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Computational Intelligence Winter Term 2012/13

Computational Intelligence Winter Term 2012/13. Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund. Plan for Today. Evolutionary Algorithms (EA) Optimization Basics EA Basics.

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Computational Intelligence Winter Term 2012/13

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  1. ComputationalIntelligence Winter Term 2012/13 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund

  2. Plan for Today • EvolutionaryAlgorithms (EA) • Optimization Basics • EA Basics G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 2 G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 2

  3. system input output ! ? ! ! ? ! ! ? ! Optimization Basics modelling simulation optimization G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 3

  4. Optimization Basics given: objective function f: X → R feasible region X (= nonempty set) objective: find solution with minimal or maximal value! x* global solution f(x*) global optimum optimization problem: find x* 2 X such that f(x*) = min{ f(x) : x 2 X } note: max{ f(x) : x 2 X } = –min{ –f(x) : x 2 X } G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 4

  5. remark: evidently, every global solution / optimum is also local solution / optimum; the reverse is wrong in general! example: f: [a,b] → R, global solution at x* x* a b Optimization Basics local solution x* 2 X : 8x 2 N(x*): f(x*) ≤ f(x) if x* local solution then f(x*) local optimum / minimum neighborhood of x* = bounded subset of X example: X = Rn, N(x*) = { x 2 X: || x – x*||2 ≤  } G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 5

  6. Optimization Basics What makes optimization difficult? • some causes: • local optima (is it a global optimum or not?) • constraints (ill-shaped feasible region) • non-smoothness (weak causality) • discontinuities () nondifferentiability, no gradients) • lack of knowledge about problem () black / gray box optimization) strong causality needed! f(x) = a1 x1 + ... + an xn→ max! with xi2 {0,1}, ai2R ) xi* = 1 if ai > 0 add constaint g(x) = b1 x1 + ... + bn xn≤ b ) NP-hard ) still harder add capacity constraint to TSP ) CVRP G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 6

  7. Optimization Basics When using which optimization method? • mathematical algorithms • problem explicitly specified • problem-specific solver available • problem well understood • ressources for designing algorithm affordable • solution with proven quality required • randomized search heuristics • problem given by black / gray box • no problem-specific solver available • problem poorly understood • insufficient ressources for designing algorithm • solution with satisfactory quality sufficient )don‘t apply EAs )EAs worth a try G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 7

  8. Evolutionary Algorithm Basics idea: using biological evolution as metaphor and as pool of inspiration ) interpretation of biological evolution as iterative method of improvement feasible solution x 2 X = S1 x ... x Sn = chromosome of individual = population: multiset of individuals multiset of feasible solutions = fitness function objective function f: X → R often: X = Rn, X = Bn = {0,1}n, X = Pn = {  :  is permutation of {1,2,...,n} } or non-cartesian sets also : combinations like X = Rn x Bp x Pq ) structure of feasible region / search space defines representation of individual G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 8

  9. Evolutionary Algorithm Basics algorithmic skeleton initialize population evaluation parentselection variation (yields offspring) evaluation (of offspring) survival selection (yields new population) N stop? Y output: best individual found G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 9

  10. parent offspring Evolutionary Algorithm Basics Specific example: (1+1)-EA in Bn for minimizing some f: Bn→ R population size = 1, number of offspring = 1, selects best from 1+1 individuals • initialize X(0)2Bnuniformlyatrandom, set t = 0 • evaluate f(X(t)) • selectparent: Y = X(t) • variation: flipeachbitof Y independentlywithprobability pm = 1/n • evaluate f(Y) • selection: if f(Y) ≤ f(X(t)) then X(t+1) = Y else X(t+1) = X(t) • if not stoppingthen t = t+1, continueat (3) no choice, here G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 10

  11. parent offspring Evolutionary Algorithm Basics Specificexample: (1+1)-EA in Rnforminimizingsomef: Rn→ R population size = 1, number of offspring = 1, selects best from 1+1 individuals compactset = closed & bounded • initialize X(0)2C ½Rnuniformlyatrandom, set t = 0 • evaluate f(X(t)) • selectparent: Y = X(t) • variation = addrandomvector: Y = Y + Z, e.g. Z » N(0, In) • evaluate f(Y) • selection: if f(Y) ≤ f(X(t)) then X(t+1) = Y else X(t+1) = X(t) • if not stoppingthen t = t+1, continueat (3) nochoice, here G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 11

  12. Evolutionary Algorithm Basics Selection • selectparentsthatgenerateoffspring → selectionforreproduction • selectindividualsthatproceedtonextgeneration→ selectionforsurvival • necessary requirements: • - selection steps must not favor worse individuals • one selection step may be neutral (e.g. select uniformly at random) • at least one selection step must favor better individuals typically : selection only based on fitness values f(x) of individuals seldom : additionally based on individuals‘ chromosomes x (→ maintain diversity) G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 12

  13. don‘t use! Evolutionary Algorithm Basics Selection methods population P = (x1, x2, ..., x) with  individuals two approaches: 1. repeatedly select individuals from population with replacement 2. rank individuals somehow and choose those with best ranks (no replacement) • uniform / neutral selection choose index i with probability 1/ • fitness-proportional selection choose index i with probability si = problems: f(x) > 0 for all x 2 X required ) g(x) = exp( f(x) ) > 0 but already sensitive to additive shifts g(x) = f(x) + c almost deterministic if large differences, almost uniform if small differences G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 13

  14. outdated! Evolutionary Algorithm Basics Selection methods population P = (x1, x2, ..., x) with  individuals • rank-proportional selection order individuals according to their fitness values assign ranks fitness-proportional selection based on ranks) avoids all problems of fitness-proportional selection but: best individual has only small selection advantage (can be lost!) • k-ary tournament selection draw k individuals uniformly at random (typically with replacement) from P choose individual with best fitness (break ties at random) ) has all advantages of rank-based selection and probability that best individual does not survive: G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 14

  15. Evolutionary Algorithm Basics Selection methods without replacement population P = (x1, x2, ..., x) with  parents and population Q = (y1, y2, ..., y) with  offspring • (, )-selection or truncation selection on offspring or comma-selection rank offspring according to their fitness select  offspring with best ranks) best individual may get lost, ≥  required • (+)-selection or truncation selection on parents + offspring or plus-selection merge offspring and parents rank them according to their fitness select  individuals with best ranks) best individual survives for sure G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 15

  16. Evolutionary Algorithm Basics Selectionmethods: Elitism Elitistselection: bestparentis not replacedbyworse individual. • Intrinsicelitism: methodselectsfromparentandoffspring, bestsurviveswithprobability 1 - Forcedelitism: ifbest individual has not survivedthenre-injectionintopopulation, i.e., replaceworstselected individual bypreviouslybestparent G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 16

  17. Evolutionary Algorithm Basics Variation operators: depend on representation mutation → alters a single individual recombination → createssingleoffspringfromtwoormoreparents maybeapplied • exclusively (eitherrecombinationormutation) chosen in advance • exclusively (eitherrecombinationormutation) in probabilisticmanner • sequentially (typically, recombinationbeforemutation); foreachoffspring • sequentially (typically, recombinationbeforemutation) withsomeprobability G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 17

  18. Evolutionary Algorithm Basics Variation in Bn Individuals2 { 0, 1 }n ● Mutation local → chooseindex k 2 { 1, …, n } uniformlyatrandom, flipbit k, i.e., xk = 1 – xk global → foreachindex k 2 { 1, …, n }: flipbit k withprobability pm2 (0,1) “nonlocal“ → choose K indicesatrandomandflipbitswiththeseindices inversion → choosestartindex ksand end indexkeatrandominvert order ofbitsbetweenstartandandindex 10011 11011 11001 00101 → 00001 k=2 ks K=2 → ke a) b) c) d) G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 18

  19. Evolutionary Algorithm Basics Variation in Bn Individuals2 { 0, 1 }n ● Recombination (twoparents) 1-point crossover → drawcut-point k 2 {1,…,n-1} uniformlyatrandom;choosefirst k bitsfrom 1st parent,choose last n-k bitsfrom 2nd parent K-pointcrossover → draw K distinctcut-pointsuniformlyatrandom;choosebits 1 to k1from 1st parent, choosebits k1+1 to k2from 2nd parent, choosebits k2+1 to k3from 1st parent, and so forth … uniform crossover → foreachindex i: choosebit i withequalprobabilityfrom 1st or 2nd parent 00 0 1 11 1 1 1001 1001 1001 0111 0111 0111 11 0 1 ) ) ) c) a) b) G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 19

  20. Evolutionary Algorithm Basics Variation in Bn Individuals2 { 0, 1 }n ● Recombination (multiparent: ½ = #parents) diagonal crossover (2 < ½ < n) → choose½ – 1 distinctcutpoints, selectchunksfromdiagonals AAAAAAAAAA BBBBBBBBBB CCCCCCCCCC DDDDDDDDDD ABBBCCDDDD cangenerate½offspring; otherwisechooseinitialchunkatrandomforsingleoffspring BCCCDDAAAA CDDDAABBBB DAAABBCCCC genepoolcrossover (½ > 2) → foreachgene: choosedonatingparentuniformlyatrandom G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 20

  21. Evolutionary Algorithm Basics Variation in Pn Individuals X = ¼(1, …, n) ● Mutation local → 2-swap / 1-translocation 5 3 2 4 1 5 3 2 4 1 5 4 2 3 1 5 2 4 3 1 global → drawnumber K of 2-swaps, apply 2-swaps K times K is positive random variable; itsdistributionmaybe uniform, binomial, geometrical, …; E[K] and V[K] maycontrolmutationstrength expectation variance G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 21

  22. Evolutionary Algorithm Basics Variation in Pn Individuals X = ¼(1, …, n) ● Recombination (twoparents) order-basedcrossover (OBX) 2 3 5 7 1 6 4 6 4 5 3 7 2 1 • - selecttwoindices k1and k2with k1 ≤ k2uniformlyatrandom • copy genes k1to k2from 1stparenttooffspring (keeppositions) • copy genes fromlefttorightfrom 2ndparent, starting after position k2 x xx7 1 6 x 5 3 2 7 1 6 4 2 3 5 7 1 6 4 6 4 5 3 7 2 1 partiallymappedcrossover (PMX) • - selecttwoindices k1and k2with k1 ≤ k2uniformlyatrandom • copy genes k1to k2from 1stparenttooffspring (keeppositions) • copy all genes not alreadycontained in offspringfrom 2ndparent (keeppositions) • fromlefttoright: fill in remaining genes from 2ndparent x xx7 1 6 x x 4 57 1 6 x 34 57 1 6 2 G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 22

  23. Evolutionary Algorithm Basics Variation in Rn Individuals X 2Rn ● Mutation additive: Y = X + Z (Z: n-dimensional randomvector) offspring = parent + mutation local → Z withboundedsupport Definition Let fZ: Rn→ R+bep.d.f. ofr.v. Z. The set { x 2Rn : fZ(x) > 0 } istermedthesupportof Z. fZ 0 x 0 nonlocal → Z withunboundedsupport fZ mostfrequentlyused! 0 x 0 G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 23

  24. Evolutionary Algorithm Basics Variation in Rn Individuals X 2Rn ● Recombination (twoparents) all crossovervariantsadaptedfromBn intermediate intermediate (per dimension) discrete simulatedbinarycrossover (SBX) → foreachdimensionwithprobability pc drawfrom: G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 24

  25. Evolutionary Algorithm Basics Variation in Rn Individuals X 2Rn ● Recombination (multiparent), ½ ≥ 3 parents intermediate whereand (all points in convexhull) intermediate (per dimension) G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 25

  26. Evolutionary Algorithm Basics Theorem Let f: Rn → Rbe a strictlyquasiconvexfunction. If f(x) = f(y) forsome x ≠ y theneveryoffspringgeneratedby intermediate recombinationisbetterthanitsparents. Proof: ■ G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 26

  27. Evolutionary Algorithm Basics Theorem Let f: Rn → Rbe a differentiablefunctionand f(x) < f(y) forsome x ≠ y. If (y – x)‘ rf(x) < 0 thenthereis a positive probabilitythat an offspringgeneratedby intermediate recombinationisbetterthanbothparents. Proof: ■ G. Rudolph: ComputationalIntelligence▪ Winter Term 2012/13 27

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