Scalable Test Problems for Evolutionary Multi-Objective Optimization

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1 Scalable Test Problems or Evolutionary Multi-Objective Optimization Kalyanmoy Deb Kanpur Genetic Algorithms Laboratory Indian Institute o Technology Kanpur PIN 8 6, India deb@iitk.ac.in Lothar Thiele, Marco Laumanns and Eckart Zitzler Computer Engineering and Networks Laboratory ETH Zürich CH-89, Switzerland {thiele,laumanns,zitzler}@tik.ee.ethz.ch TIK-Technical Report No. Institut ür Technische Inormatik und Kommunikationsnetze, ETH Zürich Gloriastrasse 5., ETH-Zentrum, CH-89, Zürich, Switzerland July 7, Abstract Ater adequately demonstrating the ability to solve dierent two-objective optimization problems, multi-objective evolutionary algorithms (MOEAs) must now show their eicacy in handling problems having more than two objectives. In this paper, we have suggested three dierent approaches or systematically designing test problems or this purpose. The simplicity o construction, scalability to any number o decision variables and objectives, knowledge o exact shape and location o the resulting Pareto-optimal ront, and introduction o controlled diiculties in both converging to the true Pareto-optimal ront and maintaining a widely distributed set o solutions are the main eatures o the suggested test problems. Because o the above eatures, they should be ound useul in various research activities on MOEAs, such as testing the perormance o a new MOEA, comparing dierent MOEAs, and better understanding o the working principles o MOEAs. Introduction Most earlier studies on multi-objective evolutionary algorithms (MOEAs) introduced test problems which were either simple or not scalable. Some test problems were too complicated to visualize the exact shape and location o the resulting Pareto-optimal ront. Schaer s (984) study introduced two single-variable test problems (SCH and SCH), which have been widely used as test problems. Kursawe s (99) test problem KUR was scalable to any number o decision variables, but was not scalable in terms o the number o objectives. The same is true with Fonseca and Fleming s (995) test problem FON. Poloni et al. s () test problem POL used only two decision variables. Although the mathematical ormulation o the problem is non-linear, the resulting Pareto-optimal ront corresponds an almost linear relationship among decision variables. Viennet s (996) test problem VNT has a discrete set o Pareto-optimal ronts, but was designed

2 or three objectives only. Similar simplicity prevails in the existing constrained test problems (Veldhuizen, 999; Deb, ). However, in 999, the irst author introduced a systematic procedure o designing test problems which are simple to construct and are scalable to the number o decision variables (Deb, 999). In these problems, the exact shape and location o the Pareto-optimal solutions are also known. The basic construction used two unctionals g and h with non-overlapping sets o decision variables to introduce diiculties towards the convergence to the true Pareto-optimal ront and to introduce diiculties along the Pareto-optimal ront or an MOEA to ind a widely distributed set o solutions, respectively. Although these test problems are used by many researchers since then, they have been somewhat criticized or the relative independence eature o the unctionals in achieving both the tasks. Such critics have grossly overlooked an important aspect o that study. The non-overlapping property o the two key unctionals in the test problems was introduced or an ease o the construction procedure. That study also suggested the use o a procedure to map the variable vector (say y) to a dierent decision variable vector (say x). This way, although test problems are constructed or two non-overlapping sets rom y, each decision variable x i (on which MOEA operators work) involves a correlation o all (or many) variables o y. Such a mapping couples both aspects o convergence and maintenance o diversity and makes the problem harder to solve. However, Zitzler, Deb, and Thiele () showed that the uncorrelated version o the test problems was even diicult to solve exactly using the then-known state-o-the-art MOEAs. In the recent past, many MOEAs have adequately demonstrated their ability to solve twoobjective optimization problems. With the suggestion o a number o such MOEAs, it is time that they must be investigated or their ability to solve problems with more than two objectives. In order to help achieve such studies, it is thereore necessary to develop scalable test problems or higher number o objectives. Besides testing an MOEA s ability to solve problems with a large number o objectives, the proposed test problems can also be used or systematically comparing two or more MOEAs. Since one such test problem can be used to test a particular aspect o multiobjective optimization, such as or convergence to the true Pareto-optimal ront or maintenance o a good spread o solutions, etc., the test problems can be used to identiy MOEAs which are better in terms o that particular aspect. For these reasons, these test problems may help provide a better understanding o the working principles o MOEAs, thereby allowing a user to develop better and more eicient MOEAs. In the remainder o the paper, we irst describe the essential eatures needed in a test problem and then suggest three approaches or systematically designing test problems or multi-objective optimization algorithms. Although most problems are illustrated or three objectives (or an ease o illustration), the test problems are generic and scalable to an arbitrary number o objectives. Desired Features o Test Problems Based on the above discussion, we suggest that the ollowing eatures must be present in a test problem suite or adequately testing an MOEA:. Test problems should be easy to construct.. Test problems should be scalable to have any number o decision variables.. Test problems should be scalable to have any number o objectives. 4. The resulting Pareto-optimal ront (continuous or discrete) must be easy to comprehend, and its exact shape and location should be exactly known. The corresponding decision variable values should also be easy to ind.

3 5. Test problems should introduce controllable hindrance to converge to the true Paretooptimal ront and also to ind a widely distributed set o Pareto-optimal solutions. The popularity o two-objective test problems suggested earlier (Deb, 999) in many research studies is partly because o the ease o constructing the test problems and the ease o illustrating the obtained set o Pareto-optimal solutions in two dimensions. Visually comparing the obtained set o solutions against the true Pareto-optimal ront provides a clear idea o the perormance o an MOEA. This can somewhat be achieved even or three objectives, with such a comparison shown in three dimensions. But or problems with more than three objectives, it becomes diicult to illustrate such a plot. Thus, or higher-objective test problems, it may be wise to have some regularity in the search space so that the Pareto-optimal surace is easily comprehensible. One o the ways to achieve this would be to have a Pareto-optimal surace symmetric along interesting hyper-planes, such as = = = M (where M is the number o objectives). This only requires a user to comprehend the interaction between M and, and the rest o the problem can be constructed by using symmetry. Another interesting approach would be to construct a problem or which the Pareto-optimal surace is a symmetric curve or at most a three-dimensional surace. Although M-dimensional, the obtained solutions can be easily illustrated parametrically in a two-dimensional plot in the case o a curve and in a three-dimensional plot in the case o the three-dimensional surace. It is now well established that MOEAs have two tasks to achieve: converging to the Paretooptimal ront and inding a good distribution o solutions on the entire Pareto-optimal ront. An MOEA can, thereore, must be tested or each o the two tasks. Thus, some test problems should test an MOEA s ability to negotiate artiicial hurdles which hinders its progress towards converging to the true Pareto-optimal ront. This can be achieved by placing some local Paretooptimal attractors or biased density o solutions away rom the Pareto-optimal ront. Some test problems must also test an MOEA s ability to ind a diverse set o solutions. This can be achieved by making the Pareto-optimal ront nonconvex, discrete, and having variable density o solutions along the ront. Although these eatures o test problems were also suggested or two-objective problems earlier (Deb, 999), they are also applicable or a large number o objectives. Moreover, the increased dimensionality associated with a large number o objectives may cause an added diiculty to MOEAs. In the ollowing sections, we suggest dierent approaches o designing test problems or multiobjective optimization. Dierent Methods o Test Problem Design Based on the above principles, there exist a number o dierent ways to systematically design test problems or multi-objective optimization. Here we discuss three dierent methods:. Multiple single-objective unctions approach,. Bottom-up approach,. Constraint surace approach. The irst approach is the most intuitive one and has been implicitly used by early MOEA researchers to construct test problems. In this approach, M dierent single-objective unctions are used to construct a multi-objective test problem. To simpliy the construction procedure, in many cases, dierent objective unctions are simply used as dierent translations o a single objective unction. For example, the problem SCH uses the ollowing two single-objective unctions or minimization (Schaer, 984): (x) =x, (x) =(x ).

4 Since the optimum x () or is not the optimum or and vice versa, the Pareto-optimal set consists o more than one solution, including the individual minimum o each o the above unctions. All other solutions which make trade-os between the two objective unctions with themselves and with the above two solutions become members o the Pareto-optimal set. In the above problem, all solutions x [, ] become member o the Pareto-optimal set. Similarly, the problem FON shown below Minimize Minimize ( (x) = exp ) n i= (x i n ), ( (x) = exp ) n i= (x i + n ), 4 x i 4 i =,,...,n. has x i = / n or all i as the minimum solution or and x i =/ n or all i as the minimum solution or. The Pareto-optimal set is constituted with all solutions in x i [ / n, / n] or all i. Veldhuizen (999) lists a number o such test problems. It is interesting to note that such a construction procedure can be extended to higher-objective problems as well (Laumanns, Rudolph, and Schweel, ). In a systematic procedure, each optimum may be assumed to lie on each o M (usually < n) coordinate directions. However, the Pareto-optimal set resulting rom such a construction depends on the chosen objective unctions, thereby making it diicult to comprehend the true nature o the Pareto-optimal ront. Moreover, even in simple objective unctions (such as in SCH (Schaer, 984)), the Pareto-optimal ront may be a combination o disconnected ronts. Thus, a test problem constructed using this procedure must be careully analyzed to ind the true Pareto-optimal set o solutions. The latter two approaches o test problem design mentioned above directly involve the Paretooptimal ront, thereby making them convenient to be used in practice. Since they require detailed discussions, we devote two separate sections or describing them. 4 Bottom-Up Approach In this approach, a mathematical unction describing the Pareto-optimal ront is assumed in the objective space and an overall objective search space is constructed rom this ront to deine the test problem. For two objectives, one such construction was briely suggested earlier (Deb, 999) and was extended or higher objectives elsewhere (Deb, ). But, here we make the idea more clear by illustrating the principle in a three-objective example test problem. Later, we suggest a generic procedure. Let us assume that we would like to have a Pareto-optimal ront where all objective unctions take non-negative values and the desired ront is the irst quadrant o a sphere o radius one (as shown in Figure ). With the help o spherical coordinates (θ, γ, andr = ), the ront can be described as ollows: (θ, γ) = cosθ cos(γ + π/4), (θ, γ) = cosθ sin(γ + π/4), (θ, γ) = sinθ, where θ π/, π/4 γ π/4. It is clear rom the construction o the above surace that i all three objective unctions are minimized, any two points on this surace are non-dominated to each other. Now, i the rest o the objective search space is constructed above this surace, we shall have a problem where the unit sphere constitutes the Pareto-optimal ront. A simple way to construct the rest o the search space is to construct suraces parallel to the above surace. This means constructing spherical suraces with radius greater than one. This can be achieved by multiplying the above three unctions with a term, which takes a value greater than or equal to one. Dierent values o the 4 () ()

5 .8 Pareto optimal ront θ..4 γ Search space.5.5 Figure : First quadrant o a unit sphere as a Pareto-optimal ront. Figure : Overall search space is bounded by the two spheres. third independent variable r (besides θ and γ) will construct dierent layers o spherical suraces on top o the Pareto-optimal sphere. Thus, the overall problem with the above three variables is as ollows: Minimize (θ, γ, r) =(+g(r)) cos θ cos(γ + π/4), Minimize (θ, γ, r) =(+g(r)) cos θ sin(γ + π/4), Minimize (θ, γ, r) =(+g(r)) sin θ, () θ π/, π/4 γ π/4, g(r). As described earlier, the Pareto-optimal solutions or the above problem are as ollows: θ π/, π/4 γ π/4, g(r )=. Figure shows the overall objective search space with any unction or g(r) with g(r). We shall discuss more about dierent g(r) unctions a little later. The above construction procedure illustrates how easily a multi-objective test problem can be constructed rom an initial choice o a Pareto-optimal surace. 4. Construction o the decision space Although the above three-objective problem requires three independent variables, the decision search space can be higher than three-dimensional. The three variables used above (θ, γ, and r) can all be considered as meta-variables and each o them can be considered as a unction o n decision variables o the underlying problem: θ = θ(x,x,...,x n ), (4) γ = γ(x,x,...,x n ), (5) r = r(x,x,...,x n ). (6) The unctions must be so chosen that they satisy the lower and upper bounds o θ, γ and g(r) mentioned in equation. Although the above construction procedure is simple, it can be used to introduce dierent modes o diiculty described earlier. In the ollowing, we describe a ew such extensions o the above construction. 5

6 4. Diiculty in converging to the Pareto-optimal ront The diiculty o a search algorithm to progress towards the Pareto-optimal ront rom the interior o the objective search space can be introduced by simply using a diicult g unction. It is clear that the Pareto-optimal surace corresponds to the minimum value o unction g. A multi-modal g unction with a global minimum at g = and many local minima at g = ν i value will introduce global and local Pareto-optimal ronts, where a multi-objective optimizer can get stuck to. Moreover, even using a unimodal g(r) unction, variable density o solutions can be introduced in the search space. For example, i g(r) =r is used, denser solutions exist away rom the Paretooptimal ront. Figure shows 5, solutions, which are randomly created in the decision variable space. On the objective space, they are shown to be biased away rom the Pareto-optimal ront. For a such a biased search space away rom the Pareto-optimal ront, multi-objective optimizers Figure : The eect o a non-linear g unction. may have diiculties in converging quickly to the desired ront. 4. Diiculties across the Pareto-optimal ront By using a non-linear mapping in equations 4 to 6, some portion o the search space can be made to have more dense solutions than the rest o the search space. In order to create a variable density o solutions on the Pareto-optimal ront, the θ and γ unctions (as in equations 4 to 5) must be manipulated. For example, Figures 4 and 5 show the problem stated in equation with θ(x )= π x, γ(x )= π x π 4. and θ(x )= π ( +[(x )] ), γ(x )= π ( +[(x )] ) π 4. respectively. In both cases, g(r) =r = x is chosen. In order to satisy the bounds in equation, we have chosen x,x,x and 5, randomly created points (in the decision space) are shown in each igure showing the objective space. The igures show the density o solutions in the search space gets aected by the choice o mapping o the meta-variables. In the second problem, there is a natural bias or an algorithm to ind solutions in middle region o the search space.in In three-objective knapsack problems, such a biased search space is observed elsewhere (Zitzler and Thiele, 999). 6

7 Figure 4: Linear mapping (5, points). Figure 5: Non-Linear mapping introduces bias in the search space (5, points). trying to solve such test problems, the task o an MOEA would be to ind a widely distributed set o solutions on the entire Pareto-optimal ront despite the natural bias o solutions in certain regions on the Pareto-optimal ront. 4.4 Generic sphere problem The ollowing is a generic problem to that described in equation, having M objectives. Minimize (θ,r)=(+g(r)) cos θ cos(θ ) cos(θ M )cos(θ M ), Minimize (θ,r)=(+g(r)) cos θ cos(θ ) cos(θ M )sin(θ M ), Minimize (θ,r)=(+g(r)) cos θ cos(θ ) sin(θ M ),.. Minimize M (θ,r)=(+g(r)) cos θ sin(θ ), Minimize M (θ,r)=(+g(r)) sin θ, θ i π/, or i =,,...,(M ), g(r). Note that the variables are mapped in a dierent manner here. The decision variables are mapped to the meta-variable vector θ (o size (M )) as ollows: θ i = π x i, or i =,,...,(M ). (8) The above mapping and the condition on θ i in equation 7 restrict each o the above x i to lie within [, ]. To simpliy the matter, the unction g(r) =r = x M (where x M [, ]) can be used. The Pareto-optimal surace occurs or the minimum o the g unction, or at x M =and the unction values must satisy the ollowing condition: M (i ) =. (9) i= As mentioned earlier, the diiculty o the above test problem can also be varied by using dierent unctionals or i and g. 7 (7)

8 4.5 Curve problem Instead o having a complete M-dimensional surace as the Pareto-optimal surace, an M-dimensional curve can also be chosen as the Pareto-optimal ront to the above problem. We realize that in this case there would be only one independent variable describing the Pareto-optimal ront. A simple way to achieve this would be to use the ollowing mapping o variables: θ i = π 4( + g(r)) ( + g(r)x i), or i =,,...,(M ), () The above mapping ensures that the curve is the only non-dominated region in the entire search space. Since g(r) = corresponds to the Pareto-optimal ront, θ i = π/4 or all but the irst variable. The advantage o this problem over the generic sphere problem as a test problem is that a two-dimensional plot o Pareto-optimal points with M and any other i will mark a curve (circular or elliptical). A plot with any two objective unctions (other than M ) will show a straight line. Figure 6 shows a sketch o the search space and the resulting Pareto-optimal curve or a three-objective version o the above problem. One drawback with this ormulation is that.5 Pareto optimal curve.5.5 Figure 6: The search space and the Pareto-optimal curve. the density o solutions closer to the Pareto-optimal curve is more than anywhere else in the search space. In order to make the problem more diicult, a non-linear g(r) unction with a higher density o solutions away rom g(r) =(suchasg(r) =/r α,whereα ) can be used. Using a multi-modal g(r) unction will also cause multiple local Pareto-optimal suraces to exist. Interestingly, this drawback o the problem can be used to create a hard maximization problem. I the all objectives are maximized in the above problem, the top surace becomes the desired Pareto-optimal ront. Since there exists less dense solutions on this surace, this problem may be a diicult maximization test problem. 4.6 Comet problem To illustrate the concept o the bottom-up approach o test problem design urther, we design one more problem which has a comet-like Pareto-optimal ront. Starting rom a widely spread region, the Pareto-optimal ront continuously reduces to thinner region. Finding a wide variety 8

9 o solutions in both broad and thin portions o the Pareto-optimal region simultaneously will be a challenging task or any MOEA: Minimize (x) =(+g(x ))(x x x 4x ), Minimize (x) =(+g(x ))(x x x +4x ), Minimize (x) =(+g(x ))x, x.5, x, g(x ). Here, we have chosen g(x )=x and x. The Pareto-optimal surace corresponds to x = and or x x with x.5. Figure 7 shows the Pareto-optimal ront on the x = surace. For a better illustration purpose, the igure is plotted with negative o () Pareto optimal ront 4 4 Figure 7: The comet problem. all i values. This problem illustrates that the entire g = g surace need not correspond to the Pareto-optimal ront. Only the region which dominates the rest o the g = g surace belongs to the Pareto-optimal ront. We have designed this unction and the curve unction or a special purpose. Because o the narrow Pareto-optimal region in both problems, we argue that the classical generating methods will require a large computational overhead in solving the above problems. Figure 8 shows the projection o the Pareto-optimal region in the - space o the comet problem. For the use o the ɛ-constraint method as a method o generating Pareto-optimal solutions (usually recommended or its convergence properties), the resulting single-objective optimization problem, which has to be solved or dierent combination o ɛ and ɛ, is as ollows: Minimize (x), subject to (x) ɛ, () (x) ɛ, x D, where D is the easible decision space. It is well known that the minimum solution or the above problem or any ɛ and ɛ ( ) is either a Pareto-optimal solution or is ineasible (Mietinnen, 9

10 5 5 5 A Figure 8: Classical generating method with the ɛ-constraint method will produce redundant single-objective optimization problems. 997). The igure illustrates a scenario with ɛ =. It can be seen rom the igure that solution o the above single-objective optimization problem or ɛ = 5 is not going to produce any new Pareto-optimal solution other than that obtained or ɛ = (or example) or or ɛ set to get the Pareto-optimal solution at A. Thus, the above generating method will resort to solving many redundant single-objective optimization problems. By calculating the area o the projected Pareto-optimal region, it is estimated that about 88% o single-objective optimization problems are redundant in the above three-objective optimization problem, i a uniorm set o ɛ vectors are chosen in the generating method. Compared to classical generating methods, MOEAs may show superior perormance in these problems in terms o overall computational eort needed in inding multiple and well distributed Pareto-optimal solutions. This is mainly because o their implicit parallel processing which enables them to quickly settle to easible regions o interest and due to their population approach which allow them to ind a wide variety o solutions simultaneously with the action o a niche-preserving operator. 4.7 Test Problem Generator The earlier study (Deb, ) suggested a generic multi-objective test problem generator, which belongs to this bottom-up approach. For M objective unctions, with a complete decision variable vector partitioned into M non-overlapping groups x (x, x,...,x M, x M ) T, the ollowing unction was suggested: Minimize (x ), Minimize (x ),.. Minimize M (x M ), Minimize M (x) =g(x M )h ( (x ), (x ),..., M (x M ),g(x M )), subject to x i R x i, or i =,,...,M. Here, the Pareto-optimal ront is described by solutions which are global minimum o g(x M )(with g ). Thus, the Pareto-optimal ront is described as () M = g h(,,..., M ). (4)

11 Since g is a constant number, the h unction (with a ixed g = g ) describes the Pareto-optimal surace. In the bottom-up approach o test problem design, the user can irst choose an h unction in terms o the objective unction values, without caring about the decision variables at irst. For example, or constructing a problem with a non-convex Pareto-optimal ront, a non-convex h unction can be chosen, such as the ollowing: ( M ) α i= h(,,..., M )= i, (5) β where α >. Figure 9 shows a non-convex Pareto-optimal ront with α = or M = and β =. A disjoint set o Pareto-optimal ront can be constructed by simply choosing a multi-modal h unction as done in the case o two-objective test problem design (Deb, 999). Figure illustrates a disconnected set o Pareto-optimal suraces (or three-objectives), which can be generated rom the ollowing generic h unction: M h(,,..., M )=M ( i +sin(π i )). (6) i= Pareto optimal patches Figure 9: A non-convex Pareto-optimal ront. Figure : regions. A disjoint set o Pareto-optimal Once the h unction is chosen, a g unction can be chosen to construct the entire objective search space. It is important to note that the g unction is deined over a set o decision variables. Recall that the Pareto-optimal ront corresponds to the global minimum value o the g unction. Any other values o g will represent a surace parallel to the Pareto-optimal surace. All points in this parallel surace will be dominated by some solutions in the Pareto-optimal surace. As mentioned earlier, the g unction can be used to introduce complexities in approaching towards the Pareto-optimal region. For example, i g has a local minimum, a local Pareto-optimal ront exists on the corresponding parallel surace. Once appropriate h and g unctions are chosen, to M can be chosen as unctions o dierent non-overlapping sets o decision variables. Using a non-linear objective unction introduces a variable density o solutions along that objective. The non-linearity in these unctions will test an MOEA s ability to ind a good distribution o solutions, despite the natural non-uniorm density

12 in solutions in the search space. Another way to make the density o solutions non-uniorm is to use an overlapping set o decision variables or objective unctions. To construct more diicult test problems, the procedure o mapping the decision variables to an intermediate variable vector, as suggested earlier (Deb, 999) can also be used here. 4.8 Advantages and disadvantages o the bottom-up approach The advantage o using the above bottom-up approach is that the exact orm o the Paretooptimal surace can be controlled by the developer. The number o objectives and the variability in density o solutions can all be controlled by choosing proper unctions. Since the Pareto-optimal ront must be expressed mathematically, some complicated Paretooptimal ronts can be diicult to write mathematically. Nevertheless, the ability to control dierent eatures o the problem is the main strength o this approach. 5 Constraint Surace Approach Unlike starting rom a pre-deined Pareto-optimal surace in the bottom-up approach, the constraint surace approach begins by a predeining the overall search space. Here, a simple geometry such as a rectangular hyper-box is assumed. Each objective unction value is restricted to lie within a predeined lower and a upper bound. The resulting problem is as ollows: Minimize (x), Minimize (x),.. (7) Minimize Subject to M (x), (L) i i (x) (U) i or i =,,...,M. It is intuitive that the Pareto-optimal set o the above problem has only one solution (the solution with the lower bound o each objective ( (L), (L),..., (L) ) T. Figure shows this problem or three objectives (with (L) i =and (U) i = ) and the resulting singleton Pareto-optimal solution =(,, ) T )isalsomarked. The problem is now made more interesting by adding a number o constraints (linear or non-linear): g j (,,..., M ), or j =,,...,J. (8) This is done to chop o portions o the original rectangular region systematically. Figure shows the resulting easible region ater adding the ollowing two linear constraints: g +, g What remains is the easible search space. The objective o the above problem is to ind the non-dominated portion o the boundary o the easible search space. Figure also marks the Pareto-optimal surace o the above problem. For simplicity and easier comprehension, each constraint involving at most two objectives (similar to the irst constraint above) can be used. In addition to the complicated shape o the Pareto-optimal ront, urther diiculties can be introduced by using varying density o solutions in the search space. This can be easily achieved by using non-linear unctionals or i with the decision variables. Interestingly, there exist twovariable and three-variable constrained test problems TNK (Tanaka, 995) and Tamaki (996) in the literature using the above concept. In this problem, only two objectives (with i = x i )and two constraints were used. The use o i = x i made a uniorm density o solutions in the search

13 Search space.8 Search space constraint constraint Pareto optimal point.8 Pareto optimal ront Figure : Entire cube is the search space. The origin is the sole Pareto-optimal solution. Figure : Two constraints are added to eliminate a portion o the cube, thereby resulting in a more interesting Pareto-optimal ront. space. As an illustration o urther diiculties through non-linear i, we construct the ollowing three-objective problem with a bias in the search space: Minimize (x )=+(x ) 5, Minimize (x )=x, Minimize (x )=x, Subject to g +, g +, x, x,x. Figure shows 5, easible solutions randomly generated in the decision variable space. The (9) Pareto optimal curve Figure : Non-Linearity in unctionals produces non-uniorm density o solutions. Pareto-optimal curve and the easible region are shown in the igure. Because o the non-linearity in the unctional with x, the search space results in a variable density o solutions along the

14 axis. Solutions are more dense near = than any other region in the search space. Although this apparent plane ( = ) is not a local Pareto-optimal ront, an MOEA may get attracted here simply because o sheer density o solutions near it. It is clear that by choosing complicated unctions o to M, more complicated search spaces can be created. Since the task o an MOEA is to reach the Pareto-optimal region (which is located at one end o the easible search space), interesting hurdles in the search space can be placed to provide diiculties to an MOEA to reach the desired Pareto-optimal ront. 5. Advantages and Disadvantages The construction process here is much simpler compared to the bottom-up approach. Simple geometric constraints can be used to construct the easible search space. Using this procedure, dierent shapes (convex, nonconvex, or discrete) o the Pareto-optimal region can be generated. Unlike the bottom-up approach, here the easible search space is not constructed by layer-wise construction rom the Pareto-optimal ront. Since no particular structure is used, the easible objective space can be derived with any non-linear mapping o the decision variable space. However, the resulting Pareto-optimal ront will, in general, be hard to express mathematically. Moreover, although the constraint suraces can be simple, the shape and continuity o the resulting Pareto-optimal ront may not be easy to visualize. Another diiculty is that since the Paretooptimal ront will lie on one or more constraint boundaries, a good constraint-handling strategy must be used with an MOEA. Thus, this approach may be ideal or testing MOEAs or their ability to handle constraints. 6 Diiculties with Existing MOEAs Most MOEA studies up until now have considered two objectives, except a ew application studies where more than two objectives are used. This is not to say that the existing MOEAs cannot be applied to problems having more than two objectives. Developers o the state-o-the-art MOEAs (such as PAES (Knowles and Corne, 999), SPEA (Zitzler and Thiele, 999), NSGA-II (Deb et al., ) and others) have all considered the scalability aspect while developing their algorithms. The domination principle, non-dominated sorting algorithms, elite-preserving and other EA operators can all be extended or handling more than two objectives. Although the niching operator can also be applied in most cases, their computational issues and ability in maintaining a good distribution o solutions are needed to be investigated in higher-objective problems. For example, a niching operator may attempt to maintain a good distribution o solutions by replacing a crowded solution with a less crowded one and the crowding o a solution may be determined by the distance rom its neighbors. For two objectives, the deinition o a neighbor along the Pareto-optimal curve is clear and involves only two solutions (let and right solutions). However, or more than two objectives, when the Pareto-optimal ront is a higher-dimensional surace, it is not clear which (and how many) solutions are neighbors o a solution. Even i a deinition can be made, inding a distance metric rom all distances o its neighbors gets computationally expensive because o the added dimensionality. Because o the higher dimensionality o the Pareto-optimal ront, more computationally eective distribution metrics may be needed. Although a widely distributed set o solutions can be ound, as using NSGA-II or SPEA (a modiied version o SPEA suggested in (Zitzler, Laumanns, and Thiele, )) shown in the next section, the obtained distribution can be ar rom being a uniormly distributed set o points on the Pareto-optimal surace. The niching method are usually designed to attain a uniorm distribution (provided that the EA operators are able to generate the needed solutions), the overall process may be much slower in problems with large number o objectives. The test problems suggested in this paper will certainly enable researchers to make such a complexity study or the state-o-the-art MOEAs. 4

15 Although the distribution o solutions is a matter to test or problems with large number o objectives, the convergence to the true Pareto-optimal ront is also important to keep track o. Because o sheer increase in the dimensionality in the objective space, interactions among decision variables may produce diiculties in terms o having local Pareto-optimal ronts and variable density o solutions. An increase in dimensionality o the objective space also causes a large proportion o a random initial population to be non-dominated to each other (Deb, ), thereby reducing the eect o selection operator in an MOEA. Thus, it is also important to test i the existing domination-based MOEAs can reach the true Pareto-optimal ront in such test problems. Since the desired ront will be known in test problems, a convergence metric (such as average distance to the ront) can be used to track the convergence o an algorithm. We discuss more about perormance metrics in the next section. 7 Perormance Metrics A number o perormance metrics or MOEA studies have been discussed in Veldhuizen (999) and Deb (). The latter study has classiied the metrics according to the aspect measures by them and suggested three dierent categories:. Metrics that evaluate closeness to the Pareto-optimal ront,. Metrics that evaluate diversity in obtained solutions, and. Metrics that evaluate both the above. Most o the existing metrics require a pre-speciied set o Pareto-optimal (reerence) solutions P. The obtained set Q is compared against P. In the irst category, almost all metrics suggested in the context o two-objective problems can be applied to problems having more than two objectives. The error ratio metric, which counts the number o solutions common to P and Q can be used in higher dimensions. The set coverage metric which calculates the proportion o solutions in Q which are dominated by P can also be used in higher dimensions. The generational distance metric, which calculates the geometric mean o the nearest Euclidean distances o solutions Q rom members o P, can also be used in higher dimensions. However, most o the metrics suggested or measuring the spread in two-objective problems may not be possible to use in higher-objective problems so easily. For example, the spacing metric, which calculates the standard deviation in the distances between consecutive solutions, would be diicult to use in higher dimensions. The concept o consecutive solutions in higher dimensions does not simply exist. Although Schott s (995) suggestion o using the minimum objective-wise distance to any other solution can be used in higher-objective problems, a Q with a good spacing metric does not necessarily mean a good distribution o solutions in the entire Pareto-optimal ront. To ensure a good distribution in the entire Pareto-optimal ront, the obtained set Q must be compared with the known Pareto-optimal ront P, so that the obtained set is evaluated or its spread with the extreme solutions o the ront. Although the spread metric, which uses consecutive distances, is a way to consider the extreme solutions in the metric evaluation, it cannot be used in problems having more than two objectives. The maximum spread metric, which measures the Euclidean distance between extreme solutions, does not reveal the true distribution o intermediate solutions anyway. However, the chi-square-like deviation measure can be used with a user-deined neighborhood parameter σ. In this metric, or each solution o P the number o points in Q which are within a σ away can be counted. Such a count vector can then be compared with a pre-deined (in most cases, a uniorm) distribution and the deviation measure can be calculated. Alternatively, the Pareto-optimal ront can represented into a number o small hyper-boxes. Thereater, the number o solutions (o Q) present in each hyper-box is counted. 5

16 Using these counts, either the chi-square-like deviation measure (rom a uniorm distribution o solutions in all hyper-boxes) or simply the number o occupied hyper-boxes can be used as a distribution metric. In the third category, the hyper-volume metric, measuring the overall hyper-volume covered by the set Q, in combination with the relative set coverage metric were designed with respect to an arbitrary number o objectives and have been used already on,, and 4-objective knapsack problems (Zitzler and Thiele, 999). The mutual non-covered hyper-volume metric (Zitzler, 999), which is similar to the set coverage metric, is applicable in higher-objective problems as well. Nevertheless, the convergence to coverage issue may not become clear rom one such metric. Furthermore, the attainment surace metric is in another method independent o the number o objectives, but repeatedly inding the intersection o cross-lines with higher-dimensional attainment suraces may be computationally expensive. From the above discussion, it is clear that some o the existing perormance metrics or evaluating the distribution o solutions can be used in higher-objective problems but they must be evaluated with caution. Further research in this direction is necessary to develop aster and better perormance metrics. 8 Test Problem Suite Using the latter two approaches o test problem design discussed in this paper, we suggest here a representative set o test problems. However, other more interesting and useul test problems can also be designed using the techniques o this paper. 8. Test Problem DTLZ As a simple test problem, we construct an M-objective problem with a linear Pareto-optimal ront: Minimize (x) = x x x M ( + g(x M )), Minimize (x) = x x ( x M )( + g(x M )),.. Minimize M (x) = x () ( x )( + g(x M )), Minimize M (x) = ( x )( + g(x M )), subject to x i, or i =,,...,n. The unctional g(x M )requires x M = k variables and must take any unction with g. We suggest the ollowing: [ g(x M ) = x M + ] (x i ) cos(π(x i )). () x i x M The Pareto-optimal solution corresponds to x M = and the objective unction values lie on the linear hyper-plane: M m= m =. A value o k = 5 is suggested here. In the above problem, the total number o variables is n = M + k. The diiculty in this problem is to converge to the hyper-plane. The search space contains ( k ) local Pareto-optimal ronts, each o which can attract an MOEA. NSGA-II nd SPEA with a population size o is run or generations using a real-parameter SBX crossover operator (η c = 5) and a variablewise polynomial mutation operator (η m = ). The crossover probability o. and mutation probability o /n are used. Ater a variable is crossed with the SBX probability distribution, they are exchanged with a probability. The perormances o NSGA-II and SPEA are shown in Figures 4 and 5, respectively. The igure shows that both NSGA-II and SPEA come close to the Pareto-optimal ront and the distributions o solutions over the Pareto-optimal ront are also 6

17 Figure 4: The NSGA-II population on test problem DTLZ. Figure 5: The SPEA population on test problem DTLZ. not bad. In this problem and in most o the latter problems, we observed a better distribution o solutions with SPEA compared to NSGA-II. However, the better distributing ability o SPEA comes with a larger computational complexity in its selection/truncation approach compared to that in the objective-wise crowding approach o NSGA-II. The problem can be made more diicult by using other diicult multi-modal g unctions (using a larger k) and/or replacing x i by non-linear mapping x i = N i (y i ) and treating y i as decision variables. For a scale-up study, we suggest testing an MOEA with dierent values o M, maybeinthe range M [, ]. It is interesting to note that or M> cases all Pareto-optimal solutions on a three-dimensional plot involving M and any two other objectives will lie on or below the above hyper-plane. 8. Test Problem DTLZ This test problem is identical to the problem described in subsection 4.4: Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) cos(x M π/) cos(x M π/), Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) cos(x M π/) sin(x M π/), Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) sin(x M π/),.. Minimize M (x) =(+g(x M )) cos(x π/) sin(x π/), Minimize M (x) =(+g(x M )) sin(x π/), x i, or i =,,...,n, where g(x M )= x i x M (x i ). () The Pareto-optimal solutions corresponds to x M = and all objective unction values must satisy the equation 9. Here, we recommend k = x M =. The total number o variables is n = M +k. NSGA-II and SPEA with identical parameter setting as in DTLZ simulation run inds Pareto-optimal solutions very close to the true Pareto-optimal ront, as shown in Figures 6 and 7, respectively. This unction can also be used to investigate an MOEA s ability to scale up its perormance 7

18 Figure 6: The NSGA-II population on test problem DTLZ. Figure 7: The SPEA population on test problem DTLZ. in large number o objectives. Like in DTLZ, or M>, the Pareto-optimal solutions must lie inside the irst quadrant o the unit sphere in a three-objective plot with M as one o the axes. To make the problem diicult, each variable x i (or i =to(m )) can be replaced by the mean value o p variables: x i = ip p k=(i )p+ x k. 8. Test Problem DTLZ In order to investigate an MOEA s ability to converge to the global Pareto-optimal ront, we suggest using the above problem with the g unction given in equation : Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) cos(x M π/) cos(x M π/), Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) cos(x M π/) sin(x M π/), Minimize (x) =(+g(x M )) cos(x π/) cos(x π/) sin(x M π/),.. Minimize M (x) =(+g(x M )) cos(x π/) sin(x π/), Minimize M (x) =(+g(x M )) sin(x π/), x i, [ or i =,,...,n, where g(x M ) = x M + ] x i x M (x i ) cos(π(x i )). () Here, we suggest k = x M =. There are a total o n = M + k decision variables in this problem. The above g unction introduces ( k ) local Pareto-optimal ronts, and one global Pareto-optimal ront. All local Pareto-optimal ronts are parallel to the global Pareto-optimal ront and an MOEA can get stuck at any o these local Pareto-optimal ronts, beore converging to the global Pareto-optimal ront (at g = ). The global Pareto-optimal ront corresponds to x M =(,...,) T. The next local Pareto-optimal ront is at g =. NSGA-II and SPEA populations ater 5 generations are shown in the true Pareto-optimal ronts in Figures 8 and 9. It is seen that both algorithms could not quite converge on to the true ront, however both algorithms have maintained a good diversity o solutions on the true ront. The problem can be 8

19 Figure 8: The NSGA-II population on test problem DTLZ. Figure 9: The SPEA population on test problem DTLZ. made more diicult by using a large k or a higher-requency cosine unction. 8.4 Test Problem DTLZ4 In order to investigate an MOEA s ability to maintain a good distribution o solutions, we modiy problem DTLZ with a dierent meta-variable mapping: Minimize (x) =(+g(x M )) cos(x α π/) cos(xα π/) cos(xα M π/) cos(xα M π/), Minimize (x) =(+g(x M )) cos(x α π/) cos(xα π/) cos(xα M π/) sin(xα M π/), Minimize (x) =(+g(x M )) cos(x α π/) cos(xα π/) sin(xα M π/),.. Minimize M (x) =(+g(x M )) cos(x α π/) sin(xα π/), Minimize M (x) =(+g(x M )) sin(x α π/), x i, or i =,,...,n, where g(x M )= x i x M (x i ). (4) The parameter α = is suggested here. Here, too, all variables x to x M are varied in [, ]. We also suggest k = here. There are n = M + k decision variables in the problem. This modiication allows a dense set o solutions to exist near the M - plane (as in Figure 5). NSGA-II and SPEA populations at the end o generations are shown in Figures and, respectively. For this problem, the inal population is dependent on the initial population. But in both methods, we have obtained three dierent outcomes: (i) all solutions are in the - plane, (ii) all solutions are in the - plane and (iii) solutions are on the entire Pareto-optimal surace. Since the problem has more dense solutions near the - and - planes, some simulation runs o both NSGA-II and SPEA get attracted to these planes. Problems with a biased density o solutions at other regions in the search space may also be created using the mapping suggested in subsection 4.. It is interesting to note that although the search space has a variable density o solutions, the classical weighted-sum approaches or other directional methods may not have any 9

20 .5 Run Run Run.5 Run Run Run Figure : The NSGA-II population on test problem DTLZ4. Three dierent simulation runs are shown. Figure : The SPEA population on test problem DTLZ4. Three dierent simulation runs are shown. added diiculty in solving these problems compared to DTLZ. Since MOEAs attempt to ind multiple and well-distributed Pareto-optimal solutions in one simulation run, these problems may hinder MOEAs to achieve a well-distributed set o solutions. 8.5 Test Problem DTLZ5 The mapping o θ i in the test problem DTLZ can be replaced with that given in equation : Minimize (x) =(+g(x M )) cos(θ π/) cos(θ π/) cos(θ M π/) cos(θ M π/), Minimize (x) =(+g(x M )) cos(θ π/) cos(θ π/) cos(θ M π/) sin(θ M π/), Minimize (x) =(+g(x M )) cos(θ π/) cos(θ π/) sin(θ M π/),.. Minimize M (x) =(+g(x M )) cos(θ π/) sin(θ π/), (5) Minimize M (x) =(+g(x M )) sin(θ π/), π where θ i = 4(+g(r)) ( + g(r)x i), or i =,,...,(M ), g(x M )= x i x M (x i ), x i, or i =,,...,n. The g unction with k = x M = variables is suggested. As beore, there are n = M + k decision variables in this problem. This problem will test an MOEA s ability to converge to a curve and will also allow an easier way to visually demonstrate (just by plotting M with any other objective unction) the perormance o an MOEA. Since there is a natural bias or solutions close to this Pareto-optimal curve, this problem may be easy or an algorithm to solve, as shown in Figure and, obtained using NSGA-II and SPEA ater generations and with other parameter setting as beore. Because o its simplicity and ease o representing the Pareto-optimal ront, we recommend that a higher-objective (M [5, ]) version o this problem be used to study the computational time complexity o an MOEA.

21 Figure : The NSGA-II population on test problem DTLZ5. Figure : The SPEA population on test problem DTLZ Test Problem DTLZ6 The above test problem can be made harder by doing similar modiication to the g unction in DTLZ5 as done in DTLZ. However, in DTLZ6, we use a dierent g unction: g(x M )= x. i. (6) x i x M The size o x M vector is chosen as and the total number o variables is identical as in DTLZ5. The above change in the problem makes NSGA-II and SPEA diicult to converge to the true Pareto-optimal ront as in DTLZ5. The population ater 5 generations o both algorithms are shown in Figures 4 and 5, respectively. The Pareto-optimal curve is also marked on the plots. It is clear rom the igures that both NSGA-II and SPEA do not quite converge to the true Figure 4: The NSGA-II population on test problem DTLZ6. Figure 5: The SPEA population on test problem DTLZ6. Pareto-optimal curve. The lack o convergence to the true ront in this problem causes these MOEAs to ind a dominated surace as the obtained ront, whereas the true Pareto-optimal ront

22 is a curve. In real-world problems, this aspect may provide misleading inormation about the properties o the Pareto-optimal ront, a matter which we discuss more in subsection 8.9 in this paper. 8.7 Test Problem DTLZ7 This test problem is constructed using the problem stated in equation. This problem has a disconnected set o Pareto-optimal regions: Minimize (x )=x, Minimize (x )=x,.. Minimize M (x M )=x M, Minimize M (x) =(+g(x M ))h(,,..., M,g), where g(x M )=+ x 9 M x i x M x i, h(,,..., M,g)=M [ ] M i i= +g ( + sin(π i)), subject to x i, or i =,,...,n. This test problem has M disconnected Pareto-optimal regions in the search space. The unctional g requires k = x M decision variables and the total number o variables is n = M + k. We suggest k =. The Pareto-optimal solutions corresponds to x M =. This problem will test an algorithm s ability to maintain subpopulation in dierent Pareto-optimal regions. Figures 6 and 7 show the NSGA-II and SPEA populations ater generations. It is clear that both (7) Figure 6: The NSGA-II population on test problem DTLZ7. Figure 7: The SPEA population on test problem DTLZ7. algorithms are able to ind and maintain stable and distributed subpopulations in all our disconnected Pareto-optimal regions. The problem can be made harder by using a higher-requency sine unction or using a multi-modal g unction as that described in equation.