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 METAHEURISTIC ALGORITHMS SIMILAR TO GENETIC ALGORITHMS LEARN AND SAMPLE A PROBABILITY DISTRIBUTION 2
Estimation of distribution algorithms Scheme Generate a set of solutions 3
Estimation of distribution algorithms Scheme Generate a set Evaluate of solutions 4
Estimation of distribution algorithms Scheme Generate a set Evaluate Select of solutions 5
Estimation of distribution algorithms Scheme Generate a set Evaluate Select of solutions Estimate the parameters of a probability distribution P ( σ ) 6
Estimation of distribution algorithms Scheme Generate a set Evaluate Select of solutions Estimate the parameters of a probability distribution P ( σ ) Sample new solutions 7
Estimation of distribution algorithms Scheme Generate a set Evaluate Select of solutions Estimate the parameters of a probability distribution P ( σ ) Evaluate Sample new solutions 8
Estimation of distribution algorithms Scheme Generate a set Evaluate Select of solutions Estimate the parameters of a probability distribution P ( σ ) Update the set Evaluate of solutions Sample new solutions 9
Combinatorial Problems UMDA [Mühlenbein, 1998] MIMIC [DeBonet, 1997] Continuous Problems FDA [Mühlenbein, 1999] UMDA c [Larrañaga, 2000] EBNA [Etxeberria, 1999] Ω MIMIC c [Larrañaga, 2000] BOA [Pelikan, 2000] EGNA [Larrañaga, 2000] EHBSA [Tsutsui, 2003] EMNA [Larrañaga, 2001] NHBSA [Tsutsui, 2006] IDEA [Bosman, 2000] TREE [Pelikan, 2007] REDA [Romero, 2009] R n S n EDAs reported Permutation Problems in the literature IDEA-ICE [Bosman, 2001] Constrained Problems MEDA [Ceberio, 2011] PLEDA [Ceberio, 2013] ? GMEDA [Ceberio, 2014] RKEDA [Ayodele, 2016] 10
Constrained Optimization Problems Definition minimising f ( x ), x = ( x 1 , . . . , x n ) subject to, g i ( x ) ≤ 0, i = 1 , . . . , r h j ( 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 … 11
Graph Partitioning Problem 2 x 3 Find a k- partition of vertices x 1 x 5 minimising the weight of edges 2 5 between sets: the cut size 1 x 6 x 4 3 x 2 4 We considered the balanced 2- n n partition GPP. X X f ( x ) = x i (1 − x j ) w ij i =1 j =1 Solutions are codified as… Objective Function x ∈ { 0 , 1 } n 12
Graph Partitioning Problem 2 2 x 3 x 3 x 1 x 1 x 5 x 5 2 5 2 5 1 1 x 6 x 6 x 4 x 4 3 3 x 2 x 2 4 4 f ( x 2 ) = 5 f ( x 1 ) = 8 x 1 x 2 0 0 1 0 1 1 0 0 0 1 1 1 x 1 x 2 x 3 x 4 x 5 x 6 x 1 x 2 x 3 x 4 x 5 x 6 n X The constraint: equal number of x i = n/ 2 zeros as ones i =1 13
Constrained Optimization Problems Why are they challenging? The search space of solutions induced by the codification is… 000000 001000 010000 011000 100000 101000 110000 111000 000001 001001 010001 011001 100001 101001 110001 111001 000010 001010 010010 011010 100010 101010 110010 111010 000011 001011 010011 011011 100011 101011 110011 111011 000100 001100 010100 011100 100100 101100 110100 111100 000101 001101 010101 011101 100101 101101 110101 111101 000110 001110 010110 011110 100110 101110 110110 111110 000111 001111 010111 011111 100111 101111 110111 111111 14
Constrained Optimization Problems Why are they challenging? The search space of solutions induced by the codification is… 000000 001000 010000 011000 100000 101000 110000 111000 000001 001001 010001 011001 100001 101001 110001 111001 000010 001010 010010 011010 100010 101010 110010 111010 000011 001011 010011 011011 100011 101011 110011 111011 000100 001100 010100 011100 100100 101100 110100 111100 000101 001101 010101 011101 100101 101101 110101 111101 000110 001110 010110 011110 100110 101110 110110 111110 000111 001111 010111 011111 100111 101111 110111 111111 The majority of the solutions are not feasible ! 15
What happens if we run a UMDA? Univariate Marginal Distribution Algorithm (UMDA) n Y P ( x ) = P ( x i ) i =1 First order marginals No dependencies are considered 16
What happens if we run a UMDA? x 1 x 2 x 3 x 4 x 5 x 6 0 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 17
What happens if we run a UMDA? x 1 x 2 x 3 x 4 x 5 x 6 0 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 Unfeasible solutions are generated… 18
Different approaches Literature review 3. Guarantee feasibility 2. Penalty functions 1. Repair solutions when sampling • Modify solutions to • Punish solutions to be • In EDAs, adapt the hold the constraints discarded when sampling to create selection feasible solutions The role of the probability model is somehow denaturalized 19
The Idea Conduct the optimisation entirely on the set of feasible solutions… 4. Use probability distributions defined on this set 20
Motivation Permutation-based Problems Travelling Salesman Problem (TSP) Combinatorial Optimization Problems 8 Whose solutions are represented as permutations 7 6 The search space consist of n ! 5 3 solutions n ! 8! = 40320 8! = 40320 2 20! = 2 . 43 × 10 18 20! = 2 . 43 × 10 18 4 1 σ = 12367854 σ = 12367854 21
Motivation Permutation-based Problems The space of permutations can be seen as a constrained space of 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 22
Motivation Permutation-based Problems The space of permutations can be seen as a constrained space of 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 23
Motivation Probability Models on Rankings 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. 24
Motivation Probability Models on Rankings 1 Distance-based ψ ( θ ) e − θ D ( σ , σ 0 ) P ( σ ) = Mallows 1 ψ ( θ ) e − P n − 1 j =1 θ j S j ( σ , σ 0 ) P ( σ ) = Generalized Mallows n − 1 w σ ( i ) Y P ( σ ) = P n j = i w σ ( j ) i =1 n − 1 Plackett-Luce n w σ ( i ) Y Y P ( σ ) = w σ ( i ) + w σ ( j ) i =1 j = i +1 Order statistics Bradley-Terry 25
Apply same idea in constrained COPs… Do probability models for constrained spaces exist? No, At least, we do not know them… 26
Designing probability models Requirements ∀ x ∈ Ω , 0 ≤ P ( x ) ≤ 1 1 | Ω | X P ( x i ) = 1 2 i =1 3 Develop efficient learning and sampling methods Sample M Estimate M’ M = M 0 27
A Square Lattice The Probability Model 1 1 1 (0,3) (1,3) (2,3) (3,3) Solutions are modelled as paths on a square lattice of (n/2+1) 2 vertices 0.25 0.34 0.5 1 0.75 0.66 0.5 Vertices: the number of ones (0,2) (1,2) (2,2) (3,2) and zeros at that stage 0.4 0.5 0.66 1 Edges: the probability of moving from one vertex to 0.6 0.5 0.34 (0,1) (1,1) (2,1) (3,1) another. 0.5 0.6 0.75 1 0.5 0.4 0.25 (0,0) (1,0) (2,0) (3,0) x = (1 , 0 , 0 , 1 , 0 , 1) Square Lattice for n=6 GPP instance π = ((0 , 1) , (1 , 1) , (2 , 1) , (2 , 2) , (3 , 2) , (3 , 3)) 28
A Square Lattice The Probability Model 1 1 1 (0,3) (1,3) (2,3) (3,3) Solutions are modelled as paths on a square lattice of (n/2+1) 2 vertices 0.25 0.34 0.5 1 0.75 0.66 0.5 Vertices: the number of ones (0,2) (1,2) (2,2) (3,2) Probability of a path and zeros at that stage n 0.4 0.5 0.66 1 P ( h i +1 − h i ) ( h i ,k i ) , ( h i +1 ,k i ) P ( k i +1 − k i ) Y Edges: the probability of P ( π ) = ( h i ,k i ) , ( h i ,k i +1) moving from one vertex to 0.6 0.5 0.34 (0,1) (1,1) (2,1) i =1 (3,1) another Horizontal step Vertical step 0.5 0.6 0.75 1 0.5 0.4 0.25 (0,0) (1,0) (2,0) (3,0) x = (1 , 0 , 0 , 1 , 0 , 1) Square Lattice for n=4 GPP instance π = ((0 , 1) , (1 , 1) , (2 , 1) , (2 , 2) , (3 , 2) , (3 , 3)) 29
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