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AIAA 2015-0381
AIAA SciTech
5-9 January 2015, Kissimmee, Florida
53rd AIAA Aerospace Sciences Meeting
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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 (mattson@byu.edu)
†
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Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
μ2
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(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.
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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.
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A.
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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
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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.
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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.
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III.
The Mechanism of L-Dominance
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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
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μ2
a1
a3
μ3
a2
0
μ1
a2
a1
0
0
μ1
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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.
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Figure 6. The effect of the userdefined parameter p on the shape of
the approximately dominated region
bounded by a Lam´e curve.
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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
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(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
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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.
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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.
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