comparing visually the results in Figure 2, w e can conclude that µ ARMOGA(4) p erforms b est, whic h is supp orted b y the b est v alues of all three metrics in T able 2. As individuals for many of the seeds are still aw a y from the global fron t for all algorithms, metrics in T able 2 do not provide a detailed insight either. The situation is further impro ved with 40 000 ev aluations, for which all algo- rithms except NSGA-I I are able to reach the global fron t for most of the seeds (Figure 3). How ev er, Figure 3 sho ws that µ ARMOGA (for all sizes of p opula- tion) pro duces the b est distribution, which is confirmed b y v alues of spacing in T able 3. Since for some seeds the individuals still are not in vicinity of the true P areto fron t, the av eraged metrics in T able 3 are still rather bad. According to T able 3, the best conv ergence is in av erage attained by µ ARMOGA(4) for this case. F or 100 000 and 200 000 ev aluations, µ ARMOGA(4) achiev es the global P areto-optimal fron t for all seeds. All the other algorithms fail to find the global fron t for some seeds, which considerably sp oils the metrics in T ables 4 and 5. Since IBEA pro duces only degenerated fronts in these cases, metrics are not ev aluated and are omitted in T ables 4 and 5. Although the distribution of fronts obtained by µ ARMOGA for all sizes of population is comparable to SPEA according to Figures 4 and 5, the met- rics in T ables 4 and 5 reveal that spacing is, in av erage, one order b etter by µ ARMOGA than b y SPEA. The b est a verage conv ergence metrics are obtained b y µ ARMOGA(4) (T ables 4 and 5). 19 5.2 DTLZ2 Problem DTLZ2 has three ob jectiv es, and the exact front corresp onds to the part of a unit sphere when restricted to the o ctant giv en b y non-negativ e v alues of all three ob jectives. This is the easiest problem for all compared algorithms and tests mainly the sp eed at which an algorithm is con verging to the exact P areto front. Already for 4 000 ev aluations, the fronts obtained b y all the compared meth- o ds are reasonably conv erged and distributed along the exact Pareto fron t. Fig- ure 6 shows that µ ARMOGA (regardless of the size of p opulation) and SPEA pro duce the best distribution of individuals along the exact front, whereas the distribution obtained by IBEA is rather p o or. This observ ation is confirmed b y the spacing metric in T able 6. The b est conv ergence is achiev ed b y IBEA according to GD and TOL5 metrics in T able 6 follow ed b y µ ARMOGA. Similar observ ations can be made from the results for 20 000 ev aluations (T able 7 and Figure 7) and 40 000 ev aluations (T able 8 and Figure 8) – the b est spacing is obtained for all sizes of p opulation by µ ARMOGA and the b est con vergence is attained by IBEA, although the distribution of individuals along the P areto front is worse. 5.3 DTLZ4 Although the definition of problem DTLZ4 is similar to DTLZ2 (cf. Section 4.1), the evolution is greatly influenced by the exp onential transformation of design v ariables, whic h maps most of the space to wards the axes in design space. This in turn pushes the evolution to the limits of the ob jectiv e space. Thus, problem DTLZ4 tests b est of the three DTLZ problems the abilit y of a genetic algorithm to obtain uniform distribution of individuals along the P areto optimal surface. F or 4 000 ev aluations, the b est distribution is p ro duced b y S PEA2 (Figure 9). 20 This is confirmed b y results of spacing in T able 9. How ever, µ ARMOGA(4) pro duces the b est conv erged results. F or 20 000 ev aluations, the distribution obtained by µ ARMOGA is already visually comparable with SPEA2 in Figure 10. Also spacing obtained by µ ARMOGA is comparable to that of SPEA2 according to T able 10 for 4 and 10 members of p opulation. Algorithms µ ARMOGA(20), IBEA, and NSGA-II produce in a verage only slightly worse con verged results. The b est GD and TOL5 v alues are attained by µ ARMOGA(20). In the case of 40 000 ev aluations, the b est distribution of individuals is at- tained b y µ ARMOGA follow ed by SPEA2 according to Figure 11 and also con- firmed by v alues of spacing in T able 11. The b est conv ergence is again obtained b y µ ARMOGA(20), follow ed by µ ARMOGA(10), µ ARMOGA(4) and IBEA, resp ectiv ely (T able 11). 5.4 WF G1 This is a difficult problem and all tested algorithms had problems with con- v ergence to the Pareto front. F or this reason, num ber of ev aluations of the ob jective function was increased to 2 000 000, after which some algorithms were able to attain the exact front. After 4 000 ev aluations, all algorithms pro duce results rather far from the P areto optimal set (T able 12). Nev ertheles, SPEA2 pro duces the most uniform distribution according to the spacing metric. After 20 000 ev aluations, µ ARMOGA(4) slightly leads in conv ergence fol- lo wed by IBEA (T able 13), pro ducing distribution with uniformity b etw een SPEA2 (b est spacing) and the rest of the algorithms. The same observ ations remain v alid for 40 000 ev aluations (T able 14. After 100 000 as well as 200 000 ev aluations, µ ARMOGA(4) dominates in 21 con vergence to the exact fron t (GD and TOL5 metrics in T ables 15 and 16), pro- ducing distribution comparable with SPEA2 (b est spacing). How ever, Figure 12 suggest, that µ ARMOGA(4) cov ers the whole P areto front, unlike SPEA2. Letting the ev olution run to 1 000 000 and 2 000 000 ev aluations, µ ARMOGA(4) dominates both in conv ergence (one order of magnitude compared to the second IBEA in T ables 15 and 16) and distribution along the exact Pareto front. In spacing metric, µ ARMOGA(4) is follo wed b y SPEA2. Figures 13 and 14 sho w that distribution of individuals by µ ARMOGA(4) uniformly cov ers the whole P areto front, while the other algorithms approaches the region around f 1 = 0 only slo wly . 5.5 Comparison of crowding distance with the new archiv- ing mechanism A set of exp eriments was run to compare µ ARMOGA with crowding distance and µ ARMOGA with the new archiving mec hanism. The p opulation of four in- dividuals w as selected for the comparison. Results for problems DTLZ1, DTLZ2, DTLZ4, and WFG1 are summarized in T ables 22–29 and Figures 17 – 28. According to these experiments, the new arc hiving approac h outp erforms cro wding distance in diversit y as is clear from Figs. 17 – 28 and spacing metric in T abs. 22 – 29. While it also has very p ositive effect on the con vergence of µ ARMOGA to the exact Pareto front for problems DTLZ1, DTLZ2, and DTLZ4 (T abs. 22–27), b oth algorithms exhibit similar conv ergence for problem WFG1 (T abs. 28 and 29). 5.6 Summary As can b e seen from ab ov e, µ ARMOGA outperforms the other metho ds in distribution of individuals along the Pareto fron t, and in many cases achiev es 22 the best conv ergence as w ell. Ho wev er, it should b e noted that the default settings of the algorithms from PISA pack age is used, which m a y not be optimal for the test problems considered. While IBEA offers exceptional conv ergence in some cases, the distribution of individuals along the exact Pareto front is usually rather p o or, with many individuals attac hed to limits of the ob jectiv e space. Our study confirms that the mechanism of cro wding distance do es not lead to uniform distribution of individuals along the Pareto fron t for more than tw o ob jectives. The same result migh t b e observed from the comparison of µ ARMOGA using the t wo archiving mec hanisms – cro wding distance and the new prop osed technique (T ables 22– 29, and Figures 17 – 28). On the other hand, SPEA2 pro duces v ery uniform distribution of individuals comparable with µ ARMOGA in some instances. Concerning the n um b er of ev aluations, DTLZ2 is the only problem, for which only 4 000 ev aluations are sufficien t to achiev e reasonable conv ergence and distri- bution of individuals on the Pareto front by all algorithms. On the other hand, for DTLZ1, even 40 000 ev aluations do not suffice to reac h the true P areto fron t for all seeds by any approac h, and results for 100 000 and 200 000 ev aluations are added for a reasonable comparison. Ev en this large num b er of ev aluations w as not sufficient to reach the proximit y of exact fron t in the case of problem WF G1, and results for 1 000 000 and 2 000 000 ev aluations are added. F or test functions DTLZ4 and WFG1, the new archiving mechanism is able to driv e the ev olution to regions, where the cov erage of the Pareto front by individuals is sparse, and reco ver nice distribution of individuals along the P areto set even for p oorly chosen initial p opulation. T o in v estigate the optimal distribution of the num b er of function ev alua- tions b et ween p opulation size and n umber of generations for micro-evolution, µ ARMOGA is run with 4, 10 and 20 individuals for DTLZ1, DTLZ2, and 23 DTLZ4. According to our experiments, the performance of the algorithm is sim- ilar for all configurations with resp ect to spacing and con vergence history and no strong dep endence is rev ealed. Ho wev er, for problem DTLZ4, the metho d tends to produce more degenerated fron ts with larger p opulation (see T able 21). Additionally , population of 4 individuals leads to the b est conv ergence metrics for problem DTLZ1, and p opulation of 20 individuals to the b est conv erged fron t for problem DTLZ4. Thus, using small p opulations and larger num b er of generations seems as the preferable approach. 6 Conclusion The goal of our study is t wofold: (a) to develop a new approac h for selecting individuals to the Pareto archiv e; (b) to explore the p otential of using small p opulation in evolutionary algorithms. The main con tribution of the paper is the presentation of a new archiv- ing mec hanism. Although its basic idea is rather simple and straightforw ard, the technique pro duces v ery promising results on all tested problems. W e are a ware of the fact that the theoretical time complexity of the mec hanism might b e rather large (quadratic in the worst case). How ever, our tests justify its usage, since the exp erimentally found complexit y is muc h more fav ourable (ap- pro ximately linear). Moreov er, it is intended to b e used in combination with small p opulation, for which such more elab orate selection mechanism is usually affordable. The prop osed selection mechanism was combined with µ ARMOGA and is compared to other three state-of-the-art algorithms (NSGA-I I, SPEA2, and IBEA) on four test problems. W e can conclude that µ ARMOGA presents Pareto sets with the same or b etter distribution as SPEA2, but usually with muc h b et- ter conv ergence to the exact front that is comparable with IBEA, th us the b est 24 com bining requiremen ts on both con vergence and distribution of individuals. A considerable impro vemen t is attained, using the new mechanism, in com- parison with the version of µ ARMOGA that uses crowding distance. Clearly , µ ARMOGA equipp ed with the new div ersity mechanism is v ery promising and ma y b e comp etitiv e with resp ect to other recent approaches. Our experiments fu rther supp ort using small p opulations (up to 10 individu- als), since runs with four individuals usually pro duces the b est results. It is well kno wn that suc h small p opulation can lead to rapid con vergence. Ho wev er, in com bination with the prop osed arc hiving mechanism, it also seems to b e more robust with resp ect to an initial p opulation. Regarding the history of con vergence to the Pareto front, in some cases as few as 4 000 ev aluations of ob jective function could b e sufficient for some problems (DTLZ2), while for other problems (multi-modal problem DTLZ1 or difficult WF G1), even 40 000 ev aluations may not b e sufficien t to approximate the true P areto front, and as many as 1 000 000 ev aluations are needed for reasonable outcome. Ac kno wledgemen t This research has b een supported by the Developmen t of Applied External Aero- dynamics Program (Ministry of Education, Y outh and Sp orts of the Czech Re- public Gran t MSM0001066901), by research pro ject A V0Z10190503 (Academy of Sciences of the Czec h Republic), and b y grant IAA100760702 (Grant Agency of the Academy of Sciences of the Czech Republic). References [1] Araka w a, M., and Hagiw ara, I. Developmen t of adaptiv e real range 25 (ARRange) genetic algorithms. JSME International Journal Series C, Me- chanic al systems, machine elements and manufacturing 41 , 4 (1998), 969– 977. [2] Bleuler, S., Laumanns, M., Thiele, L., and Zitzler, E. PISA – A platform and programming language independent interface for search algorithms. In Evolutionary Multi-Criterion Optimization (EMO 2003) (Berlin, 2003), C. M. F onseca, P . J. Fleming, E. Zitzler, K. Deb, and L. Thiele, Eds., Lecture Notes in Computer Science, Springer, pp. 494– 508. [3] Coello, C. A. C., and Pulido, G. T. A micro-genetic algorithm for m ultiob jective optimization. In Evolutionary Multi-Criterion Optimization (EMO 2001) (Berlin, 2001), E. Zitzler, K. Deb, L. Thiele, C. Co ello Co ello, and D. Corne, Eds., Lecture Notes in Computer Science, Springer, pp. 126– 140. [4] Coello, C. A. C., Pulido, G. T., and Lechuga, M. S. Handling m ultiple ob jectiv es with particle swarm optimization. IEEE T r ansactions on Evolutionary Computation 8 , 3 (2004), 256–279. [5] Deb, K. Multi-Obje ctive Optimization using Evolutionary A lgorithms . Wiley-In terscience Series in Systems and Optimization. John Wiley & Sons, Chic hester, 2001. [6] Deb, K., A gra w al, S., Pra t ap, A., and Mey ariv an, T. A fast elitist non-dominated sorting genetic algorithm for mu lti-ob jective optimization: NSGA-I I. In Par al lel Pr oblem Solving fr om Natur e – PPSN VI (Berlin, 2000), M. Schoenauer, K. Deb, G. Rudolph, X. Y ao, E. Lutton, J. J. Merelo, and H.-P . Sch w efel, Eds., Springer, pp. 849–858. 26 [7] Deb, K., Mohan, M., and Mishra, S. A fast multi-ob jective evolution- ary algorithm for finding w ell-spread pareto-optimal solutions. T ech. Rep. 2003002, KanGAL, Indian Institute of T ec hnology Kanpur, India, 2003. [8] Deb, K., Thiele, L., Laumanns, M., and Zitzler, E. Scalable multi- ob jective optimization test problems. In Congr ess on Evolutionary Compu- tation (CEC ’02) (Piscataw a y , New Jersey , 2002), IEEE Press, pp. 825–830. [9] Goldber g, D. E. Genetic Algorithms in Se ar ch, Optimization, and Ma- chine L e arning . Addison-W esley Professional, January 1989. [10] Huband, S., Barone, L., While, R. L., and Hingston, P. A scalable m ulti-ob jective test problem to olkit. In EMO (2005), C. A. C. Co ello, A. H. Aguirre, and E. Zitzler, Eds., vol. 3410 of L e ctur e Notes in Computer Scienc e , Springer, pp. 280–295. [11] Kno wles, J. D., and Corne, D. W. The Pareto archiv ed ev olution strategy : A new baseline algorithm for Pareto multiob jective optimisa- tion. In Pr o c e e dings of the 1999 Congr ess on Evolutionary Computation (CEC’99) (1999), vol. 1, pp. 98–105. [12] Kno wles, J. D., and Corne, D. W. Appro ximating the nondominated fron t using the Pareto archiv ed ev olution strategy . Evolutionary Computa- tion 8 , 2 (2000), 149–172. [13] Krishnakumar, K. Micro-genetic algorithms for stationary and non- stationary function optimization. In SPIE’s Intel ligent Contr ol and A dap- tive Systems Confer enc e (1989), G. Ro driguez, Ed., So ciet y of Photo- Optical Instrumen tation Engineers (SPIE), pp. 289–296. [14] O y ama, A. Wing Design Using Evolutionary A lgorithm . PhD thesis, T ohoku Universit y , Japan, 2000. 27 [15] Sz ¨ oll ¨ os, A., ˇ Sm ´ ıd, M., and H ´ ajek, J. Aero dynamic optimization via m ulti-ob jective micro-genetic algorithm with range adaptation, kno wledge- based reinitialization, crowding and  -dominance. A dvanc es in Engine ering Softwar e 40 , 6 (2009), 419–430. [16] V an Veldhuizen, D. A., and Lamont, G. B. Multiob jective evolu- tionary algorithm research: A history and analysis. T ech. Rep. TR-98-03, Air F orce Institute of T echnology , W right-P atterson Air F orce Base, Ohio, 2001. [17] Zitzler, E., and K ¨ unzli, S. Indicator-based selection in multiob jective searc h. In Par al lel Pr oblem Solving fr om Natur e (PPSN VIII) (Berlin, 2004), X. Y ao et al., Eds., Lecture Notes in Computer Science, Springer, pp. 832–842. [18] Zitzler, E., La umanns, M., and Thiele, L. SPEA2: Impro ving the strength pareto evolutionary algorithm for m ultiob jective optimization. In Evolutionary Metho ds for Design, Optimisation and Contr ol with Applic a- tion to Industrial Pr oblems. Pr o c e e dings of the EUR OGEN2001 Confer enc e (Barcelona, 2002), K. Giannakoglou, D. Tsahalis, J. Periaux, K. P apalil- iou, and T. F ogarty , Eds., International Center for Numerical Methos in Engineering (CIMNE), pp. 95–100. [19] Zitzler, E., and Thiele, L. Multiob jective ev olutionary algorithms: A comparative case study and the strength pareto approac h. IEEE T r ans- actions on Evolutionary Computation 3 , 4 (1999), 257–271. 28 metric NSGA-I I SPEA2 IBEA GD 3.61e+01 2.99e+01 1.80e+00 TOL5 7.53e+01 5.92e+01 2.10e+00 spacing 2.47e+00 2.82e+00 2.14e+00 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 4.17e+00 1.13e+01 1.08e+01 TOL5 5.70e+00 1.50e+01 1.50e+01 spacing 7.39e-01 7.40e-01 8.45e-01 T able 1: Problem DTLZ1, 4 000 function ev aluations Figure 1: Pareto front after 4 000 function ev aluations, problem DTLZ1, NSGA- I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with p opulation size 4 (top right), 10 (centre righ t), 20 (b ottom right). 29 metric NSGA-I I SPEA2 IBEA GD 4.96e+00 3.32e+00 5.91e-01 TOL5 8.52e+00 5.80e+00 6.64e-01 spacing 1.50e+00 2.32e+00 2.65e+00 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 2.35e-01 6.10e+00 4.21e+00 TOL5 3.05e-01 7.44e+00 5.33e+00 spacing 2.63e-01 9.85e-01 8.24e-01 T able 2: Problem DTLZ1, 20 000 function ev aluations Figure 2: P areto front after 20 000 function ev aluations, problem DTLZ1, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 30 metric NSGA-I I SPEA2 IBEA GD 3.82e+00 2.31e+00 4.93e-01 TOL5 6.36e+00 3.71e+00 4.46e-01 spacing 1.56e+00 1.71e+00 4.19e+00 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 3.61e-02 4.77e+00 3.02e+00 TOL5 5.04e-02 6.00e+00 3.79e+00 spacing 1.39e-01 4.15e-01 1.70e-01 T able 3: Problem DTLZ1, 40 000 function ev aluations Figure 3: P areto front after 40 000 function ev aluations, problem DTLZ1, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 31 metric NSGA-I I SPEA2 IBEA GD 2.33e+00 1.03e+00 - TOL5 4.01e+00 2.01e+00 - spacing 1.51e+00 1.42e+00 - metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 1.88e-03 4.60e+00 1.91e+00 TOL5 4.48e-04 5.82e+00 2.41e+00 spacing 8.50e-02 9.10e-02 1.51e-01 T able 4: Problem DTLZ1, 100 000 function ev aluations Figure 4: P areto front after 100 000 function ev aluations, problem DTLZ1, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 32 metric NSGA-I I SPEA2 IBEA GD 2.26e+00 3.76e-01 - TOL5 2.35e+00 5.64e-01 - spacing 1.43e+00 1.17e+00 - metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 1.54e-03 4.59e+00 2.08e+00 TOL5 1.74e-04 5.82e+00 2.30e+00 spacing 7.82e-02 8.33e-02 5.19e-01 T able 5: Problem DTLZ1, 200 000 function ev aluations Figure 5: P areto front after 200 000 function ev aluations, problem DTLZ1, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 33 metric NSGA-I I SPEA2 IBEA GD 1.07e-01 9.08e-02 4.96e-03 TOL5 2.51e-01 2.05e-01 7.49e-03 spacing 6.40e-01 1.83e-01 7.44e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 1.41e-02 1.77e-02 1.89e-02 TOL5 2.82e-02 3.77e-02 3.62e-02 spacing 1.30e-01 1.64e-01 1.80e-01 T able 6: Problem DTLZ2, 4 000 function ev aluations Figure 6: Pareto front after 4 000 function ev aluations, problem DTLZ2, NSGA- I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with p opulation size 4 (top right), 10 (centre righ t), 20 (b ottom right). 34 metric NSGA-I I SPEA2 IBEA GD 6.29e-02 3.50e-02 4.15e-04 TOL5 1.49e-01 7.53e-02 4.97e-04 spacing 6.45e-01 1.40e-01 6.80e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 2.59e-03 1.99e-03 2.94e-03 TOL5 4.01e-03 3.72e-03 3.29e-03 spacing 6.94e-02 8.09e-02 8.75e-02 T able 7: Problem DTLZ2, 20 000 function ev aluations Figure 7: P areto front after 20 000 function ev aluations, problem DTLZ2, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 35 metric NSGA-I I SPEA2 IBEA GD 4.64e-02 1.63e-02 4.94e-04 TOL5 1.06e-01 3.50e-02 2.71e-04 spacing 6.14e-01 1.33e-01 6.82e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 1.04e-03 1.32e-03 1.02e-03 TOL5 9.23e-04 1.02e-03 1.06e-03 spacing 6.03e-02 6.71e-02 7.03e-02 T able 8: Problem DTLZ2, 40 000 function ev aluations Figure 8: P areto front after 40 000 function ev aluations, problem DTLZ2, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 36 metric NSGA-I I SPEA2 IBEA GD 8.91e-02 1.31e-01 4.78e-03 TOL5 2.01e-01 2.81e-01 7.32e-03 spacing 6.13e-01 1.80e-01 7.30e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 1.87e-03 1.27e-02 5.45e-03 TOL5 3.42e-03 2.51e-02 1.23e-02 spacing 8.48e-01 1.88e+00 2.09e+00 T able 9: Problem DTLZ4, 4 000 function ev aluations Figure 9: Pareto front after 4 000 function ev aluations, problem DTLZ4, NSGA- I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with p opulation size 4 (top right), 10 (centre righ t), 20 (b ottom right). 37 metric NSGA-I I SPEA2 IBEA GD 2.61e-02 3.92e-02 5.60e-04 TOL5 6.25e-02 9.73e-02 2.87e-04 spacing 6.18e-01 1.21e-01 6.83e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 4.34e-04 2.34e-04 3.55e-05 TOL5 1.43e-04 1.21e-04 3.40e-05 spacing 1.46e-01 1.12e-01 3.45e-01 T able 10: Problem DTLZ4, 20 000 function ev aluations Figure 10: P areto front after 20 000 function ev aluations, problem DTLZ4, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 38 metric NSGA-I I SPEA2 IBEA GD 1.61e-02 1.91e-02 1.75e-04 TOL5 2.94e-02 3.60e-02 1.39e-04 spacing 6.42e-01 1.33e-01 6.94e-01 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) GD 2.19e-04 3.81e-04 1.44e-05 TOL5 3.48e-05 2.78e-05 1.15e-05 spacing 9.36e-02 8.01e-02 9.55e-02 T able 11: Problem DTLZ4, 40 000 function ev aluations Figure 11: P areto front after 40 000 function ev aluations, problem DTLZ4, NSGA-I I (top left), SPEA2 (centre left), IBEA (b ottom left), and µ ARMOGA with population size 4 (top right), 10 (centre righ t), 20 (b ottom right). 39 metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.86e-01 8.82e-01 7.47e-01 7.95e-01 TOL5 1.05e+00 1.04e+00 8.40e-01 9.27e-01 spacing 4.90e-01 1.60e-01 5.40e-01 5.19e-01 T able 12: Problem WFG1, 4 000 function ev aluations metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 9.01e-01 8.95e-01 7.67e-01 4.97e-01 TOL5 1.06e+00 1.05e+00 8.62e-01 5.92e-01 spacing 4.88e-01 1.67e-01 5.66e-01 2.82e-01 T able 13: Problem WFG1, 20 000 function ev aluations metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.86e-01 8.82e-01 7.47e-01 3.77e-01 TOL5 1.05e+00 1.04e+00 8.40e-01 4.56e-01 spacing 4.90e-01 1.60e-01 5.40e-01 2.24e-01 T able 14: Problem WFG1, 40 000 function ev aluations metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.80e-01 8.75e-01 7.26e-01 2.51e-01 TOL5 1.05e+00 1.04e+00 8.15e-01 3.15e-01 spacing 4.94e-01 1.62e-01 5.29e-01 1.96e-01 T able 15: Problem WFG1, 100 000 function ev aluations 40 metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.75e-01 8.79e-01 7.10e-01 1.75e-01 TOL5 1.04e+00 1.04e+00 7.99e-01 2.29e-01 spacing 4.98e-01 1.66e-01 5.47e-01 1.86e-01 T able 16: Problem WFG1, 200 000 function ev aluations Figure 12: P areto fron t after 200 000 function ev aluations, problem WF G1, NSGA-I I (top left), SPEA2 (top righ t), IBEA (b ottom left), and µ ARMOGA with population size 4 (b ottom right). 41 metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.58e-01 8.71e-01 6.81e-01 6.53e-02 TOL5 1.03e+00 1.03e+00 7.66e-01 1.14e-01 spacing 4.72e-01 1.74e-01 5.24e-01 1.53e-01 T able 17: Problem WFG1, 1 000 000 function ev aluations Figure 13: Pareto front after 1 000 000 function ev aluations, problem WF G1, NSGA-I I (top left), SPEA2 (top righ t), IBEA (b ottom left), and µ ARMOGA with population size 4 (b ottom right). 42 metric NSGA-I I SPEA2 IBEA µ ARMOGA(4) GD 8.58e-01 8.74e-01 6.69e-01 4.79e-02 TOL5 1.03e+00 1.03e+00 7.52e-01 9.69e-02 spacing 5.06e-01 1.89e-01 5.35e-01 1.54e-01 T able 18: Problem WFG1, 2 000 000 function ev aluations Figure 14: Pareto front after 2 000 000 function ev aluations, problem WF G1, NSGA-I I (top left), SPEA2 (top righ t), IBEA (b ottom left), and µ ARMOGA with population size 4 (b ottom right). 43 Initialization by LHS Stop Update archive Selection Crossover + Mutation Update population statistics Range adaptation Elitist−random reinitialization For ngen generations Every n−th generation Figure 15: Simple scheme of the µ ARMOGA algorithm. Figure 16: Exact Pareto front for problem WF G1. 44 metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) DTLZ1 0.98 1.32 1.32 DTLZ2 0.99 1.42 1.47 DTLZ4 1.05 1.35 1.35 WF G1 1.08 n/a n/a T able 19: Average n umber of up dates necessary after the addition of an indi- vidual in to archiv e, archiv e size limit 100. arc hive size 20 50 100 200 500 1 000 DTLZ4 1.06 1.05 1.06 1.06 1.02 0.94 T able 20: Average n umber of up dates necessary after the addition of an indi- vidual in to archiv e for problem DTLZ4, v ariable arc hive size. metric NSGA-I I SPEA2 IBEA DTLZ1 -/-/-/-/- -/-/-/-/- -/2/11/20/20 DTLZ2 -/-/- -/-/- -/-/- DTLZ4 11/8/5 11/11/11 6/6/6 WF G1 -/-/-/-/-/-/- -/-/-/-/-/-/- -/-/-/-/-/-/- metric µ ARMOGA(4) µ ARMOGA(10) µ ARMOGA(20) DTLZ1 -/-/-/-/- -/-/-/-/- -/-/-/-/- DTLZ2 -/-/- -/-/- -/-/- DTLZ4 -/-/- 4/4/4 4/3/3 WF G1 -/-/-/-/-/-/- n/a n/a T able 21: Num b er of degenerated Pareto fron ts for 20 seeds. Number at 4 000/20 000/40 000(/100 000/200 000(/1 000 000/2 000 000)) function ev alu- ations. ev aluations 4 000 20 000 40 000 100 000 200 000 GD 2.13e+01 2.28e+00 1.15e+00 2.41e+00 3.72e-01 TOL5 4.02e+01 1.42e+00 2.07e-01 3.99e-02 1.22e-02 spacing 3.34e+00 2.34e+00 2.73e+00 3.29e+00 2.60e+00 degenerated 0 0 0 0 0 T able 22: µ ARMOGA with crowding distance, four individuals, problem DTLZ1, a verage for tw en ty seeds. ev aluations 4 000 20 000 40 000 100 000 200 000 GD 4.17e+00 2.35e-01 3.61e-02 1.88e-03 1.54e-03 TOL5 5.70e+00 3.05e-01 5.04e-02 4.48e-04 1.74e-04 spacing 7.39e-01 2.63e-01 1.39e-01 8.50e-02 7.82e-02 degenerated 0 0 0 0 0 T able 23: µ ARMOGA with the new prop osed arc hiving algorithm, four indi- viduals, problem DTLZ1, av erage for t wen ty seeds. 45 ev aluations 4 000 20 000 40 000 100 000 200 000 GD 5.36e-02 2.54e-03 1.77e-03 2.16e-03 1.99e-03 TOL5 7.92e-02 2.92e-03 2.83e-03 3.58e-03 2.83e-01 spacing 5.74e-01 5.60e-01 5.47e-01 5.49e-01 5.41e-01 degenerated 0 0 0 0 0 T able 24: µ ARMOGA with crowding distance, four individuals, problem DTLZ2, a verage for tw en ty seeds. ev aluations 4 000 20 000 40 000 100 000 200 000 GD 1.41e-02 2.59e-03 1.04e-03 3.45e-04 1.81e-04 TOL5 2.82e-02 4.01e-03 9.23e-04 1.09e-04 2.39e-05 spacing 1.30e-01 6.94e-02 6.03e-02 4.94e-02 4.46e-02 degenerated 0 0 0 0 0 T able 25: µ ARMOGA with the new prop osed arc hiving algorithm, four indi- viduals, problem DTLZ2, av erage for t wen ty seeds. ev aluations 4 000 20 000 40 000 100 000 200 000 GD 5.46e+01 2.35e+01 1.18e+01 1.68e+01 1.48e+01 TOL5 1.46e+02 1.96e+01 1.92e-01 1.70e-01 1.67e-01 spacing 3.40e+00 6.50e+00 5.09e+00 4.73e+00 4.86e+00 degenerated 2 3 3 3 3 T able 26: µ ARMOGA with crowding distance, four individuals, problem DTLZ4, a verage for tw en ty seeds. ev aluations 4 000 20 000 40 000 100 000 200 000 GD 1.87e-03 4.34e-04 2.19e-04 9.32e-05 3.39e-05 TOL5 3.42e-03 1.43e-04 3.48e-05 1.34e-05 8.82e-06 spacing 8.48e-01 1.46e-01 9.36e-02 7.00e-02 6.35e-02 degenerated 0 0 0 0 0 T able 27: µ ARMOGA with the new prop osed arc hiving algorithm, four indi- viduals, problem DTLZ4, av erage for t wen ty seeds. ev aluations 4 000 20 000 40 000 100 000 200 000 1 000 000 2 000 000 GD 9.01e-01 3.67e-01 2.81e-01 1.93e-01 1.27e-01 4.09e-02 3.93e-02 TOL5 1.21e+00 4.26e-01 3.32e-01 2.37e-01 1.66e-01 8.01e-02 8.51e-02 spacing 1.46e+00 7.03e-01 6.85e-01 6.17e-01 6.24e-01 5.83e-01 5.28e-01 degenerated 0 0 0 0 0 0 0 T able 28: µ ARMOGA with crowding distance, four individuals, problem WFG1, a verage for tw en ty seeds. 46 ev aluations 4 000 20 000 40 000 100 000 200 000 1 000 000 2 000 000 GD 7.95e-01 4.97e-01 3.77e-01 2.51e-01 1.75e-01 6.53e-02 4.79e-02 TOL5 9.27e-01 5.92e-01 4.56e-01 3.15e-01 2.29e-01 1.14e-01 9.69e-02 spacing 5.19e-01 2.82e-01 2.24e-01 1.96e-01 1.86e-01 1.53e-01 1.54e-01 degenerated 0 0 0 0 0 0 0 T able 29: µ ARMOGA with the new prop osed arc hiving algorithm, four indi- viduals, problem WFG1, av erage for t wen ty seeds. 47 Figure 17: P areto front after 4 000 function ev aluations, problem DTLZ1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 18: P areto front after 20 000 function ev aluations, problem DTLZ1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 19: P areto front after 200 000 function ev aluations, problem DTLZ1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). 48 Figure 20: P areto front after 4 000 function ev aluations, problem DTLZ2, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 21: P areto front after 20 000 function ev aluations, problem DTLZ2, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 22: P areto front after 200 000 function ev aluations, problem DTLZ2, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). 49 Figure 23: P areto front after 4 000 function ev aluations, problem DTLZ4, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 24: P areto front after 20 000 function ev aluations, problem DTLZ4, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 25: P areto front after 200 000 function ev aluations, problem DTLZ4, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). 50 Figure 26: P areto fron t after 200 000 function ev aluations, problem WF G1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 27: Pareto front after 1 000 000 function ev aluations, problem WF G1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). Figure 28: Pareto front after 2 000 000 function ev aluations, problem WF G1, µ ARMOGA with p opulation size 4 with crowding distance (left), and with the new proposed algorithm (right). 51