Francisco Gortázar, Abraham Duarte
Dpto. de Ciencias de la Computación

Rafael Martí
Dpto. de Estadística e Investigación ...
Introduction
Path Relinking
Experimental results
Conclusions

2
Global optimization
problems
◦ Non linear function f(x)
◦ x={x1,…, xn}, l≤xi≤u

Minimize f ( x)
l ≤ x≤u
x ∈ℜ

3
Measuring success…
◦ Scalability of Evolutionary Algorithms and other

Metaheuristics for Large Scale Continuous
Optimizat...
Introduction
Path Relinking
Experimental results
Conclusions

5
1 Generate
solutions

4

2

Select
solutions both
by quality and
diversity (EliteSet)

3

Perform a
path relinking
between...
Path Relinking for Global Optimization (PR)
◦ EliteSet construction (x1,...,xb)
◦ Improve x1, and replace it with the impr...
EliteSet construction
◦ Generate diverse solutions
Sampling the solution space

◦ Based on ideas from factorial design of
...
EliteSet construction
◦ Factorial design example for
n=4, k=3
81 experiments

◦ Fractional design
Taguchi Table L9
9 exper...
EliteSet construction
◦ Taguchi method is applied to obtain diverse
solutions
◦ #values k = 3
midvalue = l + ½(u-l)
lowerv...
EliteSet construction
◦ Problem: biggest Taguchi table found: n=40

11
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

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3

3...
EliteSet construction
x1

x2

x3

x4

x5

x6

x7

x8

X9

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3...
EliteSet construction
x1

x2

x3

x4

x5

x6

x7

x8

X9

1

1

1

1

3

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3

3

3

1

2

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1

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3...
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...
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

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2

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2

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3...
EliteSet construction
x1

x2

x3

x4

x5

x6

x7

x8

X9

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3

1

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1

3...
EliteSet construction
◦ Taguchi table with 40 variables and 3 values
experiments

81

We generate 81 solutions by applying...
EliteSet construction
◦ We move the table in steps of size 20
We obtained better results
We generate 81 solutions by apply...
EliteSet construction
◦ # solutions generated with this process:
DSize=243⌈n/20⌉
◦ EliteSet is built choosing the best b s...
Improvement is performed in two stages
1. Line search based on a single variable
Grid size h=(u-l)/100
2. Simplex method
N...
Improvement method: Line searches
◦ For each variable i
We evaluate x+hei and x-hei, discarding the worst
value

◦ We orde...
Improvement method: Line searches
1. Explore the first n/2 variables in the order that was
previously calculated:
For each...
Improvement method: Line searches
1. Explore the first n/2 variables in the order that was
previously calculated:
For each...
Improvement method: Simplex
Let x be the best solution found after the first stage
(line searches)
We perturb the value of...
Relinking method
◦ Straight linking
◦ We perform the linking between three solutions
An initial solution, a
Two guiding so...
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
Relinking method
◦ We start at a=(a1,...,an)
◦ Firstly we evaluate solutions in the direction given
by the vector from a t...
Relinking method:
◦ The best solution is chosen from previous step, a(j)
◦ Then we evaluate solutions in the direction giv...
Evolutionary Path Relinking (EvoPR)
◦ [Resende y Werneck, 2004]
◦ EliteSet evolution

30
EvoPR
1. EliteSet construction
2. Do
3. Apply relinking method to solutions in EliteSet
4. If no new solution can enter El...
1. Obtain an EliteSet of b elite solutions.
2. Evaluate the solutions in EliteSet and order them. Make NewSolutions =
TRUE...
Introduction
Path Relinking
Experimental results
Conclusions

33
Benchmark
◦ 19 functions with known optimum
6 from CEC’2008
5 shifted functions
8 hybrid functions

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

...
Results
◦ Constructive methods
We compare our constructive method with the
method described in [Duarte et al., 2010]
1000 ...
Results
◦ Improvement methods
We test several methods that were reported as the best
ones for global optimization (Hvattum...
Results
◦ Path relinking methods
Static PR and Evolutionary PR
F3, F13, F17
PR

EvoPR

|ES|

10

4

8

12

50

6.2E1

7.6E...
Final experiment
◦ 19 functions, 5 dimensions, 25 runs per method
and function, 5000D evaluations
◦ Average error is repor...
Introduction
Path Relinking
Experimental results
Conclusions

40
Path Relinking for Global Optimization
◦ Constructive method based on
Fractional experiment design
Diverse solutions

◦ Li...
A balance between...
◦ Solution space exploration
◦ Limited number of evaluations

Competitive method compared with state ...
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Path relinking for high dimensional continuous optimization

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

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Path relinking for high dimensional continuous optimization

  1. 1. Francisco Gortázar, Abraham Duarte Dpto. de Ciencias de la Computación Rafael Martí Dpto. de Estadística e Investigación Operativa
  2. 2. Introduction Path Relinking Experimental results Conclusions 2
  3. 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. 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. 5. Introduction Path Relinking Experimental results Conclusions 5
  6. 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. 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. 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. 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. 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. 11. EliteSet construction ◦ Problem: biggest Taguchi table found: n=40 11
  12. 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. 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. 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. 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. 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. 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. 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. 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. 20. EliteSet construction ◦ # solutions generated with this process: DSize=243⌈n/20⌉ ◦ EliteSet is built choosing the best b solutions 20
  21. 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. 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. 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. 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. 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. 26. Relinking method ◦ Straight linking ◦ We perform the linking between three solutions An initial solution, a Two guiding solutions, x and y 26
  27. 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. 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. 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. 30. Evolutionary Path Relinking (EvoPR) ◦ [Resende y Werneck, 2004] ◦ EliteSet evolution 30
  31. 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. 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. 33. Introduction Path Relinking Experimental results Conclusions 33
  34. 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. 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. 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. 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. 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. 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. 40. Introduction Path Relinking Experimental results Conclusions 40
  41. 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. 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. 43. you! Thank you!

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