An Adaptive Evolutionary Algorithm based on Non-Euclidean Geometry for Many-Objective Optimization

An Adaptive Evolutionary Algorithm Based
on Non-Euclidean Geometry for Many-
Objective Optimization
Annibale Panichella
a.panichella@tudelf.nl
@AnniPanic
!1

Proximity and Diversity
!2
Convergence
(proximity)
Diversity
(spread)
Crowding distance
NichesGrids
Angle distance
Reference Points
Dominance
Box-dominance
θ-dominance
ε-dominance

The Challenge
• How to measure proximity and diversity depends on the
shape of the Pareto Front
• We don’t know what is the PF shape (geometry) a priori
!3
Elliptic Geometry Hyperbolic Geometry

Pareto Front Modelling
!4
(f1)p+(f2)p+…+(fM)p =1
[Martínez et al. PPSN 2014]
Family of Curves
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
MOPs with two and three objective functions.
*Image from
p 
p
p
p
p
Non-linear ﬁtting
Solved with (iterative)
numerical methods

!5
a1(f1) +a2(f2) +…+aM(fM) = K
[Tian et al. IEEE Tans. Cyber. 2018]
p1 p2 pM
a1, a2, …, aM ≥ 0
p1, p2, …, pM ≥ 0
Generalized Fitting Model (GFM)
Complexity:
O(G' *′M2(M + N))
M = #Objectives
N = Pop. Size
G' = #Iterations of the
Levenberg-Marquardt
algorithm

!6
Time(s)
0
30
60
90
120
# Objectives
M=3 M=5 M=10 M=15
NSGA-II GFM-MOEA
Problem = DTLZ1
Pop. Size = 200
# Gen. = 300
M = {3, 5, 10, 15}

AGE-MOEA
(Adaptive GEometry-based MOEA)
7

The Framework
!8
An Adaptive Evolutionary Algorithm based on Non-Euclidean Geometry GECCO ’19, July 13–17, 2019, Prague, Czech Republic
Algorithm 1: AGE-MOEA
Input: M: Number of objectives
N : Population size
Result: Final population P
1 begin
2 P RANDOM-POPULATION(N )
3 while not (stop_condition) do
4 Q GENERATE-OFFSPRING(P)
5 F FAST-NONDOMINATED-SORT(P
–
Q)
6 F NORMALIZE(F)
7 p GET-GEOMETRY(F1, M) /* Equation 8 */
8 d 1 /* First non-dominated rank */
9 while | P | + | Fd |6 N do
10 SURVIVAL-SCORE(Fd , d, p)
11 P P
–
Fd
12 d d + 1
13 SORT(Fd ) /* by survival scores */
14 P P
–
Fd [1 : (N |P |)]
15 return P
front. First, we present an overview of the proposed framework
in Section 3.1. Then, we detail the key ingredients of AGE-MOEA in
Sections 3.2, 3.3, and 3.4.
3.1 Overview
Finally, the new population of M solutions is formed by selecting
the solutions from the non-dominated fronts, one front (or level) at
a time. Therefore, the solutions from the rst front F1 are selected
rst, followed by F2, and so on. The procedure terminates when
adding the solutions of the current non-dominated front Fd would
exceed M. In this case, AGE-MOEA selects the remaining solutions
from Fd according to the descending order of their survival scores
(lines 13-14 of Algorithm 1).
The survival scores play an important role during reproduction
as well. Indeed, parents are selected from the current population P
using the binary tournament selection: a pair of solutions is randomly
selected from P; the winner of the tournament is the solution with
the best non-dominated rank (or level) or the solution with the
largest survival score at the same level of non-dominated rank.
3.2 Normalization
The rst non-dominated front F1 is rescaled and normalized by
applying the same formula used in NSGA-III [10]:
f n
i (S) = [(fi (S) zmin
i ]/ai 8S 2 F1 (4)
where fi (S) denotes the objective fi for the solution S and zmin
i is
the minimum value of the i-th objective across all solutions in the
front F1. With the numerator, the objectives are translated to have
the ideal point equal to the origin of the axes. The denominator ai
is the intercept of the M-dimensional hyperplane with the objective
AGE-MOEA inherits the main
framework on NSGA-II. The main
differences are:
1. Normalization
2. A Simple heuristic to
estimate the geometry with
complexity O(M x N)
3. Replacing the crowding
distance with the survival
score (proximity + diversity)

Normalization
!9
Population at gen. t 1. Apply the non-dominated sorting
2. Identify the ﬁrst front F1
3. Normalize F1 with the formula
fni (S) = [(fi(S) − zmini ]/ai ∀S ∈ F1
ai is the intercept of the M-dimensional
hyperplane with the objective axis fi
4. Normalize the other fronts (F2,
F3) using the values zmini and ai
computed from F1
F1
F2
F3
4
10
(2, 3)

Simplify the Fitting Equation
!10
4
10
1
1
(0,0)
Before Normalization After Normalization
Ideal Point
(1,1)
a1(f1) +a2(f2) +…+aM(fM) =Kp1 p2 pM
(f1) +(f2) +…+(fM) =1p1 p2 pM
(0,0)

Simplify the Fitting Equation
!11
(f1) +(f2) +…+(fM) =1p1 p2 pM
• Exponential equation with
no exact solutions
• We still need numerical
(iterative) methods for fitting
• We want to find the values
p1…pM such that the error of
the fitting is minimum

The Idea
!12
• Don't solve the exponential
equation
• Look at points in the front
for which the equation is
easily solvable
f1
f2
(1,1)
A
B
C(c1,c2)
(f1) +(f2) +…+(fM) =1p1 p2 pM

The Idea
!13
f1
f2
(1,1)
A
B
C(c1,c2)
(c1) +(c2) +…+(cM) =1p p p
(f1) +(f2) +…+(fM) =1p1 p2 pM
M * (c1) =1p
p = -
log(M)
log(c1)

Estimating the Geometry
!14
f1
f2
(1,1)
A
B
C
1. Find the point C(c1, c2,…, cM)
in F1 with minimum angle
distance to the bisector
2. Compute p using the formula:
0.5 1 1.5
B=(1,0)
C=(0.5, 0.5)
F=(0.9, 0.25)
f1
f central point C of a normalized non-
treme points in the front correspond to the
nt with the objective axes. After normaliza-
oint of F1 with the axis f n
i has the objective
bjectives f n
j = 0 for j , i. Furthermore, its
of the axes (ideal point) is always equal to
ent is chosen. This is because an extreme
(f n
1 (E) = 0, . . . , f n
i (E) = 1, . . . , f n
M (E) = 0)
o:
E)p
+ · · · + f n
i (E)p
+ · · · + f M
1 (E)p
⌘1/p
+ · · · + 1p
+ · · · + 0p 1/p
= 1
observation, our tting problem consists in
ystem of non-linear equations:
1)p + · · · + fM (S1)p )1/p
= 1
k )p + · · · + fM (Sk )p )1/p
= 1
(5)
whose exact solution is p = 1.
In general, given a generic central pointC that lies on the bis
of the rst quadrant ( Æ), the solution to the equation ||C||p is:
M’
i=1
C
p
i
!1/p
= 1 ! M · C
p
1 = 1
! C
p
1 = 1/Mx
! p =
lo (M)
lo (C1)
where Ci denotes the i-th coordinate of the central point C.
In the equation above, we assumed that the Ci lies exact
the bisector of the rst quadrant. However, in practice, the ce
point of a generic non-dominated front F1 may have a dist
dist?(C, Æ) 0 and, thus, its coordinates in the objective spac
not identical. For this reason, in AGE-MOEA (line 7 of Algorith
we approximate the value of the exponent p using the equati
p =
lo (M)
lo
⇣
1
M
ÕM
i=1 Ci
⌘ =
lo (M)
lo (M) lo
⇣ÕM
i=1 Ci
⌘
where M is the number of objectives; C is the central point
and computed with Equation 6; and Ci is the i-th coordina
C in the objective space. Since the coordinates of C cannot b
zero (otherwise C would coincide with the ideal point) nor all e
to ones (otherwise it would coincide with the nadir point)
summation
ÕM
i=1 Ci is always greater than zero and lower tha
This implies that the denominator in Equation 8 is always
Overall complexity O(M x N)

The Survival Score
!15
Algorithm 2: SURVIVAL-SCORE
Input:
Fd : pool of non-dominated solutions
d: index of the non-dominated front
p: exponent of estimated geometry the p-norm
1 begin
2 if d==1 then
3 score[E] +1 /* E = extreme points of F1 */
4 E /* Considered solutions */
5 Fd /* Remaining solutions */
6 for each solution S 2 do
7 proximity[S] ||f (S)||p
8 for each solution S1 2 Fd do
9 for each solution S2 2 Fd do
10 dist[S1, S2] ||f (S1) f (S2))||p
11 while | | 0 do
12 for each S 2 do
13 diversity[S] min
T 2
dist[S, T ] + min2
T 2
dist[S, T ]
14 value[S]
diversity[S]
proximity[S]
/* Select the solution with the max value */
15 S⇤ arg max
S2
value[S]
16 score[S⇤] value[S⇤]
17
–
{S⇤ } /* Considered solutions */
18 {S⇤ } /* Remaining solutions */
19 else
20 for each S 2 F1 do
21 score[S] 1/||f (S)||p
point C in Figure 1). A solution S with proximit (S) 1 dominates
parts of the unitary hypersurface of Lp (e.g., point D in Figure 1).
Equation 9, while the pairwise Lp distances between all solutions in
F1 are computed in lines 8-10. Then, the survival score is computed
within the loop in lines 12-14. In each loop, the procedure computes
the diversity score for the solutions in considering the minimum
(min) and the second minimum (min2) distances with regards to the
solution in (line 13 of Algorithm 2). In this way, the diversity of a
solution S is computed with regards to solutions that have already
been scored (or selected) in the previous iterations of the loop rather
than considering all solutions in F1. A temporary survival score
(value[S] in line 14) is then computed for each solution S 2 . The
solution S⇤ with the maximum temporary score in is selected
(line 15), and its nal survival score is assigned in line 16. Then,
the two sets and are updated in lines 17-18. The temporary
survival scores for the remaining solutions in are recomputed
in the next iterations since we need to recompute their relative
diversity with regards to the updated set .
Finally, the survival scores for the solutions in the non-dominated
fronts Fd1 are computed as the inverse of their proximity scores
(lines 19-21 in Algorithm 2). Hence, dominated solutions closer to
the unitary hypersurface induced by Lp have larger scores.
Complexity. The computational complexity of Algorithm 2 is
O(M ⇥ N2) + O(N3), where M is the number of objectives and N
is the population size. The elements of the overall complexity are:
• O(M ⇥ N) for computing the proximity scores in lines 6-7.
• O(M ⇥ N2) for computing the distances for each pair of
solutions in F1 (lines 8-10);
• O(N3) for the loop in lines 11-18. More specically, the inner
loop in line 12-14 has a complexity O(| | ⇥ | |) and it is
repeated | | (outer loop in line 11), where ✓ F1 and
✓ F1.
• O(M ⇥N) is the complexity for computing the survival score
for the fronts Fd1.
4 EMPIRICAL STUDY
Extreme points have maximum (+inf)
survival score
Proximity of each non-dominated point is
computed using the norm || · ||p
Compute the pairwise distance between
non-dominated points using the norm || · ||p
Iteratively select the solution with the
better score w.r.t. to already the selected
solutions
Score[S] =
proximity[S]
diversity[S] Normalized
diversity

Maf Benchmark
!17
Maf1 Linear, Inverted
Maf2 Concave
Maf3 Convex, Multimodal
Maf4 Concave, Multimodal
Maf5 Convex, Biased
Maf6 Concave, Degenerate
Maf7 Mixed, disconnected, Multimodal
Maf8 Linear, Degenerate
Maf9 Linear, Degenerate
Maf10 Mixed, Biased
Maf11 Convex, Disconnected, Non-separable
Maf12 Concave, Nonseparable, Biased
DeceptiveMaf13 Concave, Unimodal, Non-separable,
DegenerateMaf14 Linear, Partially separable, Large Scale
Maf15 Convex, Partially separable, Large
Scale
R. Cheng et al. A benchmark test suite for
evolutionary many-objective optimization. Complex
and Intelligent Systems 2017
True PFs for M=3

Baselines
• AR-MOEA [Y. Tian et al. 2018]
• GrEA [S. Yang et al. 2013]
• NSGA-III [Deb and Jain 2014]
• MOEA/D [Zang and Li 2007]
• θ-DEA [Yuan et al. 2016]
!18
Indicator-based with
reference point adaptation
Grid-based
θ-dominance
(relaxed dominance)
Reference points-based +
Dominance
Decomposition-based

Parameter Setting
!19
Parameters M=3 M=5 M=10
Population size N 91* 210* 275*
N. of ﬁtness evaluations N*300 27300 63000 82500
SBX probability pc = 1
SBX distributed index ηc = 30
Polynomial mutation prob. Pm = 1/n
Mutation distr. index ηc = 20
M = # Objectives
# Ref. Points
needed by
NSGA-III and
MOEA/D
n = # decision
variables
* Das and Dennis systematic approach

Results for M=3
!20
Table 3: IGD values (mean and standard deviation) achieved by the AGE-MOEA and the baselines on the Maf benchmark [2] with
M=3,5, and 10 objectives. Best performance is highlighted in grey color.
Problem M AR-MOEA GrEA NSGA-III MOEA/D -DEA AGE-MOEA
MaF1 3 4.3854e-2 (5.49e-4) # 4.2393e-2 (8.39e-4) 6.1953e-2 (2.14e-3) # 7.0473e-2 (7.88e-6) # 8.0706e-2 (7.20e-4) # 4.3056e-2 (4.25e-4)
MaF2 3 3.2100e-2 (7.71e-4) # 3.1930e-2 (4.52e-4) # 3.6179e-2 (8.14e-4) # 4.1280e-2 (1.37e-3) # 3.6522e-2 (3.46e-4) # 3.1031e-2 (6.56e-4)
MaF3 3 1.5462e+0 (1.97e+0) # 9.9568e-1 (1.98e+0) # 2.3782e+0 (3.59e+0) # 3.1098e-1 (6.76e-1) 3.3974e+0 (5.63e+0) # 5.1510e-1 (1.41e+0)
MaF4 3 1.2607e+0 (2.01e+0) # 1.3245e+0 (1.58e+0) # 3.1889e+0 (2.92e+0) # 2.1207e+0 (9.71e-1) # 1.4392e+0 (1.83e+0) # 7.5761e-1 (1.20e+0)
MaF5 3 1.0265e+0 (1.27e+0) # 9.3317e-1 (1.02e+0) # 7.1456e-1 (1.00e+0) # 1.2690e+0 (1.43e+0) # 8.1065e-1 (7.54e-1) # 3.0978e-1 (3.24e-1)
MaF6 3 5.1379e-3 (1.18e-4) 2.0989e-2 (5.84e-4) # 1.4955e-2 (1.60e-3) # 7.9072e-2 (1.26e-1) # 3.3136e-2 (2.58e-3) # 5.4330e-3 (1.12e-4)
MaF7 3 1.9735e-1 (2.33e-1) # 8.6453e-2 (4.81e-3) 7.8716e-2 (3.87e-3) 1.7749e-1 (1.18e-1) # 1.0860e-1 (6.88e-2) # 9.1365e-2 (8.64e-2)
MaF10 3 3.3521e-1 (5.23e-2) # 2.4991e-1 (4.80e-2) ⇡ 4.3074e-1 (7.67e-2) # 5.7938e-1 (9.34e-2) # 3.9627e-1 (6.32e-2) # 2.2889e-1 (3.52e-2)
MaF11 3 1.6268e-1 (1.30e-3) 2.3696e-1 (1.24e-2) # 1.6323e-1 (2.67e-3) 2.6705e-1 (6.22e-2) # 1.5653e-1 (1.51e-3) 1.7044e-1 (3.47e-3)
MaF12 3 2.2402e-1 (3.35e-3) ⇡ 2.5115e-1 (6.79e-3) # 2.3015e-1 (2.12e-2) # 2.9391e-1 (2.51e-2) # 2.2429e-1 (2.05e-3) ⇡ 2.2441e-1 (2.47e-3)
MaF14 3 9.8537e-1 (3.06e-1) ⇡ 1.3652e+0 (4.70e-1) # 1.2546e+0 (4.14e-1) # 6.1995e-1 (1.38e-1) 1.2979e+0 (4.95e-1) # 9.3860e-1 (3.08e-1)
MaF15 3 3.8100e-1 (7.39e-2) ⇡ 5.8612e-1 (8.08e-2) # 7.2037e-1 (2.10e-1) # 3.7313e-1 (9.47e-2) 9.0553e-1 (8.52e-2) # 4.2139e-1 (9.32e-2)
MaF3 5 1.5533e-1 (3.23e-1) # 1.9521e+0 (4.31e+0) # 1.3713e+0 (2.89e+0) # 1.0495e-1 (6.18e-3) # 9.3684e-2 (4.50e-3) # 5.4560e-2 (8.97e-4)
MaF4 5 2.2834e+0 (9.02e-2) # 2.3563e+0 (1.59e+0) ⇡ 3.0028e+0 (1.85e+0) # 1.1559e+1 (1.15e+0) # 2.8182e+0 (2.11e-1) # 1.8275e+0 (6.47e-2)
MaF5 5 1.9730e+0 (4.94e-3) # 1.9916e+0 (8.98e-1) # 2.0382e+0 (3.00e-1) # 7.5844e+0 (1.89e+0) # 1.9662e+0 (6.12e-3) # 1.7508e+0 (2.64e-2)
MaF7 5 2.5829e-1 (5.15e-3) ⇡ 2.3492e-1 (5.23e-3) 2.8437e-1 (5.73e-3) # 5.1550e-1 (2.40e-2) # 3.0107e-1 (2.50e-2) # 2.5782e-1 (1.72e-2)
MaF8
Mean and St. Dev. of the IGD across 50 runs
↑ means better than, ↓ worst than, and ≈ equivalent to AGE-MOEA

Maf3
!21
M = 3 objectives, N = 190 solutions, 300 iterations
IGD = 2.32e-2 IGD = 3.89e-2 IGD = 4.59e-2
IGD = 4.20e-2 IGD = 4.50e-2 IGD = 3.80e-2

Results for M=5
!22
MaF8
MaF9 5 8.8245e-2 (5.05e-3) ⇡ 1.1496e+0 (3.88e-1) # 3.5017e-1 (1.62e-1) # 1.3132e-1 (3.75e-2) # 6.7137e-1 (1.79e-1) # 8.8524e-2 (6.79e-3)
MaF11 5 3.9050e-1 (3.34e-3) # 4.9965e-1 (2.09e-2) # 3.8658e-1 (2.98e-3) ⇡ 7.7530e-1 (4.46e-2) # 3.8803e-1 (3.72e-3) ⇡ 3.8570e-1 (4.75e-3)
MaF12 5 9.4651e-1 (4.83e-3) # 9.3832e-1 (6.53e-3) # 9.3385e-1 (5.03e-3) # 1.6205e+0 (1.04e-1) # 9.2941e-1 (3.69e-3) # 9.2122e-1 (5.03e-3)
MaF14 5 8.1409e-1 (2.57e-1) # 7.5450e-1 (1.48e-1) # 8.4358e-1 (1.95e-1) # 7.8806e-1 (2.35e-1) # 1.0024e+0 (4.03e-1) # 6.5343e-1 (1.41e-1)
MaF15 5 5.6385e-1 (7.85e-2) 7.3806e-1 (5.00e-2) 1.2955e+0 (2.06e-1) # 6.3603e-1 (1.13e-1) 1.0880e+0 (1.09e-1) ⇡ 1.0901e+0 (2.33e-1)
MaF1 10 2.2537e-1 (1.69e-3) 2.3703e-1 (6.26e-3) ⇡ 2.8389e-1 (5.74e-3) # 5.3403e-1 (2.47e-2) # 3.1707e-1 (7.43e-3) # 2.3803e-1 (5.36e-3)
MaF3 10 1.5649e+0 (5.04e+0) # 1.4697e+4 (1.69e+4) # 3.2428e+3 (6.34e+3) # 1.4320e-1 (2.11e-3) # 8.5241e+0 (2.40e+1) # 6.9331e-2 (1.01e-3)
MaF4 10 9.5813e+1 (6.24e+0) # 1.6292e+2 (3.91e+1) # 9.4722e+1 (8.70e+0) # 5.3804e+2 (4.52e+1) # 1.1635e+2 (1.39e+1) # 5.7991e+1 (4.15e+0)
MaF5 10 9.7353e+1 (5.14e+0) # 4.7038e+1 (1.05e+0) ⇡ 7.7391e+1 (1.23e+0) # 3.0165e+2 (2.68e+0) # 7.7560e+1 (7.13e-1) # 4.6762e+1 (1.57e+0)
MaF6 10 3.4000e-1 (3.17e-1) ⇡ 8.6533e-1 (4.10e-1) # 6.1137e-1 (2.11e-1) # 1.1737e-1 (1.62e-1) 1.6267e-1 (2.56e-1) 3.5746e-1 (1.78e-1)
MaF7 10 1.4115e+0 (8.58e-2) # 2.6086e+0 (6.41e-2) # 1.1407e+0 (7.72e-2) # 2.4215e+0 (5.00e-1) # 9.0120e-1 (4.96e-2) # 8.4737e-1 (9.93e-3)

Results for M=10
!23
MaF8
MaF9 5 8.8245e-2 (5.05e-3) ⇡ 1.1496e+0 (3.88e-1) # 3.5017e-1 (1.62e-1) # 1.3132e-1 (3.75e-2) # 6.7137e-1 (1.79e-1) # 8.8524e-2 (6.79e-3)
MaF11 5 3.9050e-1 (3.34e-3) # 4.9965e-1 (2.09e-2) # 3.8658e-1 (2.98e-3) ⇡ 7.7530e-1 (4.46e-2) # 3.8803e-1 (3.72e-3) ⇡ 3.8570e-1 (4.75e-3)
MaF12 5 9.4651e-1 (4.83e-3) # 9.3832e-1 (6.53e-3) # 9.3385e-1 (5.03e-3) # 1.6205e+0 (1.04e-1) # 9.2941e-1 (3.69e-3) # 9.2122e-1 (5.03e-3)
MaF14 5 8.1409e-1 (2.57e-1) # 7.5450e-1 (1.48e-1) # 8.4358e-1 (1.95e-1) # 7.8806e-1 (2.35e-1) # 1.0024e+0 (4.03e-1) # 6.5343e-1 (1.41e-1)
MaF15 5 5.6385e-1 (7.85e-2) 7.3806e-1 (5.00e-2) 1.2955e+0 (2.06e-1) # 6.3603e-1 (1.13e-1) 1.0880e+0 (1.09e-1) ⇡ 1.0901e+0 (2.33e-1)
MaF1 10 2.2537e-1 (1.69e-3) 2.3703e-1 (6.26e-3) ⇡ 2.8389e-1 (5.74e-3) # 5.3403e-1 (2.47e-2) # 3.1707e-1 (7.43e-3) # 2.3803e-1 (5.36e-3)
MaF3 10 1.5649e+0 (5.04e+0) # 1.4697e+4 (1.69e+4) # 3.2428e+3 (6.34e+3) # 1.4320e-1 (2.11e-3) # 8.5241e+0 (2.40e+1) # 6.9331e-2 (1.01e-3)
MaF4 10 9.5813e+1 (6.24e+0) # 1.6292e+2 (3.91e+1) # 9.4722e+1 (8.70e+0) # 5.3804e+2 (4.52e+1) # 1.1635e+2 (1.39e+1) # 5.7991e+1 (4.15e+0)
MaF5 10 9.7353e+1 (5.14e+0) # 4.7038e+1 (1.05e+0) ⇡ 7.7391e+1 (1.23e+0) # 3.0165e+2 (2.68e+0) # 7.7560e+1 (7.13e-1) # 4.6762e+1 (1.57e+0)
MaF6 10 3.4000e-1 (3.17e-1) ⇡ 8.6533e-1 (4.10e-1) # 6.1137e-1 (2.11e-1) # 1.1737e-1 (1.62e-1) 1.6267e-1 (2.56e-1) 3.5746e-1 (1.78e-1)
MaF7 10 1.4115e+0 (8.58e-2) # 2.6086e+0 (6.41e-2) # 1.1407e+0 (7.72e-2) # 2.4215e+0 (5.00e-1) # 9.0120e-1 (4.96e-2) # 8.4737e-1 (9.93e-3)
MaF9 10 1.5808e-1 (7.75e-3) 1.4224e+0 (5.86e-2) # 4.4905e-1 (1.09e-1) # 1.3035e+0 (1.61e+0) # 7.8762e-1 (1.46e-1) # 2.9156e-1 (9.70e-3)
MaF10 10 1.5107e+0 (1.01e-1) # 1.1179e+0 (4.06e-2) # 1.6448e+0 (1.35e-1) # 1.9093e+0 (1.39e-1) # 1.1739e+0 (8.54e-2) # 9.2879e-1 (1.74e-2)
MaF11 10 9.9968e-1 (3.65e-2) # 1.1312e+0 (3.55e-2) # 1.2069e+0 (1.97e-1) # 1.9329e+0 (4.15e-2) # 1.1605e+0 (1.44e-1) # 9.7470e-1 (1.28e-2)
MaF12 10 4.4564e+0 (2.82e-2) # 4.1458e+0 (4.01e-2) # 4.3421e+0 (7.55e-2) # 8.9572e+0 (1.64e-1) # 4.3330e+0 (3.25e-2) # 3.9226e+0 (2.99e-2)
MaF15 10 8.9098e-1 (1.31e-1) 1.3888e+0 (4.96e-1) ⇡ 1.3848e+0 (2.28e-1) # 1.0341e+0 (9.57e-2) ⇡ 1.2753e+0 (1.41e-1) # 1.0738e+0 (1.46e-1)
f objectives. More specically, AGE-MOEA achieves signicantly
etter IGD values in 12 problems (out of 15) for M=3, in 14 problems
orm M=5, and in 12 problems for M=10. The largest dierence
etween the two MOEAs is observed for the Maf6 test problem
with M=5. The optimal Pareto front for Maf6 is concave (i.e., it has
Concerning -DEA, we observe that AGE-MOEA outperforms -DEA
in 13 out of 15 test problems for M=3 and M=15; and in 14 out of 15
test problems for M=10. Independently of the number of objectives
M, there are six test problems for which the IGD values achieved
by AGE-MOEA are at least one order of magnitude smaller than the

AGE-MOEA Vs. GFM-MOEA
!24
Hyperbolic Shifted
Degenerate Irregular
1
100
MaF10 MaF3 MaF4 MaF8
Problems
IGD
Algorithms
AGEMOEA
GFMMOEA
True PF for M=3
IGD values M=10, 50 runs
Pop size = 275, #Gen = 300

AGE-MOEA Vs. GFM-MOEA
!25
100
Problems
IGD
Algorithms
AGEMOEA
GFMMOEA
Running time (in s), 50 runs
10
1
100
Problems
IGD
Algorithms
AGEMOEA
GFMMOEA
IGD values M=10, 50 runs
Pop size = 275, #Gen = 300

AGE-MOEA
!26
10
https://github.com/apanichella/PlatEMO

An Adaptive Evolutionary Algorithm based on Non-Euclidean Geometry for Many-Objective Optimization

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