Dealing with constraints in estimation of distribution algorithms: - - PowerPoint PPT Presentation

Dealing with constraints in estimation of distribution algorithms: a different approach

Josu Ceberio, Alexander Mendiburu, Jose A. Lozano

Intelligent Systems Group Department of Computer Science and Artificial Intelligence University of the Basque Country (UPV/EHU)

Estimation of distribution algorithms

Introduction

2

METAHEURISTIC ALGORITHMS SIMILAR TO GENETIC ALGORITHMS LEARN AND SAMPLE A PROBABILITY DISTRIBUTION

Estimation of distribution algorithms

Scheme

3 Generate a set

• f solutions

Estimation of distribution algorithms

Scheme

4 Generate a set

• f solutions

Evaluate

Estimation of distribution algorithms

Scheme

5 Generate a set

• f solutions

Evaluate Select

Estimation of distribution algorithms

Scheme

6 Generate a set

• f solutions

Evaluate Select Estimate the parameters

• f a probability

distribution

P(σ)

Estimation of distribution algorithms

Scheme

7 Generate a set

• f solutions

Evaluate Select Sample new solutions Estimate the parameters

• f a probability

distribution

P(σ)

Estimation of distribution algorithms

Scheme

8 Generate a set

• f solutions

Evaluate Select Sample new solutions Evaluate Estimate the parameters

• f a probability

distribution

P(σ)

Estimation of distribution algorithms

Scheme

9 Generate a set

• f solutions

Evaluate Select Sample new solutions Evaluate Update the set

• f solutions

Estimate the parameters

• f a probability

distribution

P(σ)

10

EDAs reported in the literature

Permutation Problems

IDEA-ICE [Bosman, 2001] MEDA [Ceberio, 2011] PLEDA [Ceberio, 2013] GMEDA [Ceberio, 2014] RKEDA [Ayodele, 2016]

Sn

Combinatorial Problems

UMDA [Mühlenbein, 1998] MIMIC [DeBonet, 1997] FDA [Mühlenbein, 1999] EBNA [Etxeberria, 1999] BOA [Pelikan, 2000] EHBSA [Tsutsui, 2003] NHBSA [Tsutsui, 2006] TREE [Pelikan, 2007] REDA [Romero, 2009]

Ω

Continuous Problems

UMDAc [Larrañaga, 2000] MIMICc [Larrañaga, 2000] EGNA [Larrañaga, 2000] EMNA [Larrañaga, 2001] IDEA [Bosman, 2000]

Rn

Constrained Problems

?

Constrained Optimization Problems

Definition

11

minimising f(x), x = (x1, . . . , xn) subject to, gi(x) ≤ 0, i = 1, . . . , r hj(x) = 0, j = r + 1, . . . , m

Some examples à Knapsack Problem à Graph Colouring Problem à Maximum Satisfiability Problem à Capacitated Arc Routing Problem à …

Graph Partitioning Problem

12

Find a k-partition of vertices minimising the weight of edges between sets: the cut size

1 4 3 2 2 5 x1 x2 x3 x4 x5 x6

Objective Function

f(x) =

n

X

i=1 n

X

j=1

xi(1 − xj)wij

We considered the balanced 2- partition GPP. Solutions are codified as…

x ∈ {0, 1}n

1 4 3 2 2 5 1 1 1 x1 x2 x3 x4 x5 x6

x1 f(x1) = 8

x1 x2 x3 x4 x5 x6

Graph Partitioning Problem

13

The constraint: equal number of zeros as ones

x2 f(x2) = 5

1 4 3 2 2 5 1 1 1 x1 x2 x3 x4 x5 x6 x1 x2 x3 x4 x5 x6

n

X

i=1

xi = n/2

Constrained Optimization Problems

Why are they challenging?

14

The search space of solutions induced by the codification is… 000000 000001 000010 000011 000100 000101 000110 000111 001000 001001 001010 001011 001100 001101 001110 001111 010000 010001 010010 010011 010100 010101 010110 010111 011000 011001 011010 011011 011100 011101 011110 011111 100000 100001 100010 100011 100100 100101 100110 100111 101000 101001 101010 101011 101100 101101 101110 101111 110000 110001 110010 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111

000000 000001 000010 000011 000100 000101 000110 000111 001000 001001 001010 001011 001100 001101 001110 001111 010000 010001 010010 010011 010100 010101 010110 010111 011000 011001 011010 011011 011100 011101 011110 011111 100000 100001 100010 100011 100100 100101 100110 100111 101000 101001 101010 101011 101100 101101 101110 101111 110000 110001 110010 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111

Constrained Optimization Problems

Why are they challenging?

15

The search space of solutions induced by the codification is… The majority of the solutions are not feasible !

What happens if we run a UMDA?

16

P(x) =

n

Y

i=1

P(xi)

First order marginals No dependencies are considered

Univariate Marginal Distribution Algorithm (UMDA)

What happens if we run a UMDA?

17

0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 x1 x2 x3 x4 x5 x6 0.6 0.5 0.25 0.66 0.9 0.5 1 0.4 0.5 0.75 0.33 0.1 0.5

What happens if we run a UMDA?

18

Unfeasible solutions are generated…

x1 x2 x3 x4 x5 x6 0.6 0.5 0.25 0.66 0.9 0.5 1 0.4 0.5 0.75 0.33 0.1 0.5 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1

Different approaches

Literature review

19

• 1. Repair solutions
• 2. Penalty functions
• 3. Guarantee feasibility

when sampling

• Modify solutions to

hold the constraints

• Punish solutions to be

discarded when selection

• In EDAs, adapt the

sampling to create feasible solutions The role of the probability model is somehow denaturalized

The Idea

20

Conduct the optimisation entirely

• n the set of feasible solutions…
• 4. Use probability distributions

defined on this set

Motivation

Permutation-based Problems

21

Travelling Salesman Problem (TSP)

1 2 6 3 5 4 8 7 Combinatorial Optimization Problems Whose solutions are represented as permutations

σ = 12367854 n! 8! = 40320 20! = 2.43 × 1018

The search space consist of solutions

20! = 2.43 × 1018 8! = 40320 n!

σ = 12367854

Motivation

Permutation-based Problems

22

111 211 311 112 212 312 113 213 313 121 221 321 122 222 322 123 223 323 131 231 331 132 232 332 133 233 333

The space of permutations can be seen as a constrained space

• f the integers space

n = 3

111 211 311 112 212 312 113 213 313 121 221 321 122 222 322 123 223 323 131 231 331 132 232 332 133 233 333

Motivation

Permutation-based Problems

23

The space of permutations can be seen as a constrained space

• f the integers space

n = 3

Motivation

Probability Models on Rankings

24

Bibliography

à M. A. Fligner and J. S. Verducci (1998), Multistage Ranking Models, Journal of the American Statistical Association, vol. 83, no. 403, pp. 892-901. à D. E. Critchlow, M. A. Fligner, and J. S. Verducci (1991), Probability Models on Rankings, Journal of Mathematical Psychology, vol. 35, no. 3, pp. 294-318. à P. Diaconis (1988), Group Representations in Probability and Statistics, Institute of Mathematical Statistics. à M. A. Fligner and J. S. Verducci (1986), Distance based Ranking Models, Journal of Royal Statistical Society, Series B, vol. 48, no. 3, pp. 359-369. à R. L. Plackett (1975), The Analysis of Permutations, Applied Statistics, vol. 24, no. 10, pp. 193-202. à D. R. Luce (1959), Individual Choice Behaviour, Wiley. à R. A. Bradley AND M. E. Terry (1952), Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons, Biometrika, vol. 39, no. 3, pp. 324-345. à L. L. Thurstone (1927), A law of comparative judgment, Psychological Review, vol 34, no. 4, pp. 273-286.

25

P(σ) = 1 ψ(θ)e−θD(σ,σ0)

Mallows

P(σ) = 1 ψ(θ)e− Pn−1

j=1 θjSj(σ,σ0)

Generalized Mallows

P(σ) =

n−1

Y

i=1

wσ(i) Pn

j=i wσ(j)

Plackett-Luce

P(σ) =

n−1

Y

i=1 n

Y

j=i+1

wσ(i) wσ(i) + wσ(j)

Bradley-Terry

Motivation

Probability Models on Rankings

Distance-based Order statistics

Apply same idea in constrained COPs…

26

Do probability models for constrained spaces exist? No, At least, we do not know them…

Designing probability models

Requirements

27

1

∀x ∈ Ω, 0 ≤ P(x) ≤ 1

2 |Ω|

X

i=1

P(xi) = 1

3

Develop efficient learning and sampling methods

M M’

Sample Estimate

M = M 0

A Square Lattice

The Probability Model

28

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1

Square Lattice for n=6 GPP instance

Solutions are modelled as paths on a square lattice of (n/2+1)2 vertices Vertices: the number of ones and zeros at that stage Edges: the probability of moving from one vertex to another.

π = ((0, 1), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3)) x = (1, 0, 0, 1, 0, 1)

A Square Lattice

The Probability Model

29

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1

Square Lattice for n=4 GPP instance

Solutions are modelled as paths on a square lattice of (n/2+1)2 vertices Vertices: the number of ones and zeros at that stage Edges: the probability of moving from one vertex to another

π = ((0, 1), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3)) x = (1, 0, 0, 1, 0, 1)

Probability of a path

P(π) =

n

Y

i=1

P (hi+1−hi)

(hi,ki),(hi+1,ki)P (ki+1−ki) (hi,ki),(hi,ki+1)

Vertical step Horizontal step

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1

A Square Lattice

The Probability Model - Sampling

30

Square Lattice for n=6 GPP instance

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

A Square Lattice

The Probability Model - Sampling

31

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

A Square Lattice

The Probability Model - Sampling

32

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

x = (1

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

x = (1, 1

A Square Lattice

The Probability Model - Sampling

33

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

x = (1, 1, 0

A Square Lattice

The Probability Model - Sampling

34

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

x = (1, 1, 0, 0

A Square Lattice

The Probability Model - Sampling

35

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

x = (1, 1, 0, 0, 1

A Square Lattice

The Probability Model - Sampling

36

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

As labels are interchangeable, we sample first a 0

x = (1, 1, 0, 0, 1, 0)

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 stage 1 stage 2 stage 3 stage 4 stage 5 stage 6

A Square Lattice

The Probability Model - Sampling

37

Square Lattice for n=6 GPP instance

Solutions are sampled at n stages At each stage a decision has to be taken: up/right

A Square Lattice

The Probability Model – Estimating parameters

38

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1 0.5 0.4 0.25 0.6 0.5 0.34 0.75 0.66 0.5 1 1 1

Square Lattice for n=6 GPP instance

P(h,k),(h+1,k) = # solutions moved from (h, k) to (h + 1, k) # solutions reached (h, k) P(h,k),(h,k+1) = # solutions moved from (h, k) to (h, k + 1) # solutions reached (h, k)

A Square Lattice

The Probability Model – The order of variables

39

In which order should we visit the variables? Is there any difference?

1 1 1 x1 x2 x3 x4 x5 x6

x1

1 1 1 x1 x2 x3 x4 x5 x6

x1

σ = 123456 σ0 = 621345

(0,1) (1,1) (1,0) (0,0) (2,1) (2,0) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,1) (3,0) (3,2) (3,3)

20 10 10 4 1 4 10 20 9 12 10 12 9 4 4 1

Square Lattice for n=6 GPP instance

A Square Lattice

The Probability Model – The order of variables

40

Vertex close to the diagonal are more frequently visited Taking a decision on these points is largely uncertain:

At (2,2) -- the solution has 2 zeros and 2 ones. At (2,0), the preceeding positions are filled with zeros.

A Square Lattice

The Probability Model – The order of variables

41

• Map the best solution to the external border

– First visit 0000…, and then ...1111. – And the order within the subsets?

From each set, choose the item that minimizes the cut-size, in alternated rounds.

x = (1, 0, 0, 1, 1, 0) σ = ({2, 3, 6}, {1, 4, 5})

Experimental Study

Experimental Setting

42

UMDA *, TREE*1 , Lattice

(*) adapted sampling

Algorithms

Pop-size: 10n Sel-size: 5n Off-size: 10n Max evals.: 100n2 10 repetitions

Parameters

22 Instances (Johnson et al.)

G-type and U-type n=124, 250, 500, 1000

Benchmarks

Edges with no

• bservations are

assigned uniform probability Constructive at every 40 iterations Sample

Lattice Settings

x0 = 0

• 1M. Pelikan, S. Tsutsui, and R. Kalapala, Dependency Trees, Permutations and Quadratic Assignment Problem, Medal Report
• No. 2007003 Tech. Rep. 2007.

Experimental Study

Results - Performance

43

Instance Best Fitness ARPD Lattice UMDA Tree G124.02 13 0,32 0,61 0,19 G124.16 449 0,02 0,05 0,01 G250.01 31 0,33 0,49 0,20 G250.02 118 0,07 0,14 0,06 G250.04 360 0,04 0,10 0,03 G250.08 830 0,01 0,05 0,01 G500.005 61 0,30 0,40 0,08 G500.01 234 0,09 0,21 0,07 G500.02 642 0,03 0,11 0,03 G500.04 1754 0,02 0,06 0,02 G1000.0025 131 2,96 3,20 0,74 G1000.005 496 1,22 1,28 0,88 G1000.01 1420 0,56 0,66 0,62 G1000.02 3450 0,35 0,40 0,39 U500.05 23 1,17 1,89 0,57 U500.10 61 1,05 1,12 0,57 U500.20 185 0,56 0,87 0,44 U500.40 412 0,41 0,38 0,28 U1000.05 77 1,62 12,83 2,39 U1000.10 170 1,67 11,67 3,73 U1000.20 352 1,67 10,58 4,94 U1000.40 862 1,53 3,24 2,29

Experimental Study

Results - Performance

44

Instance Best Fitness ARPD Lattice UMDA Tree G124.02 13 0,32 0,61 0,19 G124.16 449 0,02 0,05 0,01 G250.01 31 0,33 0,49 0,20 G250.02 118 0,07 0,14 0,06 G250.04 360 0,04 0,10 0,03 G250.08 830 0,01 0,05 0,01 G500.005 61 0,30 0,40 0,08 G500.01 234 0,09 0,21 0,07 G500.02 642 0,03 0,11 0,03 G500.04 1754 0,02 0,06 0,02 G1000.0025 131 2,96 3,20 0,74 G1000.005 496 1,22 1,28 0,88 G1000.01 1420 0,56 0,66 0,62 G1000.02 3450 0,35 0,40 0,39 U500.05 23 1,17 1,89 0,57 U500.10 61 1,05 1,12 0,57 U500.20 185 0,56 0,87 0,44 U500.40 412 0,41 0,38 0,28 U1000.05 77 1,62 12,83 2,39 U1000.10 170 1,67 11,67 3,73 U1000.20 352 1,67 10,58 4,94 U1000.40 862 1,53 3,24 2,29

Experimental Study

Results - Performance

45

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 ARPD Instances

Average Relative Percentage Deviation

10 repetitions

Lattice UMDA Tree

46

Experimental Study

Results – Time Consumption

n=124 n=250 n=500 n=1000

Lattice UMDA Lattice Lattice Lattice UMDA UMDA UMDA TREE TREE TREE TREE

20 30 40 50 60 70 500000 1000000 1500000

Evaluations Average Type

1 2 6

Instance G124.02, 10 repetitions, Unique Solutions

Population Average Fitness

47

Experimental Study

The influence of the order

Rand Const Best

48

Experimental Study

The influence of the order

20 40 60 500000 1000000 1500000

Evaluations Avg_Distance Type

1 2 6

Instance G124.02, 10 repetitions, Unique Solutions

Avg distance of sampled solutions to the best found so far

Rand Const Best

0.5 0.6 0.7 0.8 0.9 500000 1000000 1500000

Evaluations Non_Parameter_Arc_ratio Type

1 2 6

Instance G124.02, 10 repetitions, Unique Solutions

Parameter−free arc ratio

49

Experimental Study

The influence of the order

Rand Const Best

50

Experimental Study

Convergence Comparison

n=124 n=250

Experimental Study

Results - Performance

51

Instance Best Fitness ARPD Lattice Lattice 2 Lattice 4 UMDA Tree G124.02 13 0,32

0,32 0,45

0,61 0,19 G124.16 449 0,02

0,02 0,02

0,05 0,01 G250.01 31 0,33

0,44 0,49

0,49 0,20 G250.02 118 0,07

0,09 0,05

0,14 0,06 G250.04 360 0,04

0,04 0,04

0,10 0,03 G250.08 830 0,01

0,02 0,02

0,05 0,01 G500.005 61 0,30

0,46 0,46

0,40 0,08 G500.01 234 0,09

0,12 0,13

0,21 0,07 G500.02 642 0,03

0,04 0,06

0,11 0,03 G500.04 1754 0,02

0,03 0,03

0,06 0,02 G1000.0025 131 2,96

• 3,20

0,74 G1000.005 496 1,22

• 1,28

0,88 G1000.01 1420 0,56

• 0,66

0,62 G1000.02 3450 0,35

• 0,40

0,39 U500.05 23 1,17

1,30 1,04

1,89 0,57 U500.10 61 1,05

0,81 1,20

1,12 0,57 U500.20 185 0,56

0,62 0,45

0,87 0,44 U500.40 412 0,41

0,58 0,83

0,38 0,28 U1000.05 77 1,62

• 12,83

2,39 U1000.10 170 1,67

• 11,67

3,73 U1000.20 352 1,67

• 10,58

4,94 U1000.40 862 1,53

• 3,24

2,29

Experimental Study

Results - Performance

52

Instance Best Fitness ARPD Lattice Lattice 2 Lattice 4 UMDA Tree G124.02 13 0,32

0,32 0,45

0,61 0,19 G124.16 449 0,02

0,02 0,02

0,05 0,01 G250.01 31 0,33

0,44 0,49

0,49 0,20 G250.02 118 0,07

0,09 0,05

0,14 0,06 G250.04 360 0,04

0,04 0,04

0,10 0,03 G250.08 830 0,01

0,02 0,02

0,05 0,01 G500.005 61 0,30

0,46 0,46

0,40 0,08 G500.01 234 0,09

0,12 0,13

0,21 0,07 G500.02 642 0,03

0,04 0,06

0,11 0,03 G500.04 1754 0,02

0,03 0,03

0,06 0,02 G1000.0025 131 2,96

• 3,20

0,74 G1000.005 496 1,22

• 1,28

0,88 G1000.01 1420 0,56

• 0,66

0,62 G1000.02 3450 0,35

• 0,40

0,39 U500.05 23 1,17

1,30 1,04

1,89 0,57 U500.10 61 1,05

0,81 1,20

1,12 0,57 U500.20 185 0,56

0,62 0,45

0,87 0,44 U500.40 412 0,41

0,58 0,83

0,38 0,28 U1000.05 77 1,62

• 12,83

2,39 U1000.10 170 1,67

• 11,67

3,73 U1000.20 352 1,67

• 10,58

4,94 U1000.40 862 1,53

• 3,24

2,29

Conclusions

53

The experiments support the validity of our research line: Designing probability models exclusively on the set of feasible solutions

Competitive for small instances, and better in large instances Low time complexity

Future Work

54

Many aspects to be faced in this work

Develop the idea of uncertainty of the paths in the lattice Analyze the effect of

• ther orderings

Use information of the population as Tree (mutual informations)

Ordering of the variables

Larger benchmarks Understand the dynamics

• f the Lattice for different

problem sizes

Experimentation

Square Lattice for K≥3

The probability model

Future Research Lines

Other models

55

Distance-based exponential probability models

P(x) = e−θd(x,¯

x)

Pk

i=1 e−θd(xi,¯ x)

A distance-metric The size of the search

• space. GPP:

k = ✓ n n/2 ◆

Central solution Spread parameter Normalization function

Develop efficient learning and sampling methods

Challenges

Future Research Lines

Other models

56

Dealing with constraints in estimation of distribution algorithms: a different approach

Josu Ceberio, Alexander Mendiburu, Jose A. Lozano

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