Computational Intelligence Winter Term 2018/19

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1 Computatioal Itelligece Witer Term 28/9 Prof. Dr. Güter Rudolph Lehrstuhl für Algorithm Egieerig (LS ) Fakultät für Iformatik TU Dortmud

2 Pla for Today Lecture Evolutioary Algorithms (EA) Optimizatio Basics EA Basics G. Rudolph: Computatioal Itelligece Witer Term 28/9 2

3 Optimizatio Basics Lecture modellig!!? simulatio!?! optimizatio?!! iput system output G. Rudolph: Computatioal Itelligece Witer Term 28/9 3

4 Optimizatio Basics Lecture give: objective fuctio f: X feasible regio X (= oempty set) objective: fid solutio with miimal or maximal value! optimizatio problem: fid x* X such that f(x*) = mi{ f(x) : x X } x* global solutio f(x*) global optimum ote: max{ f(x) : x X } = mi{ f(x) : x X } G. Rudolph: Computatioal Itelligece Witer Term 28/9 4

5 Optimizatio Basics Lecture local solutio x* X : x N(x*): f(x*) f(x) if x* local solutio the f(x*) local optimum / miimum eighborhood of x* = bouded subset of X example: X =, N ε (x*) = { x X: x x* 2 ε } remark: evidetly, every global solutio / optimum is also local solutio / optimum; the reverse is wrog i geeral! example: f: [a,b], global solutio at x* a x* b G. Rudolph: Computatioal Itelligece Witer Term 28/9 5

6 Optimizatio Basics Lecture What makes optimizatio difficult? some causes: local optima (is it a global optimum or ot?) costraits (ill-shaped feasible regio) o-smoothess (weak causality) strog causality eeded! discotiuities ( odifferetiability, o gradiets) lack of kowledge about problem ( black / gray box optimizatio) f(x) = a x a x max! with x i {,}, a i add costait g(x) = b x b x b x i * = iff a i > NP-hard add capacity costrait to TSP CVRP still harder G. Rudolph: Computatioal Itelligece Witer Term 28/9 6

7 Optimizatio Basics Lecture Whe usig which optimizatio method? mathematical algorithms problem explicitly specified problem-specific solver available problem well uderstood ressources for desigig algorithm affordable solutio with prove quality required radomized search heuristics problem give by black / gray box o problem-specific solver available problem poorly uderstood isufficiet ressources for desigig algorithm solutio with satisfactory quality sufficiet do t apply EAs EAs worth a try G. Rudolph: Computatioal Itelligece Witer Term 28/9 7

8 Evolutioary Algorithm Basics Lecture idea: usig biological evolutio as metaphor ad as pool of ispiratio iterpretatio of biological evolutio as iterative method of improvemet feasible solutio x X = S x... x S multiset of feasible solutios objective fuctio f: X = chromosome of idividual = populatio: multiset of idividuals = fitess fuctio ofte: X =, X = = {,}, X = = { π : π is permutatio of {,2,...,} } also : combiatios like X = x p x q or o-cartesia sets structure of feasible regio / search space defies represetatio of idividual G. Rudolph: Computatioal Itelligece Witer Term 28/9 8

9 Evolutioary Algorithm Basics Lecture algorithmic skeleto iitialize populatio evaluatio paret selectio variatio (yields offsprig) evaluatio (of offsprig) survival selectio (yields ew populatio) N stop? Y output: best idividual foud G. Rudolph: Computatioal Itelligece Witer Term 28/9 9

10 Evolutioary Algorithm Basics Lecture Specific example: (+)-EA i for miimizig some f: populatio size =, umber of offsprig =, selects best from + idividuals paret offsprig. iitialize X () uiformly at radom, set t = 2. evaluate f(x (t) ) 3. select paret: Y = X (t) 4. variatio: flip each bit of Y idepedetly with probability p m = / 5. evaluate f(y) o choice, here 6. selectio: if f(y) f(x (t) ) the X (t+) = Y else X (t+) = X (t) 7. if ot stoppig the t = t+, cotiue at (3) G. Rudolph: Computatioal Itelligece Witer Term 28/9

11 Evolutioary Algorithm Basics Lecture Specific example: (+)-EA i for miimizig some f: populatio size =, umber of offsprig =, selects best from + idividuals paret offsprig compact set = closed & bouded. iitialize X () C uiformly at radom, set t = 2. evaluate f(x (t) ) 3. select paret: Y = X (t) 4. variatio = add radom vector: Y = Y + Z, e.g. Z N(, I ) 5. evaluate f(y) o choice, here 6. selectio: if f(y) f(x (t) ) the X (t+) = Y else X (t+) = X (t) 7. if ot stoppig the t = t+, cotiue at (3) G. Rudolph: Computatioal Itelligece Witer Term 28/9

12 Evolutioary Algorithm Basics Lecture Selectio (a) select parets that geerate offsprig selectio for reproductio (b) select idividuals that proceed to ext geeratio selectio for survival ecessary requiremets: - selectio steps must ot favor worse idividuals - oe selectio step may be eutral (e.g. select uiformly at radom) - at least oe selectio step must favor better idividuals typically : selectio oly based o fitess values f(x) of idividuals seldom : additioally based o idividuals chromosomes x ( maitai diversity) G. Rudolph: Computatioal Itelligece Witer Term 28/9 2

13 Evolutioary Algorithm Basics Lecture Selectio methods populatio P = (x, x 2,..., x µ ) with µ idividuals two approaches:. repeatedly select idividuals from populatio with replacemet 2. rak idividuals somehow ad choose those with best raks (o replacemet) uiform / eutral selectio choose idex i with probability /µ fitess-proportioal selectio choose idex i with probability s i = problems: f(x) > for all x X required g(x) = exp( f(x) ) > but already sesitive to additive shifts g(x) = f(x) + c almost determiistic if large differeces, almost uiform if small differeces G. Rudolph: Computatioal Itelligece Witer Term 28/9 3

14 Evolutioary Algorithm Basics Lecture Selectio methods populatio P = (x, x 2,..., x µ ) with µ idividuals rak-proportioal selectio order idividuals accordig to their fitess values assig raks fitess-proportioal selectio based o raks avoids all problems of fitess-proportioal selectio but: best idividual has oly small selectio advatage (ca be lost!) k-ary touramet selectio draw k idividuals uiformly at radom (typically with replacemet) from P choose idividual with best fitess (break ties at radom) has all advatages of rak-based selectio ad probability that best idividual does ot survive: G. Rudolph: Computatioal Itelligece Witer Term 28/9 4

15 Evolutioary Algorithm Basics Lecture Selectio methods without replacemet populatio P = (x, x 2,..., x µ ) with µ parets ad populatio Q = (y, y 2,..., y λ ) with λ offsprig (µ, λ)-selectio or trucatio selectio o offsprig or comma-selectio rak λ offsprig accordig to their fitess select µ offsprig with best raks best idividual may get lost, λ µ required (µ+λ)-selectio or trucatio selectio o parets + offsprig or plus-selectio merge λ offsprig ad µ parets rak them accordig to their fitess select µ idividuals with best raks best idividual survives for sure G. Rudolph: Computatioal Itelligece Witer Term 28/9 5

16 Evolutioary Algorithm Basics Lecture Selectio methods: Elitism Elitist selectio: best paret is ot replaced by worse idividual. - Itrisic elitism: method selects from paret ad offsprig, best survives with probability - Forced elitism: if best idividual has ot survived the re-ijectio ito populatio, i.e., replace worst selected idividual by previously best paret method P{ select best } from parets & offsprig itrisic elitism eutral < o o fitess proportioate < o o rak proportioate < o o k-ary touramet < o o (µ + λ) = yes yes (µ, λ) = o o G. Rudolph: Computatioal Itelligece Witer Term 28/9 6

17 Evolutioary Algorithm Basics Lecture Variatio operators: deped o represetatio mutatio recombiatio alters a sigle idividual creates sigle offsprig from two or more parets may be applied exclusively (either recombiatio or mutatio) chose i advace exclusively (either recombiatio or mutatio) i probabilistic maer sequetially (typically, recombiatio before mutatio); for each offsprig sequetially (typically, recombiatio before mutatio) with some probability G. Rudolph: Computatioal Itelligece Witer Term 28/9 7

18 Evolutioary Algorithm Basics Variatio i Lecture Idividuals {, } Mutatio a) local choose idex k {,, } uiformly at radom, flip bit k, i.e., x k = x k b) global for each idex k {,, }: flip bit k with probability p m (,) c) olocal choose K idices at radom ad flip bits with these idices d) iversio choose start idex k s ad ed idex k e at radom ivert order of bits betwee start ad ed idex k=2 a) b) K=2 c) k s k e d) G. Rudolph: Computatioal Itelligece Witer Term 28/9 8

19 Evolutioary Algorithm Basics Variatio i Lecture Idividuals {, } Recombiatio (two parets) a) -poit crossover draw cut-poit k {,,-} uiformly at radom; choose first k bits from st paret, choose last -k bits from 2d paret b) K-poit crossover draw K distict cut-poits uiformly at radom; choose bits to k from st paret, choose bits k + to k 2 from 2d paret, choose bits k 2 + to k 3 from st paret, ad so forth c) uiform crossover for each idex i: choose bit i with equal probability from st or 2d paret a) b) c) G. Rudolph: Computatioal Itelligece Witer Term 28/9 9

20 Evolutioary Algorithm Basics Variatio i Lecture Idividuals {, } Recombiatio (multiparet: ρ = #parets) a) diagoal crossover (2 < ρ < ) choose ρ distict cut poits, select chuks from diagoals AAAAAAAAAA BBBBBBBBBB CCCCCCCCCC DDDDDDDDDD b) gee pool crossover (ρ > 2) ABBBCCDDDD BCCCDDAAAA CDDDAABBBB DAAABBCCCC for each gee: choose doatig paret uiformly at radom ca geerate ρ offsprig; otherwise choose iitial chuk at radom for sigle offsprig G. Rudolph: Computatioal Itelligece Witer Term 28/9 2

21 Evolutioary Algorithm Basics Variatio i Lecture Idividuals X = π(,, ) Mutatio a) local 2-swap / -traslocatio b) global draw umber K of 2-swaps, apply 2-swaps K times K is positive radom variable; its distributio may be uiform, biomial, geometrical, ; E[K] ad V[K] may cotrol mutatio stregth expectatio variace G. Rudolph: Computatioal Itelligece Witer Term 28/9 2

22 G. Rudolph: Computatioal Itelligece Witer Term 28/9 22 Evolutioary Algorithm Basics Lecture Variatio i Idividuals X = π(,, ) Recombiatio (two parets) a) order-based crossover (OBX) - select two idices k ad k 2 with k k 2 uiformly at radom - copy gees k to k 2 from st paret to offsprig (keep positios) - copy gees from left to right from 2 d paret, startig after positio k x x x 7 6 x b) partially mapped crossover (PMX) - select two idices k ad k 2 with k k 2 uiformly at radom - copy gees k to k 2 from st paret to offsprig (keep positios) - copy all gees ot already cotaied i offsprig from 2 d paret (keep positios) - from left to right: fill i remaiig gees from 2 d paret x x x 7 6 x x x

23 Evolutioary Algorithm Basics Variatio i Lecture Idividuals X Mutatio additive: Y = X + Z (Z: -dimesioal radom vector) offsprig = paret + mutatio a) local Z with bouded support Defiitio Let f Z : + be p.d.f. of r.v. Z. f The set { x Z : f Z (x) > } is termed the support of Z. x b) olocal Z with ubouded support f Z most frequetly used! x G. Rudolph: Computatioal Itelligece Witer Term 28/9 23

Evolutioary Algorithm Basics Variatio i Lecture Idividuals X Recombiatio (two parets) a) all crossover variats adapted from b) itermediate c) itermediate (per

24 Evolutioary Algorithm Basics Variatio i Lecture Idividuals X Recombiatio (two parets) a) all crossover variats adapted from b) itermediate c) itermediate (per dimesio) d) discrete e) simulated biary crossover (SBX) for each dimesio with probability p c draw from: G. Rudolph: Computatioal Itelligece Witer Term 28/9 24

25 Evolutioary Algorithm Basics Variatio i Lecture Idividuals X Recombiatio (multiparet), ρ 3 parets a) itermediate where ad (all poits i covex hull) b) itermediate (per dimesio) G. Rudolph: Computatioal Itelligece Witer Term 28/9 25

26 Evolutioary Algorithm Basics Lecture Theorem Let f: be a strictly quasicovex fuctio. If f(x) = f(y) for some x y the every offsprig geerated by itermediate recombiatio is better tha its parets. Proof: G. Rudolph: Computatioal Itelligece Witer Term 28/9 26

27 Evolutioary Algorithm Basics Lecture Theorem Let f: be a differetiable fuctio ad f(x) < f(y) for some x y. If (y x) f(x) < the there is a positive probability that a offsprig geerated by itermediate recombiatio is better tha both parets. Proof: G. Rudolph: Computatioal Itelligece Witer Term 28/9 27

General Lower Bounds for the Running Time of Evolutionary Algorithms

General Lower Bounds for the Running Time of Evolutionary Algorithms Geeral Lower Bouds for the Ruig Time of Evolutioary Algorithms Dirk Sudholt Iteratioal Computer Sciece Istitute, Berkeley, CA 94704, USA Abstract. We preset a ew method for provig lower bouds i evolutioary