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A New Memetic Strategy for the Numerical Treatment ofMulti-Objective Optimization Problems

Oliver SchuetzeCINVESTAV-IPN

Computer ScienceDepartment

Mexico City 07300,MEXICO

[email protected]

Gustavo SanchezSimon Bolivar University

Industrial TechnologyDepartment

Valle de Sartenejas, Edo.Miranda. VENEZUELA

[email protected]

Carlos A. Coello CoelloCINVESTAV-IPN

Computer ScienceDepartment

Mexico City 07300,MEXICO

[email protected]

ABSTRACT

In this paper we propose a novel iterative search procedurefor multi-objective optimization problems. The iterationprocess – though derivative free – utilizes the geometry of the directional cones of such optimization problems, and is

capable both of moving toward and along the (local) Paretoset depending on the distance of the current iterate towardthis set. Next, we give one p ossible way of integrating thislocal search procedure into a given EMO algorithm result-ing in a novel memetic strategy. Finally, we present somenumerical results on some well-known benchmark problemsindicating the strength of both the local search strategy aswell as the new hybrid approach.

Categories and Subject Descriptors

G.1.6 [Numerical Analysis]: Optimization

General Terms

Algorithms, Performance

Keywords

multi-objective optimization, memetic algorithm, hill climber,Pareto set

1. INTRODUCTIONIn general, deterministic search methods present fast (lo-

cal) convergence properties which surpass evolutionary mul-tiobjective (EMO) algorithms properties. On the other hand,many EMOs accomplish the ‘global task’ exceedingly, thatis, they tend to find very quickly a rough approximation of the entire solution set (the Pareto set ), even for highly non-linear models. Thus, to improve the overall performance of

the search procedure, it can be advantageous to hybridize

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such two methods leading to a so-called memetic EMO (ME-MO, see e.g., [18], [15], [4]).One of the first MEMOs for models on discrete domains waspresented in [14] as a ‘Multi-Objective Genetic Local Search’(MOGLS) approach. In this algorithm, the authors proposeto use the local search method after the usual variation op-erators are applied. Another historically important MEMO,called M-PAES, was proposed in [16]. Unlike the Ishibuchiapproach, M-PAES is based on a pure local search algorithm(PAES), but adding a population with a crossover operator.Two archives are also used: one that maintains the globalnon-dominated solutions and the other that is used as thecomparison set for the local search phase.The continuous case – i.e., continuous objectives defined ona continuous domain – was first explored in [11], where aneighborhood search was applied to NSGA-II [7]. In theirinitial work, the authors applied the local search only afterNSGA-II had ended. Later works compare this approachwith the same local search method being applied after everygeneration. Evidently, they found that the added computa-

tional workload impacted efficiency.In [13] a gradient-based local algorithm, sequential quadraticprogramming (SQP), was used in combination with NSGA-II and SPEA to solve the ZDT benchmark suite [26]. Theauthors stated that if there are no local Pareto fronts, thehybrid EMO has faster convergence toward the true Paretofront than the original one, either in terms of the objectiveevaluations or in terms of the CPU time consumed (recallthat a gradient-based algorithm is utilized and the sole usageof the number of function calls as basis for a comparison canbe misleading). Furthermore, they found that the hybridiza-tion technique does not decrease the solution diversity.In this work, we present a novel local search operator, theHill Climber with Sidestep (HCS). This iteration processaims to find a sequence of ‘better’ solutions (hill climber). If

no better solution is found within a certain number of trials– which indicates that the actual iterate is already ‘near’ to alocal solution – the process automatically tries to determinethe next iterates along the Pareto set (sidestep). Addition-ally, we show one possible way to integrate the HCS into agiven EMO algorithm, and thus, how to construct a novelMEMO. Unlike previous works (see [13] and [23]) the pro-posed operator does not require any gradient information,but such information can easily be integrated if provided bythe model.

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The remainder of this paper is organized as follows. In Sec-tion 2, we state some theoretical background. In Section 3,we introduce the HCS, a novel iterative procedure which isof local nature, and propose one possible way to integratethis algorithm into a global search procedure. In Section 4,we show some numerical results, and finally, some conclu-sions are drawn in Section 5.

2. BACKGROUNDIn the following we consider continuous multi-objective

optimization problems of the form

minx∈Q

{F (x)}, (MOP)

where Q ⊂Ê

n and the function F is defined as the vectorof the objective functions

F : Q → Ê

k, F (x) = (f 1(x), . . . , f k(x)),

and where each f i : Q →Ê

is continuous.The optimality of a point x ∈ Q is based on the conceptof dominance which dates back over a century and was pro-posed first by Pareto ([24]).

Def 2.1 (a) Let v, w ∈Ê

k. Then the vector v is less thanw ( v <p w), if vi < wi for all i ∈ {1, . . . , k}. Therelation ≤p is defined analogously.

(b) A vector y ∈Ê

n is dominated by a vector x ∈Ê

n

( x ≺ y) with respect to (MOP) if F (x) ≤p F (y) and F (x) = F (y), else y is called non-dominated by x.

(c) A point x ∈ Q is called Pareto optimal or a Paretopoint if there is no y ∈ Q which dominates x.

The set of all Pareto optimal solutions is called the Paretoset . This set typically – i.e., under mild regularity assump-tions – forms a (k − 1)-dimensional object. The image of thePareto set is called the Pareto front .

3. THE MEMETIC STRATEGYIn the following we propose a novel iterative local search

procedure, the HCS, and present further on one possible wayto integrate this procedure into a given EMO algorithm inorder to obtain a MEMO.

3.1 A Local Search Procedure: Hill-Climberwith Sidestep (HCS)

Some requirements on such a process, which will also beaddressed here, are as follows:

(a) The process should work with or without gradient in-

formation (whether or not provided by the model).

(b) If a given point x0 ∈ Q is ‘far away’ from the Paretoset, it is desirable that the next iterate x1 is better (i.e.,x1 ≺ x0). If x0 is already ‘near’ to the Pareto set, asearch along the solution set is of particular interest.

(c) It is desirable that the search procedure can decideautomatically between the two situations described in(b) and generate a candidate solution according to thegiven situation.

(a)

(b)

Figure 1: The descent cone (shaded) for an MOPwith 2 parameters and 2 objectives during initial(a) and final (b) stages of convergence. The descentcone shrinks to zero during the final stages of con-vergence. The figure is taken from [4].

(d) The procedure should be capable of handling (inequal-ity1) constraints.

In [3] a good insight into the structure of multi-objectiveproblems is given by analyzing the geometry of the direc-tional cones at different stages of the optimization process:when a point x0 is ‘far away’ from any local Pareto optimalsolutions, the gradients’ objectives are typically aligned andthe descent cone is almost equal to the half-spaces associ-ated with each objective. Therefore, for a randomly chosensearch direction ν , there is a nearly 50 % chance that thisdirection is a descent direction (i.e., there exits an h0 ∈

Ê +

such that F (x0+h0ν ) <p F (x0)). If on the other side a pointx0 is ‘close’ to the Pareto set, the individual gradients arealmost contradictory (compare also to the famous theoremof Kuhn and Tucker [17]), and thus the size of the descentcone is extremely narrow, resulting in a small probability fora randomly chosen vector to be a descent direction (see also

Figure 1). The Hill-Climber with Sidestep (HCS) was con-structed on the basis of these observations. In the following,the HCS algorithm is described for the case in which no gra-dient information is available, since that is more common forreal-world engineering problems which is the main area of application for EMO algorithms. Further, we describe the

1Equality constraints, which are in general much more diffi-cult to satisfy than inequality constraints, may be possibleto relax by converting them into suitable inequality con-straints with some loss of accuracy. See [5] for a thoroughdiscussion.

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unconstrained case. Possible modifications for both differ-entiable and constrained models are given below.Given a point x0 ∈ Q, the next iterate is selected as follows:a further point x1 is chosen randomly from a neighborhoodof x0, say x1 ∈ B(x0, r) with

B(x0, r) := {x ∈Ê

n : x0,i − ri ≤ xi ≤ x0,i + ri ∀i = 1, . . ,n},(1)

where r ∈Ê

n+ is a given (problem depending) radius. If

x1 ≺ x0, then ν := x1 − x0 is a descent direction2

at x0,and along it a ‘better’ candidate can be searched, for ex-ample via line search methods (see below for one possiblerealization). If x0 ≺ x1 the same procedure can be appliedto the opposite direction (i.e., ν := x0 − x1) starting withx1. If x0 is ‘far away’ from any local solution, the chance isquite high that domination occurs (see above). If x0 and x1are mutually non-dominating, the process will be repeatedwith further candidates x2, x3, . . . ∈ B(x0, r). If only mutu-ally nondominated solutions (xi, x0) are found within N ndsteps, this indicates, using the above observation, that thepoint x0 is already near to the (local) Pareto set, and henceit is desirable to search along this set. For this, we suggestto use the accumulated information by taking the averagesearch direction

ν acc =1

N nd

N ndXi=1

xi − x0xi − x0

, (2)

since with this direction one expects the maximal diversity(or ‘sidestep’) among the available directions. This directionhas previously been proposed as a local guide for a multi-objective particle swarm algorithm in [2]. Note that this isa heuristic that does not guarantee that ν acc indeed p ointsto a diversity cone. In fact, it can happen that this vectorpoints to the descent or ascent cone, though the probabilityfor this is low for points x0 ‘near’ to a local solution dueto the narrowness of these cones. However, in both casesAlgorithm 1 acts like a classical hill climber (i.e., searches

for better points) which is in the scope of the procedure.Alternatively, to prevent this and to be sure to search in adiversity direction, one can restrict (2) to points xi that liein one common descent cone (for instance, for k = 2 the cone{+, −} is the one where the values of objective f 1 are greaterthan at the given point x0 while the values of f 2 are less, seeFigure 1). However, note that there exist for k objectives2k−2 such cones (for instance, for k = 3 the cones {−, −, +},{−, +, −}, {+, −, −}, {−, +, +}, {+, −, +}, {+, +, −}), andit is ad hoc unclear which of them to favor in each situation.A pseudocode of the HCS is given in Algorithm 1, and in thefollowing we go into detail for possible realizations. Table 1summarizes the design parameters those values have to bechosen when realizing the HCS as it is proposed here.

Computation of tk. The situation is that we are given twopoints x0, x1 ∈

Ê

n such that x1 ≺ x0. That is, there existsa subsequence {i1, . . . , il} ⊂ {1, . . . , k} with

f ij (x1) < f ij (x0), j = 1, . . . , l ,

and thus, ν := x1 − x0 is a descent direction for all f ij ’sat the point x0. For this case there exist various strategies2In the sense that there exists a t ∈

Ê + such that f i(x0 +tν ) < f i(x0), i = 1, . . . , k, but not in the ‘classical’ sense, i.e.,in case f i is differentiable ∇f i(x0)T ν < 0 is not guaranteed.

Algorithm 1 Multi-Objective Hill Climber with Sidestep

Require: starting point x0 ∈ Q, radius r ∈Ê

n+

Ensure: sequence {xi}i∈ Æ

of candidate solutions1: a := (0, . . . , 0) ∈

Ê

n

2: nondom := 03: for k = 1, 2, . . . do4: set x1k := xb

k−1 and choose x2k ∈ B(x1k, r) at random5: if x1k ≺ x2k then6: ν k := x2k − x1k7: compute tk ∈ Ê + and set xn

k := x2k + tkν k.8: choose xb

k ∈ {xbk, x1k} such that f (xb

k) =min(f (xn

k), f (x1k))9: nondom := 0, a := (0, . . . , 0)

10: else if x2k ≺ x1k then11: proceed analogous to case ”x1k ≺ x2k” with12: ν k := x1k − x2k and xn

k := x1k + tkν k.13: else

14: a := a + 1

N nd

x2k−x1

k

x2k−x1

k

15: nondom := nondom + 116: if nondom = N nd then17: compute tk ∈

Ê + and set xnk := x1k + tka.

18: nondom := 0, a := (0, . . . , 0)19: end if

20: end if 21: end for

to perform the line search (see e.g., [9]). We propose toproceed in analogy to [22], where a step size control for scalaroptimization problems has been developed, as follows:for x0, x1 and f ij , j = 1, . . . , l (for simplicity denoted by f )as above define

f ν :Ê

→Ê

, f ν(t) = f (x0 + tν ) (3)

Choose e ∈ (1, 2] (the same value for all l cases) and com-pute f ν(e). If f ν(e) < f ν(1) then accept t∗ij as step size forobjective f = f i

j

. If the above condition does not hold wehave collected enough information to approximate f ν by aquadratic polynomial p(t) = at2 + bt + c with coefficientsa,b,c ∈

Ê

. Using the interpolation conditions

p(0) = f ν(0), p(1) = f ν(1), p(e) = f ν(e), (4)

we obtain all the coefficients of p. Since p(1) < p(0) and p(e) ≥ p(1) and since p is a quadratic polynomial the func-tion contains exactly one minimum at

t∗ij =−b

2a= 2

e2(f ν(1) − f ν(0)) − f ν(e) + f ν(0)

e(f ν(1) − f ν(0)) − f ν(e) + f ν(0)∈ (0, e).

(5)Finally, the question arises how this information obtainedby scalarization can be put together to select a step sizestrategy for the given multi-objective problem. The ‘safest’step size control is certainly to take the smallest value of thet∗ij ’s. In order not to get stuck due to small step sizes and tointroduce a stochastic component into the search strategywe propose to choose a step size within the range which isgiven by the t∗ij ’s, i.e.

xnew = x0 + t∗kν, (6)

where t∗k ∈ [ mini=1,...,l

t∗ij , maxi=1,...,l

t∗ij ] is taken at random.

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Computation of tk. We are given a point x0 ∈Ê

n and

the search direction a =PN nd

i=1(xi − x0)/xi − x0 with

xi ∈ B(x0, r), i = 1, . . . , N nd, and such that (x0, xi), i =1, . . . , N nd, are mutually nondominating. For this situa-tion, we propose to proceed analogously to [21], where astep size strategy for multi-objective continuation methodsis suggested: given a target value ǫ ∈

Ê + – e.g., the min-imal value which makes two solutions distinguishable froma practical point of view –, the task is to compute a newcandidate xnew = x0 + ta such that

F (x0) − F (xnew)∞ ≈ ǫ (7)

In case F is Lipschitz continuous there exists an L ≥ 0such that

F (x) − F (y) ≤ Lx − y, ∀x, y ∈ Q. (8)

This constant can be estimated around x0 by

Lx0 := DF (x0)∞ = maxi=1,...,k

∇f i(x0)1,

where DF (x0) denotes the Hessian of F at x0 and ∇f i(x0)the gradient of the i-th objective at x0. In case F is notdifferentiable the accumulated information can be used to

compute the estimation

Lx0 := maxi=1,...,N nd

F (x0) − F (xi)∞x0 − xi∞

.

Combining (7), (8) and using the estimation Lx0 leads tothe step size control

xnew = x0 +ǫ

Lx0

a

a∞. (9)

Handling constraints. In the course of the computation itcan occur that iterates are generated which are not insidethe feasible domain Q. That is, we are faced with the situ-ation that x0 ∈ Q and x1 := x0 + h0ν ∈ Q, where ν is thesearch direction. In that case we propose to proceed analo-gously to the well-known bisection method for root findingin order to track back from the current iterate x1 to thefeasible set:let in0 := x0 ∈ Q and out0 := x1 ∈ Q and m0 := in0 +0.5(out0 − in0) = x0 + h0

2ν . If m0 ∈ Q set in1 := m0,

else out1 := m0. Proceeding in an analogous way, one ob-tains a sequence {ini}i∈Æ

of feasible points which convergeslinearly to the boundary ∂Q of the feasible set. One can,for example, stop this process with an i0 ∈

Æ

such thatouti0 − ini0∞ ≤ tol, obtaining a point ini0 with maximaldistance tol to ∂Q. See Algorithm 2 for one possible real-ization. Note that by this procedure no function evaluationhas to be spent (in contrast, for instance, to penalization

methods).

Using gradient information. In case the MOP is suffi-ciently smooth, some tools from mathematical programmingcan be integrated directly into the HCS in order to increaseits performance (significantly). For instance, if all objec-tives’ gradients are available descent directions can be com-puted directly ([10], [1]) for a given point x ∈ Q. This coulde.g. replace the random search as proposed in line 4 of Al-gorithm 1.Also a movement along the Pareto set can be performed

Algorithm 2 Backtracking to Feasible Region

Require: x0 ∈ Q, x1 = x0 + h0ν ∈ Q, tol ∈Ê +

Ensure: x ∈ x0x1 ∩ Q with inf b∈∂Q b − x < tol1: in0 := x02: out0 := x13: i := 04: while outi − ini ≥ tol do5: mi := ini + 1

2(outi − ini)

6: if mi ∈ Q then7: ini+1 := mi

8: outi+1 := outi9: else

10: ini+1 := ini

11: outi+1 := mi

12: end if 13: i := i + 114: end while15: return x := ini

Table 1: Design parameters that are required forthe realization of the HCS algorithm.

Parameter Descriptionr Radius for neighborhood search (Alg. 1)

N nd Number of trials for the hill climber beforethe sidestep is performed (Alg. 1)

ǫ Desired distance (in image space) for thesidestep (7)

tol Tolerance value used for the backtrackingin Alg. 2

much more efficiently by using predictor-corrector (PC) meth-ods ([12], [20]). The direction a in line 14 of Algorithm 1 canbe viewed as a possible predictor for further efficient points

along the solution manifold, and the hill climber plays therole of the correction mechanism. However, in case classicalmulti-objective PC methods are chosen the resulting betterperformance in terms of convergence does not come for free:in this case even the second derivatives of the objectives haveto be available or approximated.

3.2 Integration of HCS into the EMO Algo-rithm

Here we present one way to integrate the HCS into anEMO algorithm. There exist certainly several ways to dothis. In this work we present a first attempt whereas theHCS acts like a typical mutation operator within a givenEMO algorithm. The advantage of this approach is that by

doing so the local search procedure can be integrated intoany existing EMO algorithm with little effort.For this particular study, we have decided to use SPEA2[27] as the basis for our memetic algorithm, together withthe SBX crossover operator [6] and a variable-wise mutationoperator. A pseudocode can be found in Algorithm 3.

If the HCS is used as local search operator within a given(global) search strategy, it seems to be wise just to computea few iterates which are used to build up the next popu-lation. For the computations in the subsequent section wehave used the following strategy: if a point x0 ∈ P k+1 is

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−5 0 5 10 15 20 25 30 35 40 45 50−5

0

5

10

15

20

25

30

35

40

45

50

f1

f2

F(x0

)

Figure 2: Numerical result of HCS for MOP CON-VEX with Q = [−5, 5]10 in objective space (uncon-strained case).

2453 function calls had to be spent in order to get this re-sult which actually represents a good approximation of thePareto front.

Next, we consider the constrained case. For the result in

Figure 3 we have chosen n = 2 and Q = [0.5, 1.5] × [1, 2]for the same problem, and thus – since the model is convex– the Pareto set is contained in the boundary of Q. Thefigures show that also in this case the HCS is capable of approaching the solution set, and moving along it furtheron. However, a total of 4211 function calls had to be spentin this setting, that is, much more than in the unconstrainedcase (note that the dimension of the model is much lower inthe latter case).Finally, we consider the problem ZDT1 (Table 2), which isa highly nonlinear model. Figure 4 shows a result in imagespace for two different initial solutions x0, z0 ∈ Q = [0, 1]10.As anticipated, the results differ significantly since the HCSis a local strategy and ZDT1 contains many local Pareto

fronts. However, the procedure is also in this case able toexplore a part of the local Pareto front which is located nearto the image of the initial solution.

4.2 Results of the EMO AlgorithmHere we make a comparison of the classical SPEA2 and

the modification of this algorithm which is enhanced by thelocal iteration process (denoted by SPEA2HCS) in order todemonstrate the advantage of the memetic strategy. Table3 displays the parameters which have been used for bothSPEA2 and SPEA2HCS. Since at the beginning of the algo-rithm’s execution, typically a global search is more effectivethan a local one, we have started the HCS after 75 % of thetotal number of generations (i.e., we have set P LS = 0 inthe first generations, see Table 3).

In order to evaluate the performance of the algorithms wehave used the following three indicators (see [25] and [19]):

Generational Distance : GD = 1

n

p Pn

i=1δ2i

Efficient Set Space : ESS =q

1

n−1

Pni=1

(di − d)2

Maximal Distance : MD = maxi,j=1,...,n

i=j

dij

Hereby, δi denotes the minimal Euclidean distance from

0.5 0.6 0.7 0.8 0.9 1

1

1.2

1.4

1.6

1.8

2

x1

x 2

x0

− 0. 02 0 0 .0 2 0 .0 4 0 . 06 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8

6

6.5

7

7.5

8

8.5

9

f1

f2

F(x0

)

Figure 3: Numerical result of HCS for MOP CON-VEX with Q = [0.5, 1.5]× [1, 2] in objective space (con-strained case).

the image F (xi) of a solution xi, i = 1, . . . , n, to the truePareto front, and

di := minj=1,...,n

i=j

dij and d :=1

n

n

Xi=1

di, (12)

where dij is the Euclidean distance between F (xi) and F (xj).In the multi-objective optimization framework, there are ingeneral three goals ([26]): (i) the distance of the result-ing nondominated set to the Pareto-optimal front shouldbe minimized, (ii) a uniform distribution of the solutionsfound is desirable, and (iii) the extent of the obtained non-dominated front should be maximized. We have chosen thethree indicators with the aim to measure the achievementof each of these goals, respectively.Table 4 shows one comparison for some DTLZ test functions([8], see also Table 2), where we have chosen n = 7 for thedimension of the parameter space and k = 3 objectives. Thenumerical results show that in almost all cases SPEA2HCS

achieves better values than SPEA2 for all three indicators.For the generational distance, which is an indicator for con-vergence, the reduction is often near 50%.The comparison of the results for DTLZ3 (with parameterα = 20) and DTLZ3* (same as DTLZ3 but with α = 2)gives some insight into the nature of memetic algorithmssuch as SPEA2HCS. The parameter α controls the numberN l of strictly local Pareto fronts and is thus an indicator forthe complexity of the model. N l is approximately equal to`α2

´n−k+1. That is, for α = 20 there exist a total of 100.000

local Pareto fronts while for α = 2 there exists merely one.

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0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

f1

f2

F(x0

)

F(z0

)

Figure 4: Numerical result of HCS for MOP ZDT1with Q = [0, 1]10 in objective space. Sequences fortwo initial solutions, x0 and z0, are shown as well asthe true Pareto front.

Table 3: Parameters for SPEA2: P size and Asize de-note the population size and the maximal cardinal-ity of the archive, and P cross, P mut, P LS denote theprobabilities for crossover, mutation and line searchrespectively.

Parameter ValueP size 100Asize 100P cross 0.8P mut 0.01P LS 0.05

The results for SPEA2HCS compared to SPEA2 are – as an-ticipated – much better for the ‘easier’ model DTLZ3* sincein that case the probability is very high that the HCS con-

verges to a global solution starting from a randomly chosenpoint, and the local search can thus contribute to increasethe overall performance. If the model gets more complicatedthe improvements achieved by the local search proceduredecreases, and global search operators get more important.Thus, it comes as no surprise that the improvements achivedby the memetic strategy for DTLZ3 are less significant thanfor DTLZ3*.It has to be noted that all the results presented here comefrom 3-objective models. This is due to the fact that for allbi-objective models that we have tested no remarkable im-provements have been achieved which indicates that SPEA2(as well as other state-of-the-art EMOs) is already very effi-cient on the well-known bi-objective benchmark suite – andcertainly on other models as well. However, since the perfor-

mance of most EMO algorithms decreases significantly withincreasing number of objectives (starting with k ≥ 3), theusage of memetic algorithms – such as the one proposed here– seems to be advantageous for the numerical treatment of MOPs which contain more than two objectives.

5. CONCLUSIONS AND FUTURE WORKIn this paper we have presented the Hill Climber with

Sidestep (HCS), a derivative free local iterative search pro-cedure. This iteration process aims either to move toward or

along the (local) Pareto set depending on the location of thecurrent iterate. Further, a novel memetic strategy has beenproposed by integrating the HCS into a given EMO. Finally,we have demonstrated the applicability and the strength of the HCS alone and within SPEA2 as a local search operatoron several benchmark optimization problems. The resultsindicate that the local search procedure can be advanta-geous in particular in the cases where a classical EMO gets‘stuck’. However, due to the nature of local search, the ad-

vantage of memetic strategies such as the one presented inthis paper decreases with increasing complexity of the un-derlying model. Thus, overwhelming results compared to‘classical’ EMOs cannot be expected in general.The particular integration of the HCS into a given EMO wehave presented has the advantage that in general every EMOcan be taken and enhanced to become memetic. However,we expect that a more sophisticated interplay of the globaland local search operators will lead to more efficient algo-rithms, and exactly this will be part of our future research.

Acknowledgements. The authors would like to thank theanonymous reviewers for their valuable comments whichgreatly helped them to improve the contents of this paper.

The third author gratefully acknowledges support from theCONACyT project no. 45683-Y.

6. REFERENCES

[1] P. A. N. Bosman and E. D. de Jong. Exploitinggradient information in numerical multi-objectiveevolutionary optimization. In Genetic and Evolutionary Computation Conference, ACM, 2005.

[2] J. Branke and S. Mostaghim. About selecting thepersonal best in multi-objective particle swarmoptimization. In Parallel Problem Solving from Nature- PPSN IX , pages 523–532, Springer Berlin /

Heidelberg, 2006.[3] M. Brown and R. E. Smith. Directed multi-objectiveoptimization. International Journal on Computers,Systems, and Signals, 6(1):3–17, 2005.

[4] C. A. Coello Coello, D. A. Van Veldhuizen, and G. B.Lamont. Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer AcademicPublishers, New York, May 2002. ISBN 0-3064-6762-3.

[5] K. Deb. Multi-Objective Optimization using Evolutionary Algorithms. Wiley, 2001.

[6] K. Deb and R. B. Agrawal. Simulated binarycrossover for continuous search space. Complex Systems, 9:115–148, 1995.

[7] K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan. Afast elitist non-dominated sorting genetic algorithmfor multi-objective optimisation: NSGA-II. In PPSN VI: Proceedings of the 6th International Conference on Parallel Problem Solving from Nature.

[8] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler.Scalable multiobjective optimization test problems. In Congress on Evolutionary Computation, volume 1,Piscataway, New Jersey , pages 825 –830, 2002.

[9] J.E. Dennis and R.B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, 1983.

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Table 4: Numerical result for SPEA2 and SPEA2HCS on the DTLZ test functions (see Table 2). For eachproblem the number of function calls was fixed to 100,000, and the two algorithms were run for 30 times.The resulting mean value and standard deviation (in brackets) are presented.

IndicatorsProblems GD ESS MD

DTLZ 1without LSwith LS

8.2042(3.3349)4.6663(1.3341)

8.0671(4.3136)5.9063(5.1751)

291.1858(96.4631)71.2207(38.4968)

DTLZ 2

without LS

with LS

0.0298(0.0094)

0.0044(0.0022)

0.0501(0.0146)

0.0306(0.0103)

2.4125(0.2907)

1.5691(0.1018)

DTLZ 3without LSwith LS

11.2395(7.9325)6.0506(3.6930)

8.4543(6.7640)14.9354(12.8529)

300.4770(185.2523)197.0717(99.7800)

DTLZ 3∗without LSwith LS

18.3752(15.0230)0.1942(0.3043)

9.2857(10.2005)0.9345(2.2947)

427.8683(318.8405)13.9865(24.0805)

[10] J. Fliege and B. F. Svaiter. Steepest descent methodsfor multicriteria optimization. Mathematical Methodsof Operations Research , 51(3):479–494, 2000.

[11] T. Goel and K. Deb. Hybrid methods formulti-objective evolutionary algorithms. InProceedings of the 4th Asia-Pacific conference on Simulated Evolution and Learning , pages 188–192,2002.

[12] C. Hillermeier. Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach . Birkhauser,2001.

[13] X. Hu, Z. Huang, and Z. Wang. Hybridization of themulti-objective evolutionary algorithms and thegradient-based algorithms. In Proceedings of the IEEE Congress on Evolutionary Computation , pages 870 –877., 2003.

[14] H. Ishibuchi and T. Murata. Multi-objective geneticlocal search algorithm. In Proc. of 3rd IEEE Int.Conf. on Evolutionary Computation, Nagoya, Japan ,pages 119–124, 1996.

[15] J. D. Knowles and D. W. Corne. Recent Advances in

Memetic Algorithms, volume 166, chapter MemeticAlgorithms for Multi-Ojective Optimization: Issues,Methods and Prospects, pages 313–352. Springer.Studies on Fuzzines and Soft Computing, 2005.

[16] J.D. Knowles and D.W. Corne. M-PAES: a memeticalgorithm for multiobjective optimization. InProceedings of the IEEE Congress on Evolutionary Computation , pages 325 – 332, 2000.

[17] H. Kuhn and A. Tucker. Nonlinear programming. InJ. Neumann, editor, Proceeding of the 2nd Berkeley Symposium on Mathematical Statistics and Probability , pages 481–492, 1951.

[18] P. Moscato. On evolution, search, optimization,genetic algorithms and martial arts: Toward memetic

algorithms. Technical Report C3P Report 826, Caltech Concurrent Computation Program , 1989.

[19] J.R. Schott. Fault tolerant design using single andmulticriteria genetic algorithm optimization. Master’sThesis, Department of Aeronautics and Astronautics,MIT. Cambridge, MA. USA, 1995.

[20] O. Schutze, A. Dell’Aere, and M. Dellnitz. Oncontinuation methods for the numerical treatment of multi-objective optimization problems. In J. Branke,K. Deb, K. Miettinen, and R. E. Steuer, editors,Practical Approaches to Multi-Objective Optimization ,number 04461 in Dagstuhl Seminar Proceedings.Internationales Begegnungs- und Forschungszentrum(IBFI), Schloss Dagstuhl, Germany, 2005.<http://drops.dagstuhl.de/opus/volltexte/2005/349>.

[21] O. Schutze, M. Laumanns, E. Tantar, C. A. CoelloCoello, and E.-G. Talbi. Convergence of stochasticsearch algorithms to gap-free Pareto frontapproximations. Proceedings of the Genetic andEvolutionary Computation Conference, 2007.

[22] O. Schutze, E.-G. Talbi, C. A. Coello Coello, L. V.Santana-Quintero, and G. Toscano Pulido. A memeticPSO algorithm for scalar optimization problems.Proceedings of IEEE Swarm Intelligence Symposium(SIS2007), 2007.

[23] D. Sharma, A. Kumar, K. Deb, and K. Sindhya.Hybridization of SBX based NSGA-II and sequential

quadratic programming for solving multi-objectiveoptimization problems. In Proceedings of the IEEE Congress on Evolutionary Computation .

[24] V. Pareto. Manual of Political Economy . TheMacMillan Press, 1971 (original edition in French in1927).

[25] D.A. Van Veldhuizen and G.B. Lamont. Evolutionarycomputation and convergence to a Pareto front. InLate Breaking papers at the Genetic Programming 1998 Conference.

[26] E. Zitzler, K. Deb, and L. Thiele. Comparison of multiobjective evolutionary algorithms: Empiricalresults. Evolutionary Computation , 8(2):173–195,2000.

[27] E. Zitzler, M. Laumanns, and L. Thiele. SPEA2:Improving the strength Pareto evolutionary algorithm.In K. Giannakoglou, D. Tsahalis, J. Periaux,P. Papailou, and T. Fogarty, editors, EUROGEN 2001. Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems,2002.

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