Constraints in Evolutionary Algorithms

Constraints: the big questions, page 233  how to evaluate and compare feasible and infeasible solutions  avoid, eliminate, repair or penalize infeasible solutions  what is the relation of infeasibility to optimality

fitness / constraints / representation: a tradeoff  some features of a problem may be included by 1.representation 2.constraints on feasible solutions 3.inclusion in fitness evaluation variables; domains; constraints; fitness

Factors in analyzing constraints 1.Representation 2.Search space 3.Neighbour definition & variation operators 4.Evaluation: fitness function 5.Selection operators

1. Representation possible strategies to address constraints:  include (some) constraints in representation  use ‘decoders’ to interpret all solutions as feasible

constraints in representation: n queens How many states?  Any queen anywhere: 16 4 = 65536(n 2 ) n  Queens on different squares: 16x15x14x13 = 43680n 2 !/(n 2 -n)!  Queens in separate columns: 4 4 = 256n n  Queens in separate cols, rows: 4x3x2x1 = 24n!

decoder makes all solutions feasible: ordinal representation of TSP recall:  transform to ordinal representation A D B C F E -> F D E B C A-> apply normal crossover and restore A D F C E B F D A B E C

decoder makes all solutions feasible: ordinal representation of TSP represent TSP of size n cities by n variables, v 1, v 2, v 3,..., v n D 1 = {1, 2, 3,..., n} D 2 = {1, 2, 3,..., n-1}... D n-1 = {1, 2} D n = {1}EXAMPLE: n=6: decodes to: F D A B E C

2. Search space  partial representation space: constraints reduce domains constraints prune subtrees BUT (for Evolutionary Algorithms)  complete representation space constraints reduce feasible domains neighbour definition & variation operations determine effect during search

Shape of complete representation search space  basic shape is multi-dimensional based on domains  neighbour definitions & variation operators determine connectivity of the space  constraints make solutions either feasible or infeasible (e.g., simplex method)  shape of feasible subspace  connected? convex?

3. Neighbour definitions & variation operators  determines connectedness of space including feasible and infeasible subspaces  properties with respect to transitions from feasible solution: is feasible space connected? are offspring of feasibles all feasible? ~~ convexity recall revised crossover definitions for TSP

Connected feasible space? example: variables v 1, v 2, v 3,..., v 6 D 1 = D 2 =...= D 6 = {1, 2, 3} constraint: values must be repeated in more than one variable feasible: {1, 2, 3, 3, 1, 2}, {3, 3, 3, 2, 2, 2} How to define alteration: crossover? mutation?

4. Evaluation: fitness function  how to include feasibility, infeasibility in fitness evaluation 1.rejection of infeasible solutions 2.repair of infeasible solutions to feasible  put repaired solution in population? 3.infeasibility penality 4.different fitness functions

4. Evaluation: fitness function 1.rejection of infeasible solutions  while creating offspring solutions, reject infeasible solutions and replace with feasible solutions v 1, v 2, v 3,..., v 6 D 1 = D 2 =...= D 6 = {1, 2, 3} constraint: values must be repeated in more than one variable feasible: {1, 2, 3, 3, 1, 2}, {3, 3, 3, 2, 2, 2}

4. Evaluation: fitness function 2.repair of infeasible solutions to feasible  when an infeasible offstring is created, repair it before evaluation  put repaired or infeasible solution in population? Which fitness feasible: {1, 2, 3, 3, 1, 2}, {3, 3, 3, 2, 2, 2}

4. Evaluation: fitness function 3.infeasibility penality  evaluate an infeasible solution for fitness but add a penalty term to its fitness value feasible: {1, 2, 3, 3, 1, 2}, {3, 3, 3, 2, 2, 2} fitness: π i=1 to 5 |v i – v i+1 | - 10*(singletons)

4. Evaluation: fitness function 4.different fitness functions  evaluate infeasible solutions and feasible ones separately how to compare them? feasible: {1, 2, 3, 3, 1, 2}, {3, 3, 3, 2, 2, 2} fitness: eval(s) = q u.eval u (s) + q f.eval f (s) eval f (s) = π i=1 to 5 |v i – v i+1 | eval u (s) = -10*(singletons)

5. Selection operators  reject infeasible solutions  return infeasible solutions to feasible space if possible (hill climbing?)  allow infeasible solutions to remain in population

evolutionary example  knapsack filling given a set of objects of varying weights, how many containers (knapsacks) are needed to carry all the objects?  W: {w 1, w 2, …,w n }, set of object weights  K: knapsack capacity  variables? domains? constraints? fitness?  a ‘decoder’ model with a feasible representation

example from stochastic search Lewis - sudoku by simulated annealing  where are the constraints? box, row, column, initial values  representation  evaluation  transformation