SUPERNOVA IMPROVEMENT FOR PLANAR LANDSCAPE AND NEEDLE MINIMUMS
Eddy Mesa, Juan David Velásquez.
Used nomenclature
Variable
𝑁
𝐱0
𝐱𝑘
𝐩𝑘
𝐬𝑘
𝑣𝑘
𝐝𝑘
𝐧𝑘𝑡
𝑟𝑖𝑗
𝐮𝑖𝑘
𝐡𝑘
𝐱∗
𝜆
𝛼
𝑇
𝜂
𝑅
𝑡, 𝑎, 𝑟
Type
Integer
Vector of 𝑁 × 1
Vector of 𝑁 × 1
Vector of 𝑁 × 1
Vector of 𝑁 × 1
Scalar, 𝑣𝑘 ≥ 0
Unitary vector of 𝑁 × 1
Vector of 𝑁 × 1
Scalar value
Unitary vector of 𝑁 × 1
Vector of 𝑁 × 1
Vector of 𝑁 × 1
Scalar
Scalar
Integer
Scalar (0, 1)
Integer
Integers
Meaning
Number of dimensions
Initial point
Current position of particle 𝑘
Previous position of particle 𝑘
Next position of the particle 𝑘
Velocity of the particle 𝑘
Current direction of the particle 𝑘
Best point visited by the particle 𝑘 until the iteration 𝑡
Distance between the particles 𝑖 and 𝑗
Direction between particles 𝑖 and 𝑘
Particle interaction for gravity
Local optimum for iteration 𝑡
Radius of the hyper-sphere containing the particles in the initial explosion
Factor for reducing the radius of explosion when the algorithm restarts
Maximum number of iterations
Reduction factor for restarts
Number of restarts
Algoritm’s counters
1
INTRODUCTION
The difficultly of real-world optimization problems has increased over time (there are shorter times to find solutions,
more variables, greater accuracy is required, etc.). Classic methods of optimization (based on derivatives, indirect
methods) are not enough to solve all optimization problems. So, direct strategies were developed, despite the fact
that they can’t guarantee an optimal solution. The first direct methods implemented were systematic searches over
solutions spaces. Step by step, they evolved to become more and more effective methods until to arrive to artificial
intelligence. Those new methods were called heuristics and metaheuristics, respectively (Himmelblau, 1972).
Metaheuristics is iterative and direct methods developed for solving certain types of optimization problems using
simple rules for search [3]. It is inspired in physical, natural or biological phenomena [6, 9]. Recently, gravity force
had inspired some metaheuristics; Its mean, for a given set of particles with mass, they will be attracted among
themselves in proportion to their mass and distance. If we assume the value of the objective function (or its inverse)
is the mass, we would have a search process capable to solve optimization problems in a similar way to hill-climbing
methods. In literature, we found the following algorithms inspired on gravity: gravitational local search (GLSA)
[10], big–bang big–crunch (BBBC) [4], gravitational search (GSA) [14] and central force optimization (CFO) [5].
These four methods differs each other. The first one is a local method while others are global. BBBC use integer
variables and calculate inertial mass point,. CFO is deterministic and GSA is random.
Following this idea, We proposed a novel metaherustic called Supernova (mesa, 2010). It is a new approach based
on the gravity analogy and inspired in the supernova phenomenon [8]. Our methodology shows those advantages:
 The method is able to find a better optimum in fewer calls to objective function for convex,quasi-convex
and even for a smoothness slopes and high rugosity functions.

It is robust for high dimensional problems, especially for functions described above.

It has a small set of easily tuned parameters.
By the way, supernova is a new metaheuristics that is still been developing. In this stage, the methodology is
immature and have weakness as: handle of constrains, unsuccessful performance in planar regions and needle
minima, control and tune of parameters, high computational costs, among others . We focus in one of them: The
performance over planar search regions and needle minima. Although there are many ways to improve the
metaheuristics, Our focus are how parameters selection and control could increase the effectiveness over planar
regions and initialization and scaled functions as tools to have more and better exploration with metaheuristic. So,
The main goal of this work is to find a method for Supernova to have a successful optimization over planar regions
with hard minimums (needles).
This paper is organized as follows. In Section 2, we present the developed strategy. In Section 3, we show the results
for a set of well-known benchmark functions and compare them with other three metaheuristics. In Section 4, we
discuss about Supernova results, and the weakness presented. In Section 5 some approach to improve other similar
metaheuristics are present. And finally, we conclude and propose future working in Section 6.
2
SUPERNOVA METHODOLOGY
This section is focused in how we develop a novel analogy for optimizing nonlinear functions inspired by Supernova
phenomenon. When a star exploits, many particles are ejected out from the star. Particles move until they reach a
steady state with the environment. This phenomena is very complex and it does not make any sense to try of simulate
the entire process with the aim of solving optimization problems [7].
In our approach, we simplify and idealize the real phenomena recreating, only, a simply dynamics to government the
particles’ interaction (expel and repel). The particles’ movement is planted as a strategy for exploring the search
space. Thus, only two forces are used in our heuristic: the attraction force generated by the gravity; and the impulse
force caused by the explosion of the star [12, 15, 17]. In order to exploit the best regions found, the strategy is
restarted with a smaller impulse; thus, the algorithm archives a combination of exploration and exploitation
necessary to global optimization [12]. In the supernovae phenomenon, the magnitudes are huger compared with
typical optimization applications, so, the constants for gravity and impulse are parameters chosen for the problem.
In our algorithm, we have 𝐾 particles. In Fig. 1, we analyze the displacement of the particle 𝑘 assuming that other
particles remain static. Black points in Fig. 1 represent the trajectory of the analyzed particle, while white points
represent the other particles.
Fig 1. Forces actuating over a single particle
In Fig. 1a, we show the displacements caused by the impulse force and attraction forces with particles 1 and 2. The
displacement 𝐢𝑘 of the particle 𝑘 caused by the impulse force is:
𝐢𝑘 = 𝐹 ∙ 𝑚𝑘 ∙ 𝑣𝑘 ∙ 𝐝𝑘
(1)
where 𝐹 is a constant parameter for the impulse force; 𝐝𝑘 is the current direction; 𝑚𝑘 is the mass and 𝑣𝑘 is the
velocity. The displacement caused by the attraction force between particles 𝑘 and 𝑖, 𝐚𝑘𝑖 , is calculated as (see Fig.
1a):
2 −1
) ∙ 𝐮𝑘𝑖
𝐚𝑘𝑖 = 𝐺 ∙ 𝑚𝑘 ∙ 𝑚𝑖 ∙ (𝑟𝑘𝑖
(2)
where 𝐺 denotes the gravitational constant, 𝑟𝑘𝑖 denotes the distance between particles 𝑘 and 𝑖, and 𝐮𝑘𝑖 denotes a
unitary vector in the same direction of particles 𝑘 and 𝑖. Thus, the net displacement 𝚫𝑘 is calculated taking into
count the contribution of the displacements caused by the attraction forces among all particles and the inertia:
𝐾
𝐾
𝚫𝑘 = 𝐢𝑘 + ∑ 𝐚𝑘𝑖 = 𝐹 ∙ 𝑚𝑘 ∙ 𝑣𝑘 ∙ 𝐝𝑘 + ∑
𝑖=1
𝑖≠𝑘
𝑖=1
𝑖≠𝑘
𝐺 ∙ 𝑚𝑘 ∙ 𝑚𝑖
∙ 𝐮𝑘𝑖
2
𝑟𝑘𝑖
(3)
Thus, the next position of the particle is calculated as the current position plus the new displacement, as show in Fig.
1b.
2.1
Algorithm description
The developed methodology is presented in the Algorithm 1. The nonlinear optimization problem is stated as 𝐱 𝐺 =
arg min 𝑓(𝐱), where 𝐱 is a vector of 𝑁 × 1 coordinates or components, and 𝑓(∙) is a bounded real-valued,
continuous and multimodal function, such that 𝑓: ℛ 𝑁 → ℛ; and 𝐱 𝐺 is a vector representing the global optimum, such
that 𝑓(𝐱 𝐺 ) ≤ 𝑓(𝐱) in a compact domain.
We have 𝐾 particles. The particle 𝑘 in the iteration 𝑡 has the following properties: a current position, 𝐱 𝑘 ; a previous
position, 𝐩𝑘 ; a next position, 𝐬𝑘 ; a mass,𝑚𝑘 ; a unitary direction, 𝐝𝑘 ; and a velocity, 𝑣𝑘 . In addition, 𝐧𝑘𝑡 denotes the
best point visited by the particle 𝑘 until the iteration 𝑡. Supernova iterates over 𝑡 = 1, … , 𝑇.
Initialization. Initial velocity, 𝑣𝑘 , and direction, 𝐝𝑘 , of each particle 𝑘 are random. For this, we use the function
runif(𝑛) (lines 3 and 4) which returns is a random number between zero and one for velocity and a random a vector
of 𝑛 components for 𝐝𝑘 .. For the initial iteration (𝑡 = 1), an initial point 𝐱 0 is required for generating the set of
expelled particles; the position of each particle 𝐬𝑘 is obtained in line 9; 𝑤ℎ𝑒𝑟𝑒 𝜆 is a scalar value representing the
radius of an hyper-sphere that contained the set of expelled particles. Thus, the star is located in 𝐱 0 when it
explodes.a
Function evaluation. In our methodology, the value of objective function evaluated in 𝐱 𝑘 is proportional to the mass
𝑚𝑘 of the particle in eq. (1)–(3) with 𝑚𝑘 > 0. Thus, for function minimization, it is necessary to use 𝑚𝑘−1 as the
mass of particle 𝑘, where:
𝑚𝑘 = 𝑓(𝐱 𝑘 ) + 1 − min 𝑓(𝐱 𝑘 )
(4)
1≤𝑘≤𝐾
Previous transformation guarantee positive values of 𝑚𝑘 . In line 18, we use 𝑓 ∗ to denote the minimum of 𝑓(∙) which
is calculated in the line 12.
Calculation of displacement. In this step, we calculate the total displacement for each particle using eq. (3). In line
20, 𝐝𝑘 is the unitary vector in direction from the previous position to the current position of the particle 𝑘. In line 21,
the unitary vector in the direction of points 𝑖 and 𝑘, 𝐮𝑖𝑘 , is calculated. In line 22, we calculate the net displacement
caused for all attraction forces (see eq. (2)); however, the accurated use of eq. (2) is not possible due to the
magnitude of the values used for measuring masses and distances in the real phenomenon are not comparable with
2 −1 2 −1
) (𝑟𝑘𝑖 ) in eq. (2) for a new expression 𝐚𝑘𝑖 in eq. (5) to guarantee a
optimization problems. Thus, we replace (𝑟𝑖𝑘
bigger influence for near particles and lower influence for high distances between particles as in the gravity fields,
without the problem of division by zero for close particles:
𝐾
−1
𝐚𝑘𝑖 = 𝐺 ∙ 𝑚𝑘 ∙ 𝑚𝑖 ∙ [1 − 𝑟𝑖𝑘 ( ∑ 𝑟𝑖𝑗 ) ] ∙ 𝐮𝑘𝑖
𝑗=1|𝑗≠𝑖
(5)
Algorithm 1. Supernova.
01:
02:
03:
𝑡 ← 1; 𝑟 ← 0; 𝜆 ← 𝜆0
loop while 𝑟 ≤ 𝑅
if 𝑡 ≠ 1 then, 𝐱 0 ← arg min 𝑓(𝐧𝑘 ), end if
1≤𝑘≤𝐾
04:
05:
06:
07:
08:
09:
10:
11:
12:
for 𝑘 =1 to 𝐾 do
𝑣𝑘 ← runif(1)
𝐝𝑘 ← 2 ∙ runif(𝑁) − 1
𝐝𝑘 ← 𝐝𝑘 /‖𝐝𝑘 ‖
𝐱𝑘 ← 𝐱0
𝐬𝑘 ← 𝐱 𝑘 + 𝜆 ∙ 𝑣𝑘 ∙ 𝐝𝑘
if 𝑡 = 1 then, 𝐧𝑘 ← arg min{𝑓(𝐱 0 ), 𝑓(𝐬𝑘 )}; end if
end for
𝑓 ∗ ← min 𝑓(𝐬𝑘 )
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
𝑎←0
loop while 𝑡 ≤ 𝑇
for 𝑘 = 1 to 𝐾 do: 𝐩𝑘 ← 𝐱 𝑘 ; 𝐱 𝑘 ← 𝐬𝑘 ; end for
for 𝑖 = 1 to 𝐾 do: for 𝑗 = 1 to 𝐾 do: 𝑟𝑖𝑗 ← ‖𝐱 𝑖 − 𝐱𝑗 ‖; end for; end for
for 𝑘 = 1 to 𝐾 do:
𝑚𝑘 ← 𝑓(𝐱 𝑘 ) + 1 − 𝑓 ∗
𝐧𝑘 ← arg min {𝑓(𝐧𝑘 ), 𝑓(𝐩𝑘 ), 𝑓(𝐱 𝑘 )}
𝐝𝑘 ← [𝐩𝑘 − 𝐱 𝑘 ]⁄‖𝐩𝑘 − 𝐱 𝑘 ‖
for 𝑖 = 1 to 𝐾 do: 𝐮𝑖𝑘 ← [𝐱 𝑖 − 𝐱 𝑘 ]⁄‖𝐱 𝑖 − 𝐱 𝑘 ‖; end for
−1
−1
𝐾
−1
𝐡 𝑘 ← ∑𝐾
𝑖=1|𝑖≠𝑘 𝐺 ∙ 𝑚𝑘 ∙ 𝑚𝑖 ∙ [1 − 𝑟𝑖𝑘 ∙ (∑𝑗=1|𝑗≠𝑖 𝑟𝑗𝑘 ) ] ∙ 𝐮𝑖𝑘
23:
24:
25:
26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:
38:
39:
40:
1≤𝑘≤𝐾
𝐬𝑘 ← 𝐱 𝑘 + m𝑘−1 𝑣𝑘 ∙ 𝐹 ∙ 𝐝𝑘 + 𝐡𝑘
end for
𝐱 ∗ ← arg min 𝑓(𝐬𝑘 ); 𝑓 ∗ ← 𝑓(𝐱 ∗ )
1≤𝑘≤𝐾
if 𝑡 = 1 or 𝑓 ∗ ≤ 𝑓(𝐱 𝐺 ) then
𝐱𝐺 ← 𝐱∗
𝑎←0
else
𝑎 ← 𝑎+1
end if
if 𝑎 > 𝜂 ∙ 𝑇 then
𝑡 ←𝑇+1
end if
𝑡 ←𝑡+1
end loop
𝑡 ←1,𝑟 ←𝑟+1
𝜆 ← 𝜆 ⁄𝛼
end loop
end algorithm
Algorithm restart. In heuristic methods, it is well-known that the initial exploration usually is insufficient for certain
multimodal regions because there are no enough focus to find the best point. When algorithm restarts, a search in a
smaller region around the best point found begins, improving the accuracy. When the algorithm reaches 𝜂 ∙ 𝑇
iterations without improving the current optimum (see lines 28 and 30), the algorithm is restarted (line 33), and a
new set of particles is created. For this, the radius of the hyper-sphere containing the initial particles is reduced (line
38); following, the best point found until the current iteration (lines10 and 19) is used as the initial point (line 03),
and a new explosion occurs (lines 4 to 10).
Stop criterion. The process finishes when the maximum number of iterations 𝑇 or restarts 𝑅 are reached (lines 2 and
14) [11].
3
TESTING
After implementation, we started to test supernova. Test had three parts. Initially, parameters were evaluated. In the
second part, some functions were rotated and move to obtain different problem’s configuration to check the
robustness. Finally, Supernova was validated against another metaheuristics. This section is divided in four parts:
Recommended values, performance for specific functions configurations, description of benchmark functions and
results.
3.1
Recommended values for the parameters
The first step of testing was to search adequate general values for the parameters of the proposed algorithm. For this,
we optimize the set of benchmark functions proposed by De Jong [2] varying each parameter. The obtained results
were:
 Number of iterations (𝑇): 100.

Number of restarts (R): 50.

Number of particles (𝐾): 60–100

Momentum parameter (𝐹): 2

Attraction parameter (G): 2

Initial radius of the hyper-sphere for generating the particles (𝜆0 ): 200

Factor for reducing the radius of the hyper-sphere in each restart (𝛼): 1.5
Complete results are reported in [8]. Those parameters are used for the complete test of Supernova, except 𝜆0 for
small search spaces, less 30 units, or huge spaces, over 200 units. For small regions, we use 𝜆0 = 30; for big regions,
we use 𝜆0 = 600.
3.2
Supernova performances over special regions
3.3
Experimental setup
Supernova has been compared with Differential Evolution (DE) [1], Adaptive Differential Evolution (ADE) [1] and
Fast Evolutionary Programming (FEP) [16] using 23 benchmark functions taken from [1] and [16]. In Table 1, the
equation of each function, the range of the search space, the number of dimensions (𝑁) and the optimal value for
each function are presented.
Table 2. Benchmark functions
𝑓
Equation
𝑓(𝐱 𝑜𝑝𝑡 )
2
𝑓1 = ∑𝑁
𝑖=1 𝑥𝑖
0.0
𝑁
𝑓2 = ∑𝑁
𝑖=1|𝑥𝑖 | + ∏𝑖=1|𝑥𝑖 |
0.0
2
𝑖
𝑓3 = ∑𝑁
𝑖=1(∑𝑗=1 𝑥𝑗 )
0.0
𝑓4
= max{|𝑥𝑖 |, 1 ≤ 𝑖 ≤ 𝑁}
0.0
2 2
2
𝑓5
= ∑𝑁−1
𝑖=1 [100(𝑥𝑖+1 − 𝑥𝑖 ) + (𝑥𝑖 − 1) ]
0.0
2
𝑓6
= ∑𝑁
𝑖=1(⌊𝑥𝑖 + 0.5⌋)
0.0
4
𝑓7
= ∑𝑁
𝑖=1 𝑖𝑥𝑖 + 𝑟𝑎𝑛𝑑𝑜𝑚[0,1)
0.0
𝑓8
= − ∑𝑁
𝑖=1(𝑥𝑖 sin(√|𝑥𝑖 |))
−12569.5
2
𝑓9
= ∑𝑁−1
𝑖=1 [𝑥𝑖 − 10 cos(2𝜋𝑥𝑖 + 10)]
0.0
𝑓10
1
1
𝑁
𝑁
𝑁
2
1
= −20 exp (−0.2√ ∑𝑁
𝑖=30 𝑥𝑖 ) − exp ( ∑𝑖=1 cos(2𝜋𝑥1 )) + 20 + e
Range
𝑁
[−500, 500]𝑁
30
[−10, 10]𝑁
30
[−100, 100]𝑁
30
[−100, 100]𝑁
30
[−30, 30]𝑁
30
[−100, 100]𝑁
30
[−1.28, 1.28]𝑁
30
[−500, 500]𝑁
30
[−5.12,5.12]𝑁
30
[−32,32]𝑁
30
[−600, 600]𝑁
30
0.0
𝑓11
𝑓12
𝑓13
= ∑𝑁
𝑖=1
𝑥𝑖2
4000
𝑥
𝑖
− ∏𝑁
𝑖=1 (cos ) + 1
√𝑖
0.0
π
2
[−50, 50]𝑁
= {10 sin2 (𝜋𝑦1 ) + ∑𝑁−1
𝑖=1 (𝑦𝑖 − 1) ∙
𝑁
0.0
[1 + 10 sin2 (𝜋𝑦𝑖+1 + 1 )] + (𝑦𝑛 − 1)2 } + ∑30
𝑖=1 𝑢(𝑥𝑖 , 10,100,4)
2
2
[−50, 50]𝑁
= 0.1{sin2 (𝜋3𝑥1 ) + ∑𝑁−1
𝑖=1 (𝑥𝑖 − 1) [1 + sin (3𝜋𝑥𝑖+1 )]
0.0
+(𝑥𝑛 − 1)2 [1 + sin2 2𝜋𝑥𝑁 ]} + ∑𝑁
𝑖=1 𝑢(𝑥𝑖 , 5,100,4)
𝑘(𝑥𝑖 − 𝑎)𝑚 , 𝑥𝑖 > 𝑎
1
0,
−𝑎 ≤ 𝑥𝑖 ≤ 𝑎 and 𝑦𝑖 = 1 + (𝑥𝑖 + 1)
𝑢(𝑥𝑖 , 𝑎, 𝑘, 𝑚) = {
4
𝑚
𝑘(−𝑥𝑖 − 𝑎) , 𝑥𝑖 < −𝑎
30
30
−1
𝑓14
=[
1
500
+
∑25
𝑗=1
1
6
𝑗+∑2
𝑖=1(𝑥𝑖 −𝑎𝑖𝑗 )
]
[−65.536, 65.536]𝑁
2
[−5, 5]𝑁
4
[−5, 5]𝑁
2
[−5,10][0,15]
2
0.0
𝑥1 (𝑏𝑖2 +𝑏𝑖 𝑥2 )
2
𝑓15
= ∑11
𝑖=1 [a i −
𝑓16
3.075 × 104
1
= 4x12 − 2.1𝑥14 + 𝑥16 + 𝑥1 𝑥2 − 4𝑥22 + 4𝑥24
3
−1.031628
𝑓17
= (x2 −
0.398
𝑏𝑖2 +𝑏𝑖 𝑥3 +𝑥4
5.1 2
𝑥
4𝜋2 1
5
]
2
+ 𝑥1 − 6) + 10 (1 −
𝜋
1
8𝜋
) cos 𝑥1 + 10
𝑓19
[−2, 2]𝑁
= [1 + (𝑥1 + 𝑥2 + 1)2 (19 − 14𝑥1 + 3𝑥12 − 14𝑥2 +6𝑥1 𝑥2 + 3𝑥22 )]
3
× [30 + (2𝑥1 − 3𝑥2 )2 × (18 − 32𝑥1 + 12𝑥12 + 48𝑥2 − 36𝑥1 𝑥2 + 27𝑥22 )]
2
[0, 1]𝑁
= − ∑4𝑖=1 𝑐𝑖 exp [− ∑𝑁
𝑗=1 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ]
𝑓20
−3.86
2
= − ∑4𝑖=1 𝑐𝑖 exp [− ∑𝑁
𝑗=1 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ]
𝑓18
𝑓21
𝑓22
𝑓23
−3.32
= ∑5𝑖=1[(x − a i )(𝑥 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1
0.0
= ∑7𝑖=1[(x − a i )(𝑥 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1
0.0
𝑇
−1
= ∑10
𝑖=1[(x − a i )(𝑥 − 𝑎𝑖 ) + 𝑐𝑖 ]
0.0
2
3
[0, 1]𝑁
6
[0, 10]𝑁
4
[0, 10]𝑁
4
[0, 10]𝑁
4
The characteristics of test set are important to know the metaheuristic. How it works agree to dimensionality,
convexity, modality and if the function are separable or not. In table 2 are describe the characteristics o this
functions. It is important to note in the Table, all characteristics are defined for the range of the function and we
declare quasi-convex functions as convex because the global behavior.
Table 2. Functions characteristics
Función Número de
dimensiones
𝑓1
30
𝑓2
30
𝑓3
30
𝑓4
30
𝑓5
30
𝑓6
30
𝑓7
30
𝑓8
30
𝑓9
30
𝑓10
30
𝑓11
30
𝑓12
30
𝑓13
30
𝑓14
2
𝑓15
4
𝑓16
2
𝑓17
2
𝑓18
2
𝑓19
3
𝑓20
6
𝑓21
4
𝑓22
4
Convexty
diferenciable
modalidad
Separabilidad
convex
convex
diferenfiable
No diferenciable
convexa
No diferenciable
diferenciable
No diferenciable
No convexa
No diferenciable
unimodal
unimodal
unimodal
unimodal
multimodal
unimodal
unimodal
Multimodal
multimodal
multimodal
multimodal
multimodal
multimodal
Multimodal
multimodal
multimodal
multimodal
multimodal
multimodal
multimodal
multimodal
multimodal
separable
Separable
No separable
separable
No separable
separable
Separable
Separable
separable
No separable
No separable
No separable
No separable
convexa
convexa
convexa
diferenciable
𝑓23
4
multimodal
̅̅̅∗ ), the standard deviation (𝜎)
For each benchmark function, we run Supernova 50 times and calculate the average (𝑓
of the best function value found in each run and the number of function calls (FO). The obtained results are reported
in Table 3.
[coello,06, DE],[a note on the extended rosenbrock function][CEC’05]
3.3.1
Results
For the 23 benchmark functions, the algorithm improve the answers (see Table 1) and have less calls to objective
function in 26% of the functions. Presented less calls to objective function in 22% of the functions for similar
minimums; Supernova presents a similar behavior in 30% of the test functions. The best behavior of Supernova is
observed for functions with a kind of descend geometry. For plane regions, Supernova shows the worst performance.
In [10], the GSA method is validated with the same test of functions but with different parameter values, such the
number of calls to the function objective is not similar and uses a smaller population to search too. Therefore, the
results presented in [10] are not comparable because the test conditions are not similar.
Table 3. Results and comparison for benchmark functions.
f
Supernova
____________________________
f∗
σ
6,3 x 10
—29
1,0 x 10
—28
1,1 x 10
—26
F.O
f1
972
f2
1812
1,1 x 10
—26
f3
852
1,0 x 10
—13
8,8 x 10
—14
f4
1212
6,1 x 10
—35
1,7 x 10
—34
2,5 x 10
—2
f5
510
28.89
f6
677
0
0
—3
—3
f7
f8
552
612
3,2 x 10
—7,4 x 10
15
f9
612
f10
972
f11
612
f12
100
f13
250
1.09
f14
100
1,9 x 10
4,28 x 10
0
4,7 x 10
—15
0
2,3 x 10
—15
0
0
1,1 x 10
1,0 x 10
5,7 x 10
1,3 x 10
2000
1,0 x 10
—24
9,7 x 10
—24
5000
3,1 x 10
—14
5,9 x 10
—14
1500
8,2 x 10
5,9 x 10
—9
9,9 x 10
—10
8,1 x 10—3
7,7 x 10
—4
—11
7,4 x 10
—11
1,6 x 10
—2
1,4 x 10
—2
1,5 x 10
6,8 x 10
5000
0
0
0
0
0.3
0.5
20000
0
0
0
0
5.06
5.87
1500
0
0
0
0
0
0
—3
—4
—3
—3
0.1415
0.3522
-12554.5
52.6
3000
9000
3,15 x 10
-12569.5
5000
1500
7,5 x 10
7,0 x 10
—12
0
7,7 x 10
—15
2000
0
1,4 x 10
—15
0
4,63 x 10
-11080.1
69.2
9,7 x 10
0
—8
1,2 x 10
574.7
38.8
4,2 x 10
0
—8
0
4,6 x 10
—2
1,2 x 10
—2
1,8 x 10
—2
2,1 x 10
—3
1,6 x 10
—2
2,2 x 10
—2
—15
8,0 x 10
—15
9,2 x 10
—6
3,6 x 10
—6
0.55
1500
5,0 x 10
—29
3,9 x 10
—29
5,1 x 10
—14
4,8 x 10
—14
1,6 x 10
—4
7,3 x 10
—5
1.39
1.2
100
2,6 x 10
—16
3,3 x 10
—16
—4
4,9 x 10
—5
4,4 x 10
-1.0315
1,36 x 10
—4
f17
204
0.39884
1,2 x 10
—3
f18
204
3.0024
2,9 x 10
—3
f19
50
-3.862
7,4 x 10
—5
f20
60
-3.26
3,68 x 10
—2
f21
510
2,17 x 10
—1
2,5 x 10
—1
3,5 x 10
—1
4
σ
—4
7,9 x 10
204
510
f*
—4
—30
f16
f23
σ
—14
7,9 x 10
404
510
f∗
—14
—30
f15
f22
σ
—28
6,6 x 10
2,5 x 10
—2
f∗
—28
F.O
1500
2,9 x 10
—2
15
ADE
DE
FEP
_____________________________ ______________________ ___________________
-8,73 x 10
—1
—8,8 x 10
—1
-1.08
4000
0.998004
4,4 x 10
—4
2,7 x 10
9,7 x 10
—4
—12
0.998004
4,5 x 10
—4
3,1 x 10
—13
1.22
0.56
3,2 x 10
—4
-1.03
4,9 x 10
—7
0.398
1,5 x 10
—7
5,0 x 10
—4
100
-1.03163
100
0.397887
100
3
100
N. A.
N. A.
N. A.
N. A.
-3.86
1,4 x 10
—5
200
N. A.
N. A.
N. A.
N. A.
-3.27
5,9 x 10
—2
100
-10.1532
100
100
-10.4029
-10.5364
2,3 x 10
1,7 x 10
—8
-1.03163
3,3 x 10
—4
0.397887
—15
2,2 x 10
—6
4,9 x 10
—7
5,8 x 10
—6
3
9,9 x 10
2,0 x 10
—9
—15
2,5 x 10
—6
-10.4029
3,9 x 10
—7
-10.5364
1,9 x 10—7
-10.1532
DISCUSSION
Agree with the descriptions given, for the functions, on section 4.3. We can conclude that:
 Planar regions and penalties (f5,f12,f13,f16,f17,18)
 Needle minimums (f5,f14,f21,f23,f24)
Are problematic topologies for Supenova.
3.02
0.11
-5.52
1.59
-5.52
2.12
-6.57
3.14
5. HOW HAD BEEN OTHER METAHEURISTICS IMPROVED?
Generally, there is not a specific landscape associated to specific improve over performance. Regularly, an improved
is present numerically for a set of functions. Follow the functions with low performance, we found a improvement of
the performance for rosenbrock function (f5) with particle swarm optimization (PSO). And have a second idea to
improve the performance having better control of the parameters. Both ideas will be exposed below in this section.
5.1. PSO with LDS
5.2. Parameters
Stochastic methods have not the same pathway each run of algorithms. Also, the intelligence immerse in
metaheuristics change dynamically from the characteristics found in the search. Parameters give to metaheuristics
robustness and flexibility (Talbi, 2009).
To choose parameters is not a simple task because a wrong set of parameters would make that metaheuristics have
premature convergence or even no convergence at all (Bäck, 1996; Talbi, 2009).
Metaheuristics need to have enough parameters to be flexible, but each parameter increase the complexity to the tune
and control of the method and the necessary changes to optimize a new problem. Each metaheuristic is a complete
different world, has different parameters, and they influence the metaheuristic in different ways. There is not a
unique right to choice parameters in metaheuristics. In literature two points of view are related, but they are only for
Evolutionary Computing (EC).
Traditionally, EC includes genetic algorithms (GA), evolution strategies (ES), and evolutionary programming (EP).
By the way, other metaheuristics met the conditions propose by Angeline (1995) to be an EC metaheuristic, method
that iterated until specific stopping criteria. So, we would extend this classification to another metaheuristics and
heuristics in general, but they do not use the same operators that EC ( Fogel, D. B., et al, 1991). Specifically, most
part of another heuristics has not operators formalized as operators, they use a complete rules with parameters inside,
so maybe it is not a direct analogy (Zapfel, et al., 2010). In the next two sections the formal parameter classification
proposed in literature was discussed and follow an example of other heuristic adaptation are presented.
5.2.1. PARAMETERS CLASSIFICATION
Each metaheuristic has a parameter’s set to control the search. Although there is no consensus about a unique
classification to metaheuristic parameters, one approximation had been developed for EC (Angeline, 1995;
Hinterding,et al., 1997; Kramer, 2008). According with the definition provided by Angeline (1995) this classification
could be extended to others paradigms.
The most complete classification is proposed by Kramer (2008) who links Angeline (1995) point of view and
Hinterding et al. (1997)
5.2.1.1 Exogenous.
These parameters are those whose are affected by metaheuristic performance, but they are external to it, for example:
constrain changes, problems with parts functions.
5.2.1.2 Endogenous
These are internal to the method and could be change by the user or method itself. Even though, Endogenous
parameters are our focus, we cannot forget the exogenous ones because they will affect their selection
Population level: In this level the parameters are global for the optimization. Operator of EC that use this type or
parameters control the next generation. For example: population size, stopping criteria, etc.
Individual level .This kind of parameters only affects each individual, for example: It could be the pass for each
individual.
Component level. In this level, the parameters affect part of the individual like a gene of a chromosome in Generic
Algorithms (GA).
Also, it is important to notice that, authors propose this classification, but they do not talk about a right or unique
manner to adapt and automatize each level of parameters. The techniques are addressed by next classification.
5.2.2. PARAMETERS SETTINGS
In last section, parameters were changed according to their level of sort in the metaheuristic. In this case, the
parameters are chosen in two stages the first is before to use metaheuristic, and it is called “to tune” and the second
one is called “to control” and both have different ways to be select.
5.2.2.1. Tuning.
Parameters are tuned before to use metaheuristic. Those initial parameters could be chosen by three different levels
accord Eiben, et al (1999).
Manually: The initial parameters could be chosen by and expert ad hoc it is a right manner but lately are not the most
recommended (Kramer, 2008; Talbi, 2009)
Design of experiments (DOE): It implies design test to show the behavior, and use or define a metric analyze the
result and take a decision about a value or a range of values for each parameter. (Kramer, 2008; Talbi, 2009).
Metaevolutionary: It means that parameters are chosen by other metaheuristics, in literature a good example is given
by bacterial chemotaxis method proposed in (Müller, et al., 2000). Additionally, Kramer (2008) extend this
procedure to control like in Hyper-metaheuristics (Hamadi, et al., 2011)
5.2.2.2. Control
Parameters can change while metaheuristic is running. Control process is this change in “real time”. Accord to
Hinterding et al. (1997) and Eiden et al (1999), Kramer (2008) describes and discusses about three methods to
control parameters.
Deterministic: It could be static or dynamics. Statics means there is no change at all and dynamic it change with a
specific rule like a dynamic penalty than change with the distance to feasible zone (Kramer, 2008; Mesa, 2010)
Adaptive: In this case, parameter changes agree a rule like if it happens, then do this. A classic example of adaptive
parameters is the 1/5th rule for mutation use in ES (Bäck, 1996; Kramer, 2008).
Self-adaptive: In this type of control, parameters evolves agree to the problem for example the self-adaptation of the
mutation step-size with the direction to mutation (Kramer, 2008; Schwefel, 1995)
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