differential evolution with self-adaptation and local search for constrained multiobjective...

Differential Evolution with Self-adaptation and Local Search forConstrained Multiobjective Optimization

Ales Zamuda, Student Member, IEEE, Janez Brest, Member, IEEE, Borko Boskovic, Student Member, IEEE,Viljem Zumer, Member, IEEE

AbstractThis paper presents Differential Evolution withSelf-adaptation and Local Search for Constrained Multiob-jective Optimization algorithm (DECMOSA-SQP), which usesthe self-adaptation mechanism from DEMOwSA algorithmpresented at CEC 2007 and a SQP local search. The constrainedhandling mechanism is also incorporated in the new algorithm.Assessment of the algorithm using CEC 2009 special sessionand competition on constrained multiobjective optimizationtest functions is presented. The functions are composed ofunconstrained and constrained problems. Their results areassessed using the IGD metric. Based on this metric, algorithmstrengths and weaknesses are discussed.

I. INTRODUCTIONIn the last decade, there has been a growing interest in

applying randomized search algorithms such as evolutionaryalgorithms, simulated annealing, and tabu search to multi-objective optimization problems in order to approximate theset of Pareto-optimal solutions [1]. Various methods havebeen proposed for this purpose, and their usefulness hasbeen demonstrated in several application domains [2]. Sincethen a number of performance assessment methods havebeen suggested. Due largely to the nature of multiobjectiveevolutionary algorithms (MOEAs), their behaviors and per-formances are mainly studied experimentally. Most of theexisting simulation studies comparing different evolutionarymultiobjective methodologies are based on specific perfor-mance measures.

Several continuous multiobjective optimization problem(MOP) test suites have been proposed by the evolutionarycomputation community [2], which have played a crucialrole in developing and studying MOEAs. However, moretest instances are needed to resemble complicated real-life problems and thus stimulate the MOEA research. Thereport [3] suggests a set of unconstrained (bound constrained)MOP test instances and a set of general constrained testinstances for the CEC09 MOP algorithm contest. It alsoprovides performance assessment guidelines.

This paper presents a hybrid algorithm for multiobjectiveoptimization which combines our self-adaptive differentialevolution mechanism [4], [5], a local search technique, anda constraint handling technique. The self-adaptation mecha-nism used was introduced in algorithm SA-DE [4] and lateron used in the DEMOwSA algorithm [5]. The used local

The authors are with the Computer Architecture and Languages Lab-oratory, Institute of Computer Science, Faculty of Electrical Engineeringand Computer Science, University of Maribor, Smetanova ulica 17, SI-2000Maribor, Slovenia (phone: ++386 2 220 7404; fax: ++386 2 220 7272; email:{ales.zamuda, janez.brest, borko.boskovic, zumer}@uni-mb.si).

search method is implemented by an algorithm of sequentialquadratic programming SQP [6]. SQP and SA-DE methodsare applied to the DEMO [7] algorithm to create newcandidate solutions in optimization process. Hybridization ofboth methods is done by improving some candidate solutionsgenerated by differential evolution with SQP. The solutionobtained with SQP replaces the candidate solution only ifit dominates it. The multiobjective evolutionary optimizercore was based on the DEMOwSA algorithm [5], [7], [8],[9]. Among the other optimizers using Differential Evolution(DE) [10] in multiobjective optimization [11] the DEMOalgorithm is using DE for the candidate solution creation.Environmental selection in DEMO can be chosen to be eitherthe NSGA-II [12], SPEA2 [13], or IBEA [14] algorithms en-vironmental selection. Our previous experience showed thatDE with self-adaptation, which is well known in evolutionstrategies [15], [16], [17], can lead to faster convergencethan the original DE [18], [19], [20], [21]. Therefore itwas decided in [5] to extend the DEMO algorithm byincorporating the self-adaptation mechanism in DE controlparameters.

Assessment of the new algorithm is performed on thetest functions from the CEC 2009 special session [3] withthe therein provided performance assessment metrics. Thetest functions suite is divided among constrained and uncon-strained problems. Some of the unconstrained problems aresimilar to [22], e.g. two DTLZ [23] problems and the hardestCEC 2007 competition problem WFG1 [24]. DTLZ andWFG problems are composed of 5-objective functions. Theremaining unconstrained problems have 3 or less objectives.The 10 constrained problems have 3 or less objectives and2 or less constraints on these objectives.

This paper is organized as follows. In the next section,more related work is presented, such as DEMOwSA al-gorithm, sequential quadratic programming method, and aconstraint handling mechanism. In the third section, our newalgorithm DECMOSA-SQP is described. The fourth sectiondescribes the experiments and the obtained results on theCEC 2009 special session test suite. The results are assessedby accompanying IGD performance metric. The last sectionconcludes with final remarks and prepositions for futurework.

II. RELATED WORKIn this section we give an overview of previous work. The

Differential Evolution for Multiobjective Optimization withSelf-adaptation (DEMOwSA) algorithm [5] is presented. The

self-adaptive mechanism used in our DEMOwSA algorithmis briey outlined. In a following subsection the SQP lo-cal search procedure [25] is described. The algorithm wasextended to handle constrained multiobjective optimizationproblems from the method used in jDE-2 [26] which isbased on [18], [19] and its constraint handling mechanism isexplained.

A. Differential EvolutionDifferential Evolution (DE) [27], [10] is a floating-point

encoding evolutionary algorithm for global optimization overcontinuous spaces [28], [29], which can also work withdiscrete variables. It was proposed by Storn and Price [10]and since then it has been modified and extended severaltimes with new versions being proposed.

In [30], a new DE mutation operator was introduced. Aliand Torn in [31] also suggested some modifications to theclassical DE to improve its efficiency and robustness.

DE is a population based algorithm and vector ~xi,G, i =1, 2, ...,NP is an individual in the population. NP denotespopulation size and G the generation. During one generationfor each vector, DE employs mutation, crossover and selec-tion operations to produce a trial vector (offspring) and toselect one of those vectors with best fitness value.

By mutation for each population vector a mutant vector~vi,G+1 is created. One of the most popular DE mutationstrategy is rand/1/bin [28], [10]:

~vi,G+1 = ~xr1,G + F (~xr2,G ~xr3,G), (1)where the indexes r1, r2, r3 represent the random and mu-tually different integers generated within the range [1,NP ]and also different from index i. F is an amplification factorof the difference vector within the range [0, 2], but usuallyless than 1. Another DE strategy is best/1/bin [28], [10]:

~vi,G+1 = ~xbest,G + F (~xr2,G ~xr3,G), (2)where ~xbest,G denotes the best evaluated vector in generationG. Newer DE strategy rand/1/either-or [28] is:~vi,G+1 = ~xr1,G+0.5(Fi,G+1 +1) (~xr2,G+~xr3,G2~xr1,G).

(3)DE was also introduced for multiobjective optimiza-

tion [32], [7], [5], [33]. The original DE algorithm is welldescribed in literature [28], [10], and therefore, we will skipdetailed description of the whole DE algorithm.

B. Self-adaptation in DEEvolution strategies [15], [16], [17], [34] include a self-

adaptive mechanism [35], [36], [37], encoded directly in eachindividual of the population to control some of the parame-ters [38] in the evolutionary algorithm. Self-adaptation allowsan evolution strategy to adapt itself to any general class ofproblems, by reconfiguring itself accordingly, and to do thiswithout any user interaction [16], [34].

An evolution strategy (ES) has a notation /, -ES,where is parent population size, is the number of parentsfor each new individual, and is child population size.

An individual is denoted as ~a = (~x,~s, f(~x)), where ~x aresearch parameters, ~s are control parameters, and f(~x) isthe evaluation of the individual. Although there are othernotations and evolution strategies, we will only need a basicversion.

DE creates new candidate solutions by combining theparent individual and a few other individuals of the samepopulation. In each iteration, a candidate replaces the parentonly if it has better fitness value. DE has three controlparameters: amplification factor of the difference vector F , crossover control parameter CR, and population size NP .

The original DE algorithm keeps all three control param-eters fixed during the optimization process. However, therestill exists a lack of knowledge how to find reasonably goodvalues for the control parameters of DE, for a given function[39], [40]. Based on the experiment in [18], the necessity ofchanging control parameters during the optimization processwas confirmed. In [26] a performance comparison of certainselected DE algorithms, which use different self-adapting oradapting control parameter mechanisms, was reported.

C. Differential Evolution for Multiobjective Optimizationwith Self Adaptation (DEMOwSA)

In [5] we use the idea of self-adaptation mechanismfrom evolution strategies and apply this idea to the controlparameters of the candidate creation in the original DEMOalgorithm [7]. We named the new constructed version ofDEMO algorithm, DEMOwSA algorithm.

Each individual of DEMOwSA algorithm [5] is extendedto include self-adaptive F and CR control parameters toadjust them to the appropriate values during the evolutionaryprocess. The F parameter is mutation control parameter andCR is the crossover control parameter.

For each individual in the population, a trial vector is com-posed by mutation and recombination. The mutation proce-dure is different in the DEMOwSA algorithm in comparisonto the original DEMO. For adaptation of the amplificationfactor of the difference vector Fi for trial individual i, fromparent generation G into child generation G+ 1 for the trialvector, the following formula is used:

Fi,G+1 = FGi eN(0,1),where denotes learning factor and is usually proportional to = 1/(

2D), D being a dimension of the problem. N(0, 1)

is a random number with a Gauss distribution. The FGidenotes averaging the parameter F of the current individuali and the randomly chosen individuals r1, r2, and r3 fromthe generation G:

FGi = Fi,G + Fr1,G + Fr2,G + Fr3,G4

,

where indices r1, r2, and r3 are the indices of individualsalso being used in the chosen DE strategy for the mutationprocess. The mutation process for i-th candidate ~vi,G+1 forgeneration G+ 1 is defined as:

~vi,G+1 = ~xr1,G + Fi,G+1 (~xr2,G ~xr3,G),

where ~xr1,G, ~xr2,G, and ~xr3,G are search parameter valuesof the uniform randomly selected parents from generation G.

An analogous formula is used for the CR parameter:

CRi,G+1 = CRGi eN(0,1),

where is same as in the adaptation of F parameter andCRGi denotes averaging:

CRGi = CRi,G + CRr1,G + CRr2,G + CRr3,G4

.

Recombination process is taken from the strategyrand/1/bin used in the original DEMO [7] or the originalDE [10], and the adapted CRi,G+1 is used to create amodified candidate ui,j,G+1 by binary crossover:

ui,j,G+1 =

vi,j,G+1 if rand(0, 1) CRi,G+1 or

j = jrand

xi,j,G otherwise,

where j [1,D] denotes the j-th search parameter,rand(0, 1) [0, 1] denotes a uniformly distributed randomnumber, and jrand denotes an uniform randomly chosenindex of the search parameter, which is always exchanged.

The selection principle also helps in adapting F andCR parameters, because only the individuals adapting goodcontrol and search parameter values survive. With the moreappropriate values for the self-adaptive control parameters,the search process attains better objective space parameters.Therefore, the search process converges faster to better indi-viduals which, in turn, are more likely to survive and produceoffspring and, hence, propagate these better parameter values.

D. Sequential Quadratic ProgrammingSequential Quadratic Programming (SQP) [41], [25] is a

non-linear optimization method based on gradient compu-tation. It is a generalization of Newtons method to mul-tiple dimensions, incorporating an quadratic approximationmodel for the objective function given an initial guessfor the solution. Great strength of the SQP method is itsability to solve problems with nonlinear constraints. Theapproximation model is solved at each iteration to yield astep toward the solution of the original problem. However,it is not necessary that the solution of the approximationmodel always improves the original problem. As with mostoptimization methods, SQP is not a single algorithm, butrather a conceptual method from which numerous specificalgorithms have evolved [25].

The algorithm for SQP we have used is FSQP-AL and itsimplementation in CfSQP [6]1. The norm of descant directionwas set to = 1.e 08. Constraints were enforced onlyin terms of bounds to search parameters, i.e. linear bound-constraints were used.

1We used package CfSQP with academic license.

E. Constraint HandlingOver the last few years several methods have been pro-

posed for handling constraints by genetic algorithms forparameter optimization problems. These methods have beengrouped by Michalewicz et al. [42], [43] into four categories:

1) Methods based on preserving the feasibility of solu-tions. The idea behind the method is based on special-ized operators which transform feasible parents intofeasible offspring.

2) Methods based on penalty functions. Many evolu-tionary algorithms incorporate a constraint-handlingmethod based on the concept of exterior penalty func-tions which penalize infeasible solutions.

3) Methods which make a clear distinction between feasi-ble and infeasible solutions. There are a few methodswhich emphasize the distinction between feasible andinfeasible solutions in the search space.

4) Other hybrid methods. These methods combine evo-lutionary computation techniques with deterministicprocedures for numerical optimization problems.

Most constrained problems can be handled by the penaltyfunction method. A measure of the constraint violation isoften useful when handling constraints. A solution ~x isregarded as feasible with regards to inequality constraintsgi(~x) 0 and equality constraints hj(~x) = 0 if

gi(~x) 0, i = 1, ..., q, (4)|hj(~x)| 0, j = q + 1, ...,m, (5)

where equality constraints are transformed into inequalities.The mean value of all constraints violations v is defined as:

v =(q

i=1 Gi(~x) +m

j=q+1 Hj(~x))

m, (6)

where

Gi(~x) =

{gi(~x), gi(~x) > 0,

0, gi(~x) 0,(7)

Hj(~x) =

{|hj(~x)|, |hj(~x)| > 0,0, |hj(~x)| 0.

(8)

The sum of all constraint violations is zero for feasiblesolutions and positive when at least one constraint is violated.An obvious application of the constraint violation is to use itto guide the search towards feasible areas of the search space.There has been quite a lot of work on such ideas and otherconstraint techniques within the EA-community. A summaryof these techniques can be found in [44], [45], which alsocontains information on many other stochastic techniques.

The recent -jDE algorithm [46] follows the jDE-2 algo-rithm [19] and emphasizes constraints as follows. It comparestwo solutions, say i and j, during the selection operation:

~xi,G+1 =

~xj,G if (vi,G > vj,G),

~xj,G else if (vj,G = 0) (f(~xi,G) > f(~xj,G)),~xi,G otherwise.

(9)

The algorithm distinguishes between feasible (v = 0) andinfeasible individuals: any feasible solution being better thanany infeasible one.

F. Controlling the LevelThe jDE-2 algorithm had difficulties when solving con-

strained optimization problems with equality constraints.Takahama and Sakai in [47] pointed out that for problemswith equality constraints, the level should be controlledproperly in order to obtain high quality solutions.

The -jDE algorithm uses level controlling. In theproposed method level constraint violation precedes theobjective function. A method of controlling the level isdefined according to equations (10)(13). The level is up-dated until the number of generations G reaches the controlgeneration Gc. After the number of generations exceeds Gc,the level is set to 0 to obtain solutions with minimumconstraint violation

0 = (10)v0 = v(~x) (11)

vG =

{1vG1, 2v(~x) < vG1, 0 < G < Gc

vG1, otherwise(12)

G =

{max{vG(1 GGc )cp , }, 0 < G < Gc0, G Gc

(13)

where ~x is the top -th individual and = 0.3NP . Note,that (0) = 0 when mean violation v(~x) is calculated.Similarly, ~x is the top -th individual and = 0.7NP . cpis a parameter to control the speed of constraints reducingrelaxation, while parameters 1 < 1 and 2 > 1 adaptivelycontrol vG value, which also controls the speed of theconstraints reducing relaxation. Parameters 1 and 2 canonly decrease the vG value by a small amount when top individuals have mean violations v(~x) multiplied by 1less than vG. Using this adaptation, the level could reach0 before G Gc.

III. DIFFERENTIAL EVOLUTION WITH SELF-ADAPTATIONAND LOCAL SEARCH FOR CONSTRAINED

MULTIOBJECTIVE OPTIMIZATION ALGORITHM(DECMOSA-SQP)

Hybridization of differential evolution in the algorithmDEMOwSA and of the local search SQP was done so thatusing SQP some of the individuals in DEMOwSA populationwere chosen and local search was performed on them. TheDEMOwSA algorithm searched over the criteria space glob-ally, while local search tried to improve locally in the criteriaspace, using an approximation model for the problem criteriaspace. The local search was enforced only each genLS = 25generations and for SQP a maximum of FEsLS = 5000function evaluations was allowed per whole population. Fromthe population up to indLS = 10% (parameter value obtainedempirically) randomly chosen individuals were processedusing local search.

We also enforced another criteria for selection of theseindividuals. Namely, an individual was considered for localsearch only if was improved since last considered for localsearch. Using this criteria same solution was not sent to localsearch twice, since it would produce same result and onlywaste function evaluations.

Local search method is performed on each individual upto two times after being selected for local search. First,miter1 = 3 iterations of SQP are performed. Then, ifthe obtained individual dominates the original individual insolution space, miter2 = 10 more iterations of SQP are doneon the improved individual. After the second local searchsimilar, only if the new individual dominates the originalindividual, the original individual is replaced.

The constraint handling was introduced in theDECMOSA-SQP algorithm by altering the dominationprinciple. This affected the 1) candidate replacementstrategy used in DEMO for instant update of currentpopulation and 2) multiobjective selection strategy forpruning of non-dominated individuals after each generation.Both of the strategies affected use the domination principle.The selection strategy uses the domination principle whencalculating fitness value for SPEA2 [13].

IV. RESULTSIn this section results are given in the format specified for

CEC 2009 Special Session and Competition on ConstrainedMultiobjective Optimization given in [3].A. PC Configuration

All tests were performed on the following PC configura-tion. System: Linux, 2.6.17-1.2142 FC4smp. CPU: Twice 64-bit Dual Core AMD Opteron(tm) Processor 280 (2390.651MHz, 4787.74 bogomips). RAM: 8GB. Language: C++.B. Algorithmic Parameter Settings

The DECMOSA-SQP algorithm has many parameters,among which the most important are the selection algorithmand candidate creation strategy. We have chosen SPEA2 asthe selection algorithm and our candidate creation strategybased on self-adaptation mechanism. The usual dynamicrange of F parameter is F [0, 1], and for CR it isCR [0, 1]. With the applied adaptation mechanism, somecontrol parameter constraints were used, as will be describedbelow. The estimated cost of parameter tuning in terms ofnumber of function evaluations is zero only a few additionalrandom numbers and simple calculations are conducted toexpress the adapted control parameters. No search is requiredand one control parameters adaptation is executed beforeeach function evaluation.

In the presented experiments the following parametersettings were used for our algorithm. The global lower andupper bounds for control parameter F were 0.1 F 0.9,and for control parameter CR they were 1/24 CR 0.9.Their initialization was random value between 0 and 1, forboth F and CR, respectively. The parameter was set to = 2/

2D, which conforms to the recommendation from

[15]. The population size parameter NP was set equal toapproximation set size for each of the functions, i.e. 100,150, and 800, respectively.

For the constraint handling method, following parameterswere chosen: = 0.3NP , = 0.7NP , cp = 5, 1 = 0.8,2 = 2.0, and Gc=0.2Gmax.

C. Obtained ResultsFor 30 runs on the test function suite [3] we obtained the

IGD metric values given in Figure 1. The subfigures showevolution of the means and standard deviations of IGD metricvalues of the approximate sets obtained with the number offunction evaluations (generations) for all test instances. Forall functions, mean IGD value gradually improves with somedeteriorations due to the multiobjective selection tradeoffdifferences in IGD and SPEA2 metrices.

Sometimes the SPEA2 multiobjective selection metricused in our algorithm, assessed an approximation set differ-ently to the IGD metric. While the SPEA2 metric seemed toimprove the approximation sets and the standard deviationof the measured approximation sets was improving, theirIGD metric assessment value got worse. This effect can beobserved at ends of the optimization runs for functions UF2,UF3, UF4, R2 DTLZ3 M5, CF1, CF4, CF8, and CF10.

Standard deviation also follows the mean IGD values givenin Figure 1, except for the functions CF1 and WFG1. Forfunction WFG1, standard deviation increases each time thelocal search is activated. For function CF1, the standarddeviation follows the mean values of IGD, but starts tofluctuate more after nearly 0.1 IGD value is attained whichagain seems to be due to differences in selection metrices.

The average IGD metric vales of the 30 final approx-imation sets obtained for each test problem are given inTable I. As seen from the assessment of results, the Paretofronts of the unconstrained function instances were all ap-proximated to the extent of IGD metric value being smallerthan 0.38, except for the R2 DTLZ3 M5 and WFG1 M5problems. The best approximated unconstrained functionsseem to be UF1UF4 and UF7. The constrained functionswere all approximated to the extent of feasible region, withthe exception of CF3 test problem. The IGD metric values ofthe remaining constrained test problems was below 0.2605for CF1, CF2, CF4, and CF6CF9. For test problem CF3,no feasible solution was obtained.

The best attained approximation sets, i.e. those withminimal IGD metric values, are shown in Figure 2. As itwas also evidenced from the Table I and Figure 1, it isclearly seen that the best attained function was UF4. Forfunctions CF1 and CF2, many non-feasible solutions werefound and therefore also, the evolution of their standarddeviations for IGD metric in Figure 1 is fluctuating so much.The algorithm had most problems with the constrained testproblem instances. For functions CF1 and CF3CF7, therewere only few non-dominated solutions in the approximationsets. For the functions CF8CF10, there were more solutionsin the approximation sets, but still quite sparse which again

TABLE ITHE AVERAGE IGD METRIC VALES OF THE 30 FINAL APPROXIMATION

SETS OBTAINED AND CPU TIME IN SECONDS USED FOR EACH TESTPROBLEM.

Function name IGD mean IGD std IGD best IGD worst CPU timeUF1 0.0770281 0.039379 0.055126 0.0880129 4.434UF2 0.0283427 0.0313182 0.0173361 0.040226 7.17267UF3 0.0935006 0.197951 0.0305453 0.168162 8.17833UF4 0.0339266 0.00537072 0.0316247 0.035643 5.74567UF5 0.167139 0.0895087 0.133012 0.237081 3.48233UF6 0.126042 0.561753 0.0579174 0.589904 3.41167UF7 0.024163 0.0223494 0.0198913 0.0427502 6.206UF8 0.215834 0.121475 0.0989388 0.228895 16.6217UF9 0.14111 0.345356 0.0772668 0.332909 10.2383

UF10 0.369857 0.65322 0.238279 0.580852 6.818CF1 0.107736 0.195921 0.0581246 0.186248 28.1633CF2 0.0946079 0.294282 0.037197 0.257011 4.85267CF3 1000000 0 1e+06 1e+06 3.574CF4 0.152659 0.466596 0.0533448 0.301675 4.08333CF5 0.412756 0.590779 0.0963843 0.537761 4.598CF6 0.147824 0.124723 0.0823086 0.186999 6.81233CF7 0.26049 0.259938 0.170495 0.373809 4.586CF8 0.176344 0.625776 0.0977472 0.420665 16.7807CF9 0.127132 0.145787 0.108369 0.265246 13.4307

CF10 0.507051 1.19892 0.284772 0.811085 16.7113R2 DTLZ2 M5 0.383041 0.11695 0.336664 0.418682 104.06R2 DTLZ3 M5 943.352 551.654 745.167 1206.18 156.276

WFG1 M5 1.9178 1.42433 1.83629 3.31374 170.543

shows that the constrained function subset was much harderto solve than the unconstrained subset.

The average CPU time used for each test problem was alsomeasured. The function runs were performed in parallel, buteach run was run in sequence, from random seeds takingvalues of 1 to 30. The average user mode CPU times for theprocesses of each optimization run are given in Table I. Asseen from the results reported, optimization of test problemsR2 DTLZ2 M5, R2 DTLZ3 M5, and WFG1 M5 take themost CPU time. All the remaining test problems optimiza-tions finished below 30 seconds, i.e. less than 15 minutes for30 runs of a test problem. The total execution time of theparallelized optimization and performance assessment withthe IGD metric takes less than 4 hours on the system used.

V. CONCLUSIONSWe have presented an algorithm for constrained mul-

tiobjective optimization. Its performance assessment wasobtained using the test function suite from special sessionon multiobjective optimization at CEC 2009. On 23 testproblems 30 runs are performed and results are assessedusing the IGD metric. The results indicate that the approachused may be considered for constrained optimization aswell the unconstrained multiobjective optimization. Futurework could assemble in more instrumentation of the controlparameters of the presented DECMOSA-SQP algorithm.

0 0.2 0.4 0.6 0.8

1 1.2 1.4

0 500 1000 1500 2000 2500

IGD mean for UF1IGD standard deviation for UF1

(a) UF1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 500 1000 1500 2000 2500 3000


(b) UF2 0

0.2 0.4 0.6 0.8

1 1.2

0 500 1000 1500 2000 2500 3000


(c) UF3

0 0.02 0.04 0.06 0.08 0.1

0.12 0.14 0.16 0.18 0.2

0 500 1000 1500 2000 2500


(d) UF4 0 1

2

3 4

5 6

0 500 1000 1500 2000 2500


(e) UF5 0 1

2

3 4

5 6

0 500 1000 1500 2000 2500


(f) UF6

0 0.2 0.4 0.6 0.8

1 1.2 1.4

0 500 1000 1500 2000 2500


(g) UF7 0

0.5 1

1.5 2

2.5 3

3.5

0 200 400 600 800 1000 1200 1400 1600 1800 2000


(h) UF8 0

0.5 1

1.5 2

2.5 3

3.5

0 200 400 600 800 1000 1200 1400 1600 1800 2000


(i) UF9

0 2 4 6 8

10 12 14 16

0 200 400 600 800 1000 1200 1400 1600 1800


(j) UF10 0

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

0 50 100 150 200 250 300 350 400

IGD mean for R2_DTLZ2_M5IGD standard deviation for R2_DTLZ2_M5

(k) R2 DTLZ2 M5 0

500 1000 1500 2000 2500 3000

0 50 100 150 200 250 300 350 400

IGD mean for R2_DTLZ3_M5IGD standard deviation for R2_DTLZ3_M5

(l) R2 DTLZ3 M5

0 0.5

1 1.5

2 2.5

3 3.5

4

0 50 100 150 200 250 300 350

IGD mean for WFG1_M5IGD standard deviation for WFG1_M5

(m) WFG1 0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45

0 500 1000 1500 2000 2500 3000

IGD mean for CF1IGD standard deviation for CF1

(n) CF1 0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6

0 500 1000 1500 2000 2500 3000


(o) CF2

0

200000

400000

600000

800000

1e+06

0 500 1000 1500 2000 2500 3000


(p) CF3 0

0.5 1

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2.5 3

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(q) CF4 0 2 4 6 8

10 12

0 500 1000 1500 2000 2500 3000


(r) CF5

0 1

2

3 4

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(s) CF6 0 2 4 6 8

10 12 14

0 500 1000 1500 2000 2500 3000


(t) CF7 0 1 2 3 4 5 6 7 8 9

0 200 400 600 800 1000 1200 1400 1600 1800 2000


(u) CF8

0 0.5

1 1.5

2 2.5

3

0 200 400 600 800 1000 1200 1400 1600 1800 2000


(v) CF9 0 2 4 6 8

10 12

0 200 400 600 800 1000 1200 1400 1600 1800 2000


(w) CF10

Fig. 1. Evolution of the means and standard deviations of IGD metric values of the approximate sets obtained with the generation number for all testinstances.

0 0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2

UF1 approximation setUF1 Pareto front

(a) UF1 0

0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(b) UF2 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1


(c) UF3

0 0.2 0.4 0.6 0.8

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0 0.2 0.4 0.6 0.8 1 1.2


(d) UF4 0

0.2

0.4

0.6

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1

0 0.2 0.4 0.6 0.8 1


(e) UF5 0

0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(f) UF6

0 0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(g) UF7

0 0.2

0.4 0.6

0.8 1

1.2 1.4

1.6 0 0.5 1

1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 1.2

(h) UF8 0

0.5 1

1.5 2

2.5 0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

(i) UF9

0 0.2

0.4 0.6

0.8 1

1.2 0 0.2 0.4

0.6 0.8 1 1.2

1.4 0 0.2

0.4 0.6 0.8 1 1.2

(j) UF10 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

CF1 approximation setCF1 Pareto front

(k) CF1 0

0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(l) CF2

0 0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(m) CF3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 1.1

0 0.2 0.4 0.6 0.8 1


(n) CF4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 1.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


(o) CF5

0 0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


(p) CF6 0

0.2 0.4 0.6 0.8

1 1.2

0 0.2 0.4 0.6 0.8 1 1.2


(q) CF7

0 0.5

1 1.5

2 2.5

3 3.5 0

0.5 1 1.5 2

2.5 3 0 0.5 1 1.5 2 2.5 3

(r) CF8

0 0.5

1 1.5

2 2.5

3 0 0.5 1

1.5 2 2.5

0 1 2 3 4 5 6

(s) CF9 0

0.2 0.4

0.6 0.8

1 1.2 0

5 10 15 20

25 30 35

0 0.2 0.4 0.6 0.8 1 1.2

(t) CF10

Fig. 2. Pareto front and best attained approximation set for all test instances.

ACKNOWLEDGMENT

We thank the authors of CfSQP for providing us with theacademic license of their software.

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