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Introduction to Evolutionary Algorithms and Optimization Techniques

This lecture series by Prof. Dr. Günter Rudolph delves into the fundamentals of Computational Intelligence, focusing on Evolutionary Algorithms (EA) and optimization basics. It covers key concepts such as objective functions, feasible regions, and the challenges in optimization, including local vs. global optima and constraints. The discussion includes the algorithmic skeleton of EAs, illustrating how biological evolution inspires iterative improvement methods. Examples such as the (1+1)-EA showcase practical applications of these methodologies in optimization problems.

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Introduction to Evolutionary Algorithms and Optimization Techniques

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  1. ComputationalIntelligence Winter Term 2013/14 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 2013/14 2 G. Rudolph: ComputationalIntelligence▪ Winter Term 2013/14 2

  3. system input output ! ? ! ! ? ! ! ? ! Optimization Basics modelling simulation optimization G. Rudolph: ComputationalIntelligence▪ Winter Term 2013/14 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 2013/14 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 2013/14 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 iffai > 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 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 2013/14 27

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