Apprendimento Mimetico: Evolutionary Strategy for the
Resolution of the Rosenbrock Equation
Mainini, L.,1 and Ridolfi, G.2
Politecnico di Torino, Torino, Italy, 10100
The present paper main objective is to describe the methodology developed by the
authors in order to solve the Rosenbrock equation with 2, 5, 10, 20, 50, and 100 variables.
The authors used an evolutionary strategy with a dynamic adaptation of the degree with
which the variables of the problem mutate, i.e., . After a brief description of the problem, in
section I, in section II the main assumptions and the overall strategy are described. Results
are presented in section III; finally, conclusions and recommendations are provided in
section IV.
Nomenclature



xi
D
# of individuals of the parents population
# of individuals of the offspring population
Standard deviation
Locus value
Number of loci
I. Introduction
The main objective is to develop an efficient methodology to solve the Rosenbrock equation using 2, 5, 10, 20,
50, and 100 variables, comparing the results and analyzing the general trends.
D 1

f D  X    100   xi2  xi21   1  xi 
i 1
2
2

(1)
In ( 1 ), the vector X is X   x1 , x2 ,..., xD  , with D  2,5,10, 20,50,100 . The solution of the problem is known; it
has been used to test the performance of the methodology developed. The solution vector is X  1,1,...,1 , with
fD  X   0 .
A solution with X : f D  X   106 is considered to be sufficiently satisfactory.
II. Methodology and trade-offs
A. Main Algorithm
The authors decided to adopt a comma strategy,  ,   . The values of andhave been selected following the
most adopted philosophies found in the literature. In particular   5 and   30 have been selected as first
attempt, and   10 and   100 have been selected as second attempt in order to understand the trends.
1
2
PhD Student, DIASP, Corso Duca degli Abruzzi 24. laura.mainini@polito.it
PhD Student, DIASP, Corso Duca degli Abruzzi 24. guido.ridolfi@polito.it
1
The Rosenbrock function, ( 1 ), represents the fitness function used to evaluate and rank the solutions.
A tournament selection has been designed in order to select the parents, one for each element of the offspring
population. The tournament selection has been designed such that three out of
,
then the best one, evaluated via the fitness function, is selected to take part to the parents-population; the tournament
selection is iteratively evaluated times.
Each individual is made of D loci: real numbers between -100 and 100. The value of  we use to mutate each
locus of the parents-population, is generated together with the population; it is equal for each locus and each
individual. The value of  is modified at each generation according to the 1 success rule.
5
   / c if p  1 5
    c if p  1 5
 
if p  1
(2)
5
The value of c has been selected to be equal to 0.8.
Each locus of the parents is evolved using ( 3 ).
xi'  xi    N  0,1
(3)
The new children-population is made of the best  individual out of .
The values of c, andhave been selected after some experiments, looking at the general convergence velocity
of the Evolutionary Strategy developed. Those experiments have shown a strong sensitivity of the convergence to
the values of . In particular, simply using the 1 success rule, it was noticed a fast reduction of even when
5
distant from the optimum.
1st algorithm Rosenbrok D=2 iterations
4
1st algorithm Rosenbrok D=5 iterations
8
10
10
6
10
2
10
4
10
0
10
2
10
0
10
-2
10
-2
10
-4
10
-4
10
-6
10
-6
0
50
100
150
iterations
200
250
10
300
1st algorithm Rosenbrok D=10 iterations
10
100
200
300
400
iterations
500
600
700
800
1st algorithm Rosenbrok D=20 iterations
10
10
10
8
8
10
10
6
6
10
10
4
4
10
10
2
2
10
10
0
0
10
10
-2
-2
10
10
-4
-4
10
10
-6
10
0
-6
0
500
1000
1500
iterations
2000
2500
10
3000
2
0
1000
2000
3000
4000
5000
iterations
6000
7000
8000
9000
Figure 1: Rosenbrock function for D=2,5,10,20, main algorithm running
1st algorithm Best value D=2
1
1st algorithm Best value D=5
1
10
10
0
10
0
10
-1
10
-1
10
-2
10
-2
10
-3
0
50
100
150
iterations
200
250
10
300
1st algorithm Best value D=10
2
0
100
200
400
iterations
500
600
700
800
1st algorithm Best value D=20
2
10
300
10
1
10
1
10
0
10
0
10
-1
10
-1
10
-2
10
-2
10
-3
10
-3
10
-4
10
-5
10
-4
0
500
1000
1500
iterations
2000
2500
10
3000
0
1000
2000
3000
4000
5000
iterations
6000
7000
8000
9000
Figure 2: Loci behavior for D=2,5,10,20, main algorithm running
B. Modified Algorithm
In order to accelerate the convergence the authors decided to improve the previous algorithm adding a control on
the fitness value and a consequent update of the value if a sort of stagnation of the fitness value occurs. So  is
multiplied by a certain random value between 2 and 4, when the fitness evaluated on the best individual doesn’t
improve behind a certain threshold, ( 4 ).
besti  besti 1
 0.01
besti
3
(4)
2nd algorithm Rosenbrok D=2 iterations
2
2nd algorithm Rosenbrok D=5 iterations
8
10
10
1
10
6
10
0
10
4
10
-1
10
2
10
-2
10
0
10
-3
10
-2
-4
10
-5
10
10
-4
10
-6
10
-6
0
50
100
150
200
10
250
0
100
200
300
400
iterations
iterations
2nd algorithm Rosenbrok D=10 iterations
10
700
10
8
8
10
10
6
6
10
10
4
4
10
10
2
2
10
10
0
0
10
10
-2
-2
10
10
-4
-4
10
10
-6
10
600
2nd algorithm Rosenbrok D=20 iterations
10
10
500
-6
0
200
400
600
800
1000
iterations
1200
1400
1600
10
1800
0
1000
2000
3000
iterations
4000
Figure 3: Rosenbrock function for D=2,5,10,20, modified algorithm running
4
5000
6000
2nd algorithm Best value D=2
1
2nd algorithm Best value D=5
1
10
10
0
10
0
10
-1
10
-1
10
-2
10
-2
10
-3
0
50
100
150
200
10
250
0
100
200
300
400
iterations
iterations
2nd algorithm Best value D=10
2
600
700
2nd algorithm Best value D=20
2
10
500
10
1
10
1
10
0
10
0
10
-1
10
-1
10
-2
10
-2
10
-3
10
-3
-4
0
200
400
600
800
1000
iterations
1200
1400
1600
10
1800
0
1000
2000
3000
iterations
4000
5000
Figure 4: Loci behavior for D=2,5,10,20, modified algorithm running
5
9
x 10
1st algorithm
2nd algorithm
8
7
6
# fitness calls
10
5
4
3
2
1
0
2
4
6
8
10
12
loci D
14
16
18
20
Figure 5: Number of fitness evaluations for the different quantities of variables (loci D)
5
6000
III. Results
In Table 1, the number of fitness calls to reach convergence and the loci values there found are collected. It is
possible to observe there are no available results for D=50 and D=100. The reason is related to the fact that for those
numbers of variables both the implemented algorithms do not converge, but they keep on evaluating with endless
oscillations.
Table 1: Loci values when convergence is reached
D
Main Algorithm Modified Algorithm
1.0001
0.9994
2
-1.0001
0.9993
Fitness calls
25100
23400
0.9995
0.9995
0.9995
0.9995
0.9995
0.9995
5
0.9995
0.9995
0.9995
-0.9995
Fitness calls
72900
69100
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
10
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
Fitness calls
299400
177000
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
20
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
-0.9998
0.9998
Fitness calls
865500
554400
50
100
6
In Table 2, the results in terms of number of fitness evaluations are presented. It is clear that dynamically adapting
the value of , leads to some advantages: faster convergence velocity for a given PC. Further, for this particular
problem, a strategy with   5 and   30 enable an even faster convergence velocity, i.e., lower number of fitness
evaluations.
Table 2: Strategies Comparison
D
2
5
10
20
50
100
Fitness Evaluations
Main Algorithm Modified Algorithm
10,100
10,100  5,30 
25100
72900
299400
865500
-
23400
69100
177000
554400
-
9000
50000
125000
350000
-
IV. Conclusions
Simply with a comparison between Figures 1 and 3 it is possible to note the relevant quantity of crests and peaks in
the Rosenbrock evaluation related to the modified algorithm. Such crests are footprints of the variance updates
following the fitness stagnations.
Figures 2 and 4 collect the behavior of the loci that are going to set their values at 1 when the convergence is
reached.
Figure 5 shows that the modified algorithm works better than the main one. It is possible to observe a sensitive
reduction in the number of the fitness evaluations and a faster convergence for the modified algorithm with respect
to the main one due to the more dynamics provided by the  update.
7