AIAA 2015-0381 AIAA SciTech 5-9 January 2015, Kissimmee, Florida 53rd AIAA Aerospace Sciences Meeting Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 L-Dominance: A New Mechanism Combining -Dominance and Pareto Knee Exploitation in Evolutionary Multi-objective Optimization Braden J. Hancock∗, Tim B. Nysetvold†, and Christopher A. Mattson‡ Brigham Young University, Provo, UT, 84602, USA In Evolutionary Multi-objective Optimization (EMO), the mechanism of -dominance has received a lot of attention because of its ability to guarantee convergence near the Pareto frontier and maintain diversity among solutions at a reasonable computational cost. A significant weakness of this mechanism is its inability to also identify and exploit knee regions of the Pareto frontier, which are frequently the regions of the frontier that are most interesting to the user. We therefore propose a new mechanism—L-dominance, based on the Lam´e curve—as an alternative to -dominance in EMO. The geometry of the Lam´e curve naturally supports a greater concentration of solutions in directions of high tradeoff between objectives than a corresponding -box of the same area. This adaptable resolution of solutions in knee regions of the Pareto frontier will result in significant gains in computational efficiency for complex optimization routines in large n-objective design scenarios. I. Introduction Nearly all design problems include multiple objectives. Often, these objectives are in conflict with each other. In recent years, one of the most popular approaches for performing numerical optimization in such scenarios has been Evolutionary Multi-objective Optimization (EMO). In EMO methods, multiple potential solutions evolve simultaneously toward the Pareto frontier—the region of the design space that represents all non-dominated solutions. An introduction to EMO methods and principles can be found in (Deb 2008).5 The goals of these methods are typically to ∗ Research Assistant, Department of Mechanical Engineering, and AIAA Student Member Research Assistant, Department of Mechanical Engineering, and AIAA Student Member ‡ Associate Professor, Department of Mechanical Engineering, and AIAA Member ([email protected]) † 1 of 11 Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. μ2 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 (a) converge near the Pareto frontier, (b) maintain diversity among discovered solutions, and (c) achieve the first two goals in a computationally efficient manner.4 The mechanism of -dominance, first introduced by Laumanns et al., has received a lot of attention in the discipline in recent years because of its ability to achieve all three of these goals when implemented in conjunction with an EMO algorithm.11 As shown in Fig. 1, -dominance divides the design space into boxes with dimensions equal to . No more than one solution is allowed per box, and the solutions in all nondominated boxes are guaranteed a place in each successive generation. Limiting the number of solutions in each region ensures diversity, and convergence is guaranteed once all of the boxes that contain the Pareto frontier have been occupied. The calculations associated with this mechanism are computationally benign, and so the third EMO goal is achieved as well. However, one very significant weakness of -dominance is its apathy concerning the tradeoff properties of the Pareto frontier. Multiple studies have shown that designers tend to sePareto knees lect solutions from knee regions of the Pareto frontier, where rapid meaningful tradeoff of objectives occurs (for example, two knees have been identified in Fig. 1).1–3 In these regions of high tradeoff, greater resolution of solutions (and therefore ε smaller -boxes) are required to adequately capture the geometry of the Pareto frontier. However, using this same resolution ε over the entire frontier could result in great computational inμ1 efficiencies, particularly in high-dimensional spaces. In this paper, we propose the application of a new mecha- Figure 1. A Pareto set of solutions resulting from the use of -dominance. nism that provides the benefits of -dominance while also alTwo knee regions of the Pareto fronlowing for efficient knee region exploitation. This mechanism tier have also been identified. employs a shape known as a Lam´e curve that was introduced and utilized in a previous publication by the authors.6 By preserving diversity through the use of Lam´e curves centered on solutions rather than -boxes oriented with respect to the objective axes, the mechanism of L-dominance achieves the three primary goals of EMO methods while naturally promoting a greater density of solutions in regions of high tradeoff on the Pareto frontier. The remainder of this paper is organized as follows: we begin in Sec. II by briefly reviewing the concepts of -dominance and Pareto knee exploitation. In Sec. III we introduce the mechanism of L-dominance, describe how it can be applied, and discuss its advantages. In Sec. IV we show how L-dominance can result in significant efficiency gains over -dominance. Finally, in Sec. V we offer concluding remarks. 2 of 11 II. Technical Preliminaries This section reviews the theory and proposed applications of two concepts that serve as foundations for the L-dominance mechanism. The first is -dominance. In Sec. IIA we provide the definition of -dominance as it was originally introduced. We then briefly describe four variations that have been proposed in order to improve its ability to provide varying resolution over the design space. The second concept is Pareto knee exploitation. In Sec. IIB we review the motivation and common approaches for increasing the density of solutions in knee regions of the Pareto frontier. -dominance The concept of domination in a design space is well-known: a solution is dominated if there exists at least one other solution that is at least as good as the solution in question in every objective, and better than that solution in at least one objective. -dominance allows for a solution to be “approximately” dominated. The user provides a value for that represents the minimum amount of change in an objective that he or she considers to be significant. Then using intervals of size , the design space can be partitioned into a grid of -boxes (as shown earlier in Fig. 1). Each solution is contained within one of these -boxes, and the coordinates of that box become the values by which -dominance is assessed. Consider, for example, the design space presented in Fig. 2, wherein all objectives are to be minimized. While solution A does not dominate solution B (not all coordinates of A are less than or equal to the coordinates of B), A does -dominate B, since the box coordinates of A (2, 2) are all less than or equal to the coordinates of B (2, 3), and at least one box coordinate is less than the corresponding box coordinate of B. -dominance is referred to as a mechanism rather than a method because the mere existence of -boxes is not sufficient to B generate a Pareto frontier. Rather, calculations of -dominance are used within existing EMO methods when determining which soA lutions to archive or allow to pass on information to the following generation. This flexibility has allowed -dominance to be incorporated into a wide variety of methods. A few of these include ε -MOEA,4 -NSGA-II,9 -MyDE,13 and Adaptive- Box.10 ε As was mentioned in Sec. I, a very apparent weakness of μ1 dominance as it was originally defined is its inability to adjust Figure 2. Solution A does not the resolution of generated solutions within the design space. It dominate Solution B, but does maintains diversity equally over all regions of the Pareto frontier, -dominate it regardless of whether objectives are changing rapidly in that area or hardly at all. Consequently, a number of -dominance variations have been proposed in the years since its inception. μ2 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 A. 3 of 11 x1p + x2p + ... + xnp = 1 : 0<p<∞ (1) This equation is then used to compute values i that vary throughout the space according to a geometric sequence, creating smaller -boxes where a single knee of the Pareto frontier is projected to be (see Fig. 3(b)).7 However, this method requires prior knowledge of the shape of the Pareto frontier (i.e., function calls spent on creating an approximation), assumes a symmetric geometry, and cannot be adjusted to account for multiple knees. Clearly, the ability of an -dominance to identify and exploit Pareto knee regions is very desirable within the discipline. However, a satisfactory solution has yet to be found. (b) μ2 (a) μ2 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 The original creators of -dominance recommended using a different value of i for each objective i in the design space. This potentially changes the -boxes from squares to rectangles, but still clearly lacks the ability to increase the resolution of solutions only where it is needed. Jin and Wong proposed using a grid of dynamically adjusted -boxes (see Fig. 3(a)). This approach, called Adaptive Rectangle Archiving (ARA), allows for the grid lines that define -boxes to move along the objective axes in order to adjust the density of solutions in the design space.8 However, according to Laumanns et al., “for ensuring the convergence property it is important not to move or translate any of the box limits; in other words, the assignment of the elements to the boxes must stay the same.”11 Thus, the improvement in solution density offered by this method comes at the cost of one of the three primary objectives of EMOs—a guarantee of convergence. Five years after the introduction of -dominance, Hern´andez-D´ıaz et al. introduced the concept of Pareto-adaptive -dominance, wherein an approximation of the Pareto frontier is obtained and associated with an equation of the form: μ1 μ1 Figure 3. Pareto sets resulting from (a) Adaptive Rectangle Archiving with dynamically adjusted -boxes, and (b) Pareto-adaptive -dominance. 4 of 11 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 B. Pareto knee exploitation According to Bechikh et al., “knee regions are parts of the Pareto front presenting the maximal trade-offs between objectives.”1 Understandably, these portions of the design space are generally particularly interesting to the designer and require closer inspection than other regions with less tradeoff occuring. The desirability of obtaining solutions in the knee regions of the Pareto frontier can be attested to by many experts in the field of multi-objective optimization. In 1999, Das commented that ”from practical experience...the user or designer usually picks a point...where the Pareto surface bulges out the most,” referring to convex knees.3 In 2004, Branke et al. introduced a modifed EMO algorithm which was capable of discovering and exploiting multiple knee regions in a design space, noting once again that ”the solutions at the knee are most likely to be the optimal choice of the decision maker.”2 And in 2010, Bechikh et al. proposed the use of mobile reference points to identify knee regions of the Pareto frontier.1 The search for EMO methods capable of satisfactorily exploiting knee regions continues in the discipline today. Outside of EMO, other methods for exploiting Pareto μ2 knees have also been proposed. One such method is the use ∆r1 of a smart Pareto filter following the generation of a welldistributed Pareto set, typically through gradient-based techniques. By systematically removing all solutions that lie in ∆r2 ∆t2 a region of “practically insignificant tradeoff” of an already accepted solution (see Fig. 4), a smart Pareto set is gener∆t1 μ1 ated, which is a Pareto set that includes a greater concentration of solutions in knee regions. Producing the requisite welldistributed set can be computationally nontrivial, however. Thus, in a previous work of the authors of this paper, a method Figure 4. The shaded region repfor directly generating smart Pareto sets was proposed. This resents the region of “practically inmethod, called the Smart Normal Constraint (SNC) method, significant tradeoff” (defined by the employed a series of single-objective optimizations in con- user through parameters r and t) that junction with linear constraints that were calculated to produce surrounds a smart Pareto solution. solutions in regions of greater interest to the designer, based on Other solutions which reside in this area will be discarded when using a the tradeoff properties of the Pareto frontier.6 The tradeoff of smart Pareto filter. objectives occuring in the design space was approximated using equations corresponding to the shape of a Lam´e curve, which will be described in greater detail in the following section. Using the developments described in this paper, the Lam´e curve can now be used to generate smart Pareto sets in EMO methods as well as gradient-based ones. 5 of 11 III. The Mechanism of L-Dominance Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 In this section, we will first introduce the mathematical definition of the Lam´e curve, then demonstrate how it can be used to enable L-dominance calculations and discuss what the advantages of L-dominance are over -dominance.The fundamental concept behind L-dominance is the replacement of -boxes with a shape that naturally lends itself to a greater density of solutions in regions of high tradeoff. This is done by redistributing the Using Lam´e curves defined with respect to archived Pareto solutions, rather than -boxes defined with respect to the objective axes, the diversity and convergence guaranteed by epsilon-dominance are maintained, but Pareto knee exploitation is also enabled. A. The Mathematical Definition of the Lam´e Curve The Lam´e curve is defined by the equation n 1p X |Ai,i (xi − di )| p = 1 (n ≥ 2) (2) i=1 where 1 . . . 0 a1 . . A = .. . . . .. 1 0 . . . an (a > 0) (3) and d is a vector from the origin to the center point of the Lam´e curve. Algebraically, this represents the p-norm12 of x, offset by d (based on its location in the design space), multiplied by a transformation matrix A (for scaling purposes), and set equal to 1. When used as an EMO mechanism, the user-defined variables a and p of the Lam´e curve allow the user to determine the distribution of the Pareto solutions that will be generated for that particular problem. Each value ai corresponds to objective i in the problem and may be interpreted as the amount of change in that objective that would constitute a significant difference between two solutions in the user’s mind if all other objectives remain practically unchanged (see Fig. 5). Accordingly, larger values for ai will result in larger approximate domination boundaries, and therefore fewer solutions in the final Pareto set. The parameter p affects the curvature of the approximate domination boundaries and therefore controls the extent to which high tradeoff between objectives is required in order for two solution nearby each other to both remain in the Pareto set. The effect of p on the shape of the approximately dominated region is illustrated in Fig. 6. While the method will work for any value of p between 0 and 2, it is assumed that for most purposes, the user will select a value between 0 and 1. At p = 0, the area of the approximately dominated region approaches 0. At p = 2, the Lam´e curve becomes 6 of 11 μ2 a1 a3 μ3 a2 0 μ1 a2 a1 0 0 μ1 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 μ2 (a) (b) Figure 5. The Lam´e curves for a design space in a design space with (a) 2 objectives and (b) 3 objectives a circle (or higher dimensional version of a circle), resulting in an approximately even distribution of solutions over the entire Pareto frontier. B. How to apply L-Dominance Calculations involving L-dominance occur in the same place where -dominance calculations occur within any -based method. This means that, like -dominance, the L-dominance mechanism can be easily incorporated into a wide variety of different algorithms. Essentially, wherever dominance calculations are being made, a two-stage dominance check takes place. First, an approximate dominance check occurs. If a new solution lies within the Lam´e curve of an existing Pareto solution, the new solution is discarded automatically (just as a second solution in a single -box would be discarded). If no overcrowding of a single Lam´e curve is discovered, then a regular dominance check occurs. In most methods employing -dominance, if the new solution is neither approximately dominated nor actually dominated, then that solution will be added to an archived set of solutions that will progress further in the overall optimization routine. 7 of 11 Figure 6. The effect of the userdefined parameter p on the shape of the approximately dominated region bounded by a Lam´e curve. Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 C. Advantages of L-Dominance The advantages of applying the L-dominance mechanism within a method are numerous. Like -based methods, it guarantees convergence near the Pareto frontier. Once the Pareto frontier is entirely contained within Lam´e curves, no new solutions may be added. This stagnation of the Pareto set can easily be used as a termination condition for the algorithm. L-dominance also guarantees diversity—based on the user-defined values for a and p, a certain minimum difference between solutions specific to the problem at hand will be required for two nearby solutions to remain in the Pareto set. Furthermore, like the -dominance mechanism, L-dominance is computationally benign. Because the Lam´e is describable with a single equation, the determination of whether or not a new solution resides in an existing Lam´e curve is a very simple check to perform. Aside from satisfying the three objectives of EMOs, L-dominance provides additional benefits above and beyond -dominance and its proposed variations. As has already been mentioned, Ldominance encourages a greater density of solutions in knee regions of the Pareto frontier. There is also no limit to how many knees it can identify, and it requires no prior information about the shape of the Pareto frontier to work. Unlike -dominance, where adjacent solutions can be located immediately on either side of the line between -boxes, there is a minimum distance that adjacent solutions are guaranteed to have between them in L-dominance. Finally, the approximate domination region of a Pareto solution under the L-dominance method has a finite area, rather than extending out to infinity in the directions away from the utopia point, which is the case with -dominance. As a result, solutions on the edges of the Pareto frontier—which are also considered knee solutions by many definitions—are more likely to be discovered when using L-dominance. IV. Simulation Results One way to assess the benefits of using L-dominance over -dominance is to compare the resulting Pareto sets when approximate domination regions of equal area are used. Consider, for example, Fig. 7-8, which show the Pareto sets generated for two two-objective problems when the approximate domination mechanisms formerly described are utilized within the -MOEA algorithm developed by Deb et al.4 In Fig. 7, the design space is in the shape of a supercircle of order 4, centered on the point (1,1). The more complicated Pareto frontier in Fig. 8 comes from the TNK problem introduced by Tanaka et al.14 In both problems, the objectives are to be minimized, and the objectives have been scaled so that they are bounded by the interval [0, 1]. The Pareto set in Fig. 7(a) was generated using L-dominance with a PIT region defined by a = [0.65, 0.65] and p = 0.33. This results in an approximately dominated area (the area within √ the Lam´e curve) of 0.083. Using -boxes of equal area (so that i = 0.083 = 0.288 ), Fig. 7(b) was generated. As was mentioned in Sec. IIIC, the dominated regions of a given -box expand to positive infinity in each objective direction behind the -box. Thus, when a Pareto frontier has a 8 of 11 (a) (b) μ2 ε2 0 μ1 (c) 1 μ2 1 μ2 1 ε2 0 1 ε1 μ1 0 1 ε1 μ1 1 Figure 7. The Pareto sets generated for a design space in the shape of a supercircle using the mechanisms of (a) L-dominance, (b) -dominance, and (c) finite -dominance. The areas of the regions of approximate domination are equal for all mechanisms and the same number of function calls were performed for each approach. (a) (b) ε2 0 μ1 11 (c) 1 μ2 1 μ2 1 μ2 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 very tight knee, such as the one shown here, the resulting set may have a very small number of solutions directly on the knee (a single solution, in this case) and nearly no solutions anywhere else on the Pareto frontier. Aside from being a poor representation of the Pareto frontier at large, this sparcity of solutions makes comparison between mechanisms difficult. Thus, for the sake of illustration, the Pareto set resulting from a variation on -dominance called finite -dominance is also shown in Fig. 7(c). With finite -dominance, the -dominated region of a solution extends only to the bounds of its own -box, thereby maintaining a limit of one solution per box while preventing the dominated space from being infinitely large. Looking at Fig. 7(c), it is evident that while coverage of the Pareto frontier has improved with finite -dominance, the Pareto knee remains severly underrepresented. ε2 0 ε1 μ1 1 0 ε1 μ1 1 Figure 8. The Pareto sets generated for the TNK problem (Tanaka et al, 1995) using the mechanisms of (a) L-dominance, (b) -dominance, and (c) finite -dominance. The areas of the regions of approximate domination are equal for all mechanisms and the same number of function calls were performed for each approach. 9 of 11 Downloaded by BRIGHAM YOUNG UNIVERSITY on April 27, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-0381 For both of these problems, the same number of function calls were used to generate the Pareto sets corresponding to each mechanism. Thus, for equal computational cost, the approximately dominated area in the shape of a Lam´e curve resulted in not only better representation of the Pareto frontier at large, but also a much finer resolution in the knee regions. Here, resolution refers to the proximity of solutions in these regions where a greater concentration of solutions is desired. In order to increase the resolution and representation of the -based methods, a smaller value for might be selected. For example, in Fig. 7 an -box of approximately one-fourth the current size (used with the finite -dominance variation) would be capable of capturing approximately the same resolution in the knee as the the L-dominance method. However, a smaller approximately dominated region will result in a greater required number of function calls to reach convergence. Furthermore, as the number of dimensions of a problem increases linearly, the number of approximately dominated regions increases exponentially. For high-dimensional problems, using any form of -dominance to achieve the same resolution as L-dominance quickly becomes computationally prohibitive. V. Conclusion The mechanism of -dominance provides many benefits in EMO, including guaranteed convergence, diversity preservation, and computational efficiency. However, without the ability to increase the density of solutions discovered in knee regions of the Pareto frontier, the methods employing -dominance are handicapped in many scenarios. In this paper, we introduce a new mechanism, L-dominance, which maintains the positive qualities of -dominance while adding the ability to exploit Pareto knees. This is accomplished by reshaping the area of the approximately dominated regions from -boxes to Lam´e curves, which naturally support a greater resolution of solutions in regions of high tradeoff. The value of the proposed mechanism has been shown theoretically and by simulation results. Based on these results, we confidently recommend the use of Lam´e-dominance over -dominance as a diversity/convergence mechanism, especially in highdimensional EMO problems. VI. Acknowledgements This research was made possible by a Research Experience for Undergraduates grant by the National Science Foundation. 10 of 11 References 1 S. Bechikh, L. B. Said, and K. Ghedira. Searching for knee regions in multi-objective optimization using mobile reference points. 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