Una presentación de un algoritmo evolutivo para resolver problemas de optimización global que di en el Metaheuristics International Conference de 2011 en Udine, Italia.

1. Francisco Gortázar, Abraham Duarte
Dpto. de Ciencias de la Computación
Rafael Martí
Dpto. de Estadística e Investigación Operativa

2. Introduction
Path Relinking
Experimental results
Conclusions
2

3. Global optimization
problems
◦ Non linear function f(x)
◦ x={x1,…, xn}, l≤xi≤u
Minimize f ( x)
l ≤ x≤u
x ∈ℜ
3

4. Measuring success…
◦ Scalability of Evolutionary Algorithms and other
Metaheuristics for Large Scale Continuous
Optimization Problems, M. Lozano and F. Herrera
(Eds.), http://sci2s.ugr.es/eamhco/CFP.php
◦ 19 continuous functions
◦ Number of variables D={50,100,200,500,1000}
◦ Maximum number of evaluations: 5000D
The search space is continuous…
◦ Key issue: for which points will f(x) be evaluated?
4

5. Introduction
Path Relinking
Experimental results
Conclusions
5

6. 1 Generate
solutions
4
2
Select
solutions both
by quality and
diversity (EliteSet)
3
Perform a
path relinking
between each
pair of solutions
Return the best solution found so far
GRASP with PR for the MMDP
6

7. Path Relinking for Global Optimization (PR)
◦ EliteSet construction (x1,...,xb)
◦ Improve x1, and replace it with the improved
solution
◦ NewSubsets = (x,y), pair set of EliteSet solutions
◦ While NewSubsets ≠ ∅
Select next pair (x,y) from NewSubsets
Remove (x,y) from NewSubsets
Perform a path relinking with (x,y) →w
Improve w
If f(w) < f(x1) then x1 = w
7

8. EliteSet construction
◦ Generate diverse solutions
Sampling the solution space
◦ Based on ideas from factorial design of
experiments
Factorial design kn
n, #variables
k, #values
Genichi Taguchi
8

9. EliteSet construction
◦ Factorial design example for
n=4, k=3
81 experiments
◦ Fractional design
Taguchi Table L9
9 experiments
(34)
Exp.
Exp.
Factores
1
2
3
4
1
1
1
1
1
2
1
2
2
2
3
1
3
3
3
4
2
1
2
3
5
2
2
3
1
6
2
3
1
2
7
3
1
2
3
8
3
2
1
3
9
3
3
2
1
9

10. EliteSet construction
◦ Taguchi method is applied to obtain diverse
solutions
◦ #values k = 3
midvalue = l + ½(u-l)
lowervalue = l + ¼(u-l)
uppervalue = l + ¾(u-l)
◦ #variables n∈{50,100,200,500,1000}
10

11. EliteSet construction
◦ Problem: biggest Taguchi table found: n=40
11

12. EliteSet construction
x1
x2
x3
x4
x5
x6
x7
x8
X9
1
1
1
1
1
1
1
1
1
1
2
2
2
1
1
1
1
1
1
3
3
3
1
1
1
1
1
2
1
2
3
1
1
1
1
1
2
2
3
1
1
1
1
1
1
2
3
1
2
1
1
1
1
1
3
1
2
3
1
1
1
1
1
3
2
1
3
1
1
1
1
1
3
3
2
1
1
1
1
1
1
12

13. EliteSet construction
x1
x2
x3
x4
x5
x6
x7
x8
X9
1
1
1
1
2
2
2
2
2
1
2
2
2
2
2
2
2
2
1
3
3
3
2
2
2
2
2
2
1
2
3
2
2
2
2
2
2
2
3
1
2
2
2
2
2
2
3
1
2
2
2
2
2
2
3
1
2
3
2
2
2
2
2
3
2
1
3
2
2
2
2
2
3
3
2
1
2
2
2
2
2
13

14. EliteSet construction
x1
x2
x3
x4
x5
x6
x7
x8
X9
1
1
1
1
3
3
3
3
3
1
2
2
2
3
3
3
3
3
1
3
3
3
3
3
3
3
3
2
1
2
3
3
3
3
3
3
2
2
3
1
3
3
3
3
3
2
3
1
2
3
3
3
3
3
3
1
2
3
3
3
3
3
3
3
2
1
3
3
3
3
3
3
3
3
2
1
3
3
3
3
3
14

15. EliteSet construction
x1
x2
x3
x4
x5
x6
x7
x8
X9
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
1
1
1
1
1
1
3
3
3
1
1
1
1
1
2
1
2
3
1
1
1
1
1
2
2
3
1
1
1
1
1
1
2
3
1
2
1
1
1
1
1
3
1
2
3
1
1
1
1
1
3
2
1
3
1
1
1
1
1
3
3
2
1
1
1
1
15

16. EliteSet construction
x1
x2
x3
X4
x5
x6
x7
x8
X9
2
2
1
1
1
1
2
2
2
2
2
1
2
2
2
2
2
2
2
2
1
3
3
3
2
2
2
2
2
2
1
2
3
2
2
2
2
2
2
2
3
1
2
2
2
2
2
2
3
1
2
2
2
2
2
2
3
1
2
3
2
2
2
2
2
3
2
1
3
2
2
2
2
2
3
3
2
1
2
2
2
16

17. EliteSet construction
x1
x2
x3
x4
x5
x6
x7
x8
X9
3
3
1
1
1
1
3
3
3
3
3
1
2
2
2
3
3
3
3
3
1
3
3
3
3
3
3
3
3
2
1
2
3
3
3
3
3
3
2
2
3
1
3
3
3
3
3
2
3
1
2
3
3
3
3
3
3
1
2
3
3
3
3
3
3
3
2
1
3
3
3
3
3
3
3
3
2
1
3
3
3
17

18. EliteSet construction
◦ Taguchi table with 40 variables and 3 values
experiments
81
We generate 81 solutions by applying this table to the first
40 variables (x1,..,x40) and setting 1 to all other variables
We generate 81 solutions by applying this table to the first
40 variables (x1,..,x40) and setting 2 to all other variables
We generate 81 solutions by applying this table to the first
40 variables (x1,..,x40) and setting 3 to all other variables
◦ Totalizing 243 solutions
18

19. EliteSet construction
◦ We move the table in steps of size 20
We obtained better results
We generate 81 solutions by applying the table to the
variables x21, … ,x60 and value 1 to the remaining
variables
We generate 81 solutions by applying the table to the
variables x21, … ,x60 and value 2 to the remaining
variables
We generate 81 solutions by applying the table to the
variables x21, … ,x60 and value 3 to the remaining
variables
19

20. EliteSet construction
◦ # solutions generated with this process:
DSize=243⌈n/20⌉
◦ EliteSet is built choosing the best b solutions
20

21. Improvement is performed in two stages
1. Line search based on a single variable
Grid size h=(u-l)/100
2. Simplex method
Not limited to the grid
21

22. Improvement method: Line searches
◦ For each variable i
We evaluate x+hei and x-hei, discarding the worst
value
◦ We order the variables i={1, … ,n} according to
these values
x-hei x
x+hei
22

23. Improvement method: Line searches
1. Explore the first n/2 variables in the order that was
previously calculated:
For each variable i, evaluate solutions with the form
x+khei
k=[-20,20]
l ≤ x+khei ≤ u
HOW? Randomly & using a first-improvement approach
x-4hei
x-hei x x+hei
x+4hei
23

24. Improvement method: Line searches
1. Explore the first n/2 variables in the order that was
previously calculated:
For each variable i, evaluate solutions with the form
x+khei
k=[-20,20]
l ≤ x+khei ≤ u
HOW? Randomly & using a first-improvement approach
2. Re-evaluate the contribution of each variable, and reReorder
We explore again the first n/2 variables
3. Repeat it for 10 iterations, or until no further
improvement is possible
24

25. Improvement method: Simplex
Let x be the best solution found after the first stage
(line searches)
We perturb the value of each variable:
x=(x1, ... ,xi + α, ... ,xn)
α is generated by a uniform probability in [-1,1]
The simplex method is applied to these solutions
25

26. Relinking method
◦ Straight linking
◦ We perform the linking between three solutions
An initial solution, a
Two guiding solutions, x and y
26

27. Relinking method
◦ Straight linking
y2
y
x2
x
a(2k-1)
a(k+1)
a(k-1)
a(j)
a(1)
a2
a
a1
x1
y1
27

28. Relinking method
◦ We start at a=(a1,...,an)
◦ Firstly we evaluate solutions in the direction given
by the vector from a to x
1
a (1) = a + ( x − a )
k
1
a ( 2) = a +
( x − a)
k −1
...
1
a ( k − 1) = a + ( x − a )
2
28

29. Relinking method:
◦ The best solution is chosen from previous step, a(j)
◦ Then we evaluate solutions in the direction given by
the vector from a(j) to y
1
a (k ) = a ( j ) + ( y − a( j ))
k
1
a (k + 1) = a ( j ) +
( y − a( j ))
k −1
...
1
a (2k − 2) = a( j ) + ( y − a ( j ))
2
29

30. Evolutionary Path Relinking (EvoPR)
◦ [Resende y Werneck, 2004]
◦ EliteSet evolution
30

31. EvoPR
1. EliteSet construction
2. Do
3. Apply relinking method to solutions in EliteSet
4. If no new solution can enter EliteSet → rebuild EliteSet
5. Until max number evaluations is reached
31

32. 1. Obtain an EliteSet of b elite solutions.
2. Evaluate the solutions in EliteSet and order them. Make NewSolutions =
TRUE and GlobalIter=0.
while ( NumEvaluations < MaxEvaluations ) do
3. Generate NewSubsets, which consists of the sets (a, x, y) of
solutions in EliteSet that include at least one new solution. Make
NewSolutions = FALSE and Pool = ∅.
while ( NewSubsets ≠ ∅ ) do
4. Select the next set (a, x, y) in NewSubSets.
5. Apply the Relinking Method to produce the sequence from a to x
and y.
6. Apply the Improvement Method to the best solution in the
sequence. Let w be the improved solution. Add w to Pool.
7. Delete (a, x, y) from NewSubsets
end while
for (each solution w ∈ Pool)
8. Let xw be the closest solution to w in EliteSet
if ( f(w) < f(x1) or ( f(w) < f(xb) & d(w, xw)>dthresh) then
9. Make xw = w and reorder EliteSet
10. Make NewSolutions = TRUE
end if
end for
11. GlobalIter = GlobalIter +1
If ( GlobalIter = MaxlIter or NewSolutions= FALSE)
12. Rebuild the RefSet. GlobalIter =0
end if
end while
32

33. Introduction
Path Relinking
Experimental results
Conclusions
33

34. Benchmark
◦ 19 functions with known optimum
6 from CEC’2008
5 shifted functions
8 hybrid functions
◦ Requirements
5 dimensions: D={50,100,200,500,1000}
25 executions per algorithm and function
Maximum number of evaluations: 5000D
34

35. Requirements
◦ Experiments report the error:
f (x)-f (op)
x is the best solution found
op is the optimum of the function
35

36. Results
◦ Constructive methods
We compare our constructive method with the
method described in [Duarte et al., 2010]
1000 constructions
F3, F8, F13
50
100
200
500
1000
Frequency
5.3E10
1.1E11 2.7E11
7.3E11 1.5E12
Taguchi
3.7E10
6.0E10 1.3E11
3.7E11 7.6E11
36

37. Results
◦ Improvement methods
We test several methods that were reported as the best
ones for global optimization (Hvattum et al., 2010)
100 constructions (Taguchi) + Improvement + Simplex
F3, F13, F17
50
100
200
500
1000
CS [Kolda et al., 2003]
8.2E13
2.8E14
8.0E14
1.7E15
2.1E16
SW [Solis y Wets, 1981]
3.2E10
6.2E10
1.4E11
1.9E11
8.4E11
TLS [Duarte et al., 2010]
1.7E9
5.2E9
1.3E10
2.3E11
9.7E10
TSLS
1.8E3
4.1E3
1.2E4
2.2E10
4.6E4
37

38. Results
◦ Path relinking methods
Static PR and Evolutionary PR
F3, F13, F17
PR
EvoPR
|ES|
10
4
8
12
50
6.2E1
7.6E1
5.2E1 7.9E1
100
1.3E2
1.8E2
2.5E2 6.1E2
200
5.2E2
4.8E2
9.8E2 1.7E3
500
2.3E3
1.6E3
2.7E3 4.3E3
1000
5.8E3
3.9E3
6.7E3 8.9E3
Average
1.7E3
1.3E3
2.1E3 3.1E3
38

39. Final experiment
◦ 19 functions, 5 dimensions, 25 runs per method
and function, 5000D evaluations
◦ Average error is reported
DE
CHC
G-CMACMAES
STS
EvoPR
50
3.1E0
2.4E5
1.0E2
1.3E2
1.4E1
100
3.0E1
5.8E6
2.3E2
6.2E2
6.3E1
200
3.5E2
1.4E8
5.4E2
2.8E3
1.9E2
500
3.7E3
3.4E9
2.2E255
1.9E4
3.7E2
1000
1.5E4
2.0E10
-
1.4E4
1.2E3
Average
3.7E3
4.8E9
5.6E254
7.2E3
3.7E2
39

40. Introduction
Path Relinking
Experimental results
Conclusions
40

41. Path Relinking for Global Optimization
◦ Constructive method based on
Fractional experiment design
Diverse solutions
◦ Linking strategy with more than two solutions
◦ Improvement method
Line search with grid
First-improvement approach
Random evaluation of points
41

42. A balance between...
◦ Solution space exploration
◦ Limited number of evaluations
Competitive method compared with state of
the art
◦ EvoPR performs better in high dimensions
42

43. you!
Thank you!