Path relinking for high dimensional continuous optimization
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Transcript of Path relinking for high dimensional continuous optimization
Francisco Gortázar, Abraham Duarte
Dpto. de Ciencias de la ComputaciónDpto. de Ciencias de la Computación
Rafael Martí
Dpto. de Estadística e Investigación Operativa
� IntroductionIntroductionIntroductionIntroduction
� Path Relinking
Experimental results� Experimental results
� Conclusions
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� Global optimization problems◦ Non linear function f(x)
◦ x={x1,…, xn}, l≤xi≤u◦ x={x1,…, xn}, l≤xi≤u
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ℜ∈≤≤
x
uxl
xfMinimize )(
� Measuring success…◦ Scalability of Evolutionary Algorithms and other
Metaheuristics for Large Scale Continuous Optimization Problems, M. Lozano and F. Herrera 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?
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� Introduction
� PathPathPathPath RelinkingRelinkingRelinkingRelinking
Experimental results� Experimental results
� Conclusions
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Generate solutions
Selectsolutions bothby quality and
diversity (EliteSet)
Perform a path relinkingbetween each
pair of solutions
1 2 3
GRASP with PR for the MMDP 6
Return the best solution found so far4
� Path Relinking for Global Optimization (PR)◦ EliteSet construction (x1,...,xb)
◦ Improve x1, and replace it with the improved solutionsolution
◦ 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
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� EliteSet construction◦ Generate diverse solutions
� Sampling the solution space
◦ Based on ideas from factorial design of ◦ Based on ideas from factorial design of experiments
� Factorial design kn
� n, #variables
� k, #values
� Genichi Taguchi
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� EliteSet construction◦ Factorial design example for
n=4, k=3
� 81 experiments
ExpExpExpExp.... FactoresFactoresFactoresFactores
1111 2222 3333 4444
1 1 1 1 1
2 1 2 2 2� 81 experiments
◦ Fractional design
� Taguchi Table L9(34)
� 9 experiments
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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
� EliteSet construction◦ Taguchi method is applied to obtain diverse
solutions
◦ #values k = 3◦ #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}
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� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
1 1 1 1 1 1 1 1 1
1 2 2 2 1 1 1 1 1
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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
� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
1 1 1 1 2 2 2 2 2
1 2 2 2 2 2 2 2 2
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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
� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
1 1 1 1 3 3 3 3 3
1 2 2 2 3 3 3 3 3
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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
� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
1 1 1 1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
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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
� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 X4X4X4X4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
2 2 1 1 1 1 2 2 2
2 2 1 2 2 2 2 2 2
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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
� EliteSet construction
x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x6x6x6x6 x7x7x7x7 x8x8x8x8 X9X9X9X9
3 3 1 1 1 1 3 3 3
3 3 1 2 2 2 3 3 3
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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
� EliteSet construction◦ Taguchi table with 40 variables and 3 values � 81
experiments
� We generate 81 solutions by applying this table to the first 40 variables (x ,..,x ) and setting 1 to all other variables40 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
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� 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 x , … ,x and value 1 to the remaining 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
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� EliteSet construction◦ # solutions generated with this process:
� DSize=243⌈n/20⌉
◦ EliteSet is built choosing the best b solutions◦ EliteSet is built choosing the best b solutions
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� Improvement is performed in two stagestwo stagestwo stagestwo stages1. Line search based on a single variable
� Grid size h=(u-l)/100
2. Simplex method2. Simplex method
� Not limited to the grid
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� Improvement method: Line searches◦ For each variable i� We evaluate x+hei and x-hei, discarding the worst
valuevalue
◦ We order the variables i={1, … ,n} according to these values
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xx-hei x+hei
� Improvement method: Line searches
1. ExploreExploreExploreExplore the first n/2 variables in the order that was previously calculated:
For each variable i, evaluateevaluateevaluateevaluate solutions with the form
� x+khei
� k=[-20,20]� k=[-20,20]
� l ≤ x+khei ≤ u
HOW?HOW?HOW?HOW? Randomly & using a first-improvement approach
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xx-hei x+hei x+4heix-4hei
� Improvement method: Line searches
1. ExploreExploreExploreExplore the first n/2 variables in the order that was previously calculated:
For each variable i, evaluateevaluateevaluateevaluate solutions with the form
� x+khei
� k=[-20,20]� k=[-20,20]
� l ≤ x+khei ≤ u
HOW?HOW?HOW?HOW? Randomly & using a first-improvement approach
2. ReReReRe----evaluateevaluateevaluateevaluate the contribution of each variable, and re-order
� We explore again the first n/2 variables
3. RepeatRepeatRepeatRepeat it for 10 iterations, or until no further improvement is possible
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� 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]� α is generated by a uniform probability in [-1,1]
� The simplex method is applied to these solutions
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� Relinking method◦ Straight linking
◦ We perform the linking between three solutions
� An initial solution, a� An initial solution, a
� Two guiding solutions, x and y
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� Relinking method◦ We start at a=(a1,...,an)
◦ Firstly we evaluate solutions in the direction given by the vector from a to xby the vector from a to x
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)(2
1)1(
...
)(1
1)2(
)(1
)1(
axaka
axk
aa
axk
aa
−+=−
−−
+=
−+=
� 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 ythe vector from a(j) to y
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))((2
1)()22(
...
))((1
1)()1(
))((1
)()(
jayjaka
jayk
jaka
jayk
jaka
−+=−
−−
+=+
−+=
� EvoPR1. EliteSet construction
2. Do
3. Apply relinking method to solutions in EliteSet3. Apply relinking method to solutions in EliteSet
4. If no new solution can enter EliteSet → rebuild EliteSet
5. Until max number evaluations is reached
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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
∈∈∈∈
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7. Delete (a, x, y) from NewSubsetsend whilefor (each solution w ∈∈∈∈ Pool)
8. Let xw be the closest solution to w in EliteSetif ( f(w) < f(x1) or ( f(w) < f(xb) & d(w, xw)>dthresh) then
9. Make xw = w and reorder EliteSet10. Make NewSolutions = TRUE
end ifend for11. GlobalIter = GlobalIter +1If ( GlobalIter = MaxlIter or NewSolutions= FALSE)
12. Rebuild the RefSet. GlobalIter =0end if
end while
� Introduction
� Path Relinking
Experimental Experimental Experimental Experimental resultsresultsresultsresults� Experimental Experimental Experimental Experimental resultsresultsresultsresults
� Conclusions
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� Benchmark◦ 19 functions with known optimum
� 6 from CEC’2008
� 5 shifted functions� 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
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� Requirements◦ Experiments report the error:
� f (x)-f (op)
� x is the best solution found� x is the best solution found
� op is the optimum of the function
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� Results◦ Constructive methods
� We compare our constructive method with the method described in [Duarte et al., 2010]method described in [Duarte et al., 2010]
� 1000 constructions
� F3, F8, F13
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50505050 100100100100 200200200200 500500500500 1000100010001000
Frequency 5.3E10 1.1E11 2.7E11 7.3E11 1.5E12
Taguchi 3.7E103.7E103.7E103.7E10 6.0E106.0E106.0E106.0E10 1.3E111.3E111.3E111.3E11 3.7E113.7E113.7E113.7E11 7.6E117.6E117.6E117.6E11
� Results◦ Improvement methods
� We test several methods that were reported as the best ones for global optimization (Hvattum et al., 2010)ones for global optimization (Hvattum et al., 2010)
� 100 constructions (Taguchi) + Improvement + Simplex
� F3, F13, F17
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50505050 100100100100 200200200200 500500500500 1000100010001000
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 1.8E31.8E31.8E31.8E3 4.1E34.1E34.1E34.1E3 1.2E41.2E41.2E41.2E4 2.2E102.2E102.2E102.2E10 4.6E44.6E44.6E44.6E4
� Results◦ Path relinking methods
� Static PR and Evolutionary PR
� F3, F13, F17� F3, F13, F17
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PRPRPRPR EvoPREvoPREvoPREvoPR
|ES| 10101010 4444 8888 12121212
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 1.3E31.3E31.3E31.3E3 2.1E3 3.1E3
� Final experiment◦ 19 functions, 5 dimensions, 25 runs per method
and function, 5000D evaluations
◦ Average error is reported◦ Average error is reported
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DEDEDEDE CHCCHCCHCCHC GGGG----CMACMACMACMA----ESESESES
STSSTSSTSSTS EvoPREvoPREvoPREvoPR
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
� Introduction
� Path Relinking
Experimental results� Experimental results
� ConclusionsConclusionsConclusionsConclusions
40
� Path Relinking for Global Optimization◦ Constructive method based on
� Fractional experiment design
� Diverse solutions� Diverse solutions
◦ Linking strategy with more than two solutions
◦ Improvement method
� Line search with grid
� First-improvement approach
� Random evaluation of points
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� A balance between...◦ Solution space exploration
◦ Limited number of evaluations
� Competitive method compared with state of � Competitive method compared with state of the art◦ EvoPR performs better in high dimensions
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