Introduction. Nature-Inspired Computing. Terminology. Problem Types. Constraint Satisfaction Problems - CSP. Free Optimization Problem - FOP

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1 Nature-Ispired Computig Hadlig Costraits Dr. Şima Uyar September 2006 Itroductio may practical problems are costraied ot all combiatios of variable values represet valid solutios feasible solutios ifeasible solutios costrait hadlig ot straightforward most variatio operators blid to costraits Termiology problem give i terms of variables (v,...,v ) each variable has domais (D,...,D ) free search space: S=D xd 2 x...xd problems distiguished by presece / absece of a objective fuctio costraits Problem Types Yes No Yes Costraied Optimizatio Poblem Costrait Satisfactio Problem No Free Optimizatio Problem --- Free Optimizatio Problem - FOP defied by the pair <S,f> S: free search space f: objective fuctio o S solutio of a FOP is s S with optimal f Costrait Satisfactio Problems - CSP defied by the pair <S, Φ > S: free search space Φ : a formula (Boolea fuctio o S) usually called a feasibility coditio typically compoud etity derived from more elemetary costraits solutio to a CSP: s S with φ( s) = True

2 CSP Examples graph three-colorig problem G=(N,E), E NxN color the odes of a graph G with 3 colors i such a way that o eighbour odes have the same color eighbour ode: odes coected by a edge boolea satisfiability problem (SAT) CSP mai challeges o objective fuctio to defie fitess extreme case of eedle-i-a-haystackproblem large plateaus at zero level (False) some sigular peaks (True) basic approach trasform costraits ito optimizatio objectives Costraied Optimizatio Problem - COP combiatio of FOP ad CSP defied by a triple <S,f,φ> S: free search space f: objective fuctio o S φ: a formula (Boolea fuctio o S) COP Example - TSP cities C=(c,c 2,...,c ) S=C objective fuctio: miimizatio d( s, s ) with s i = i i+ + f ( s) = defied as s COP similar to CSP trasform costraits ito optimizatio fuctio becomes FOP idirect costrait hadlig doe before NIH ru leave costraits as costraits explicitly esure feasibility of solutios direct costrait hadlig eforced explicitly durig ru Hadlig Costraits most CSPs are discrete two types of COPs discrete COPs (combiatorial optimizatio) cotiuous COPs costrait hadlig basically same 2

3 Costrait Types Search Space iequality / equality costraits liear / o-liear costraits F F S F Costrait Hadlig three mai types idirect costrait hadlig direct costrait hadlig mappig costrait hadlig Costrait Hadlig Methods idirect costrait hadlig pealty fuctios direct costrait hadlig repairig ifeasible solutio cadidates preservig feasibility through special represetatios ad operators seperatio of costraits ad objectives mappig costrait hadlig usig decoder fuctios Pealty Fuctios Aim: trasformig a costraied optimizatio problem ito a ucostraied oe by addig / subtractig a value to / from the objective fuctio based o the amout of costrait violatio Types of Pealty Fuctios death pealty static pealty dyamic pealty adaptive pealty f p = f ± p 3

4 Death Pealty reject all ifeasible solutios easiest ad computatioally most efficiet o eed to estimate degree of ifeasibility oly advisable if the feasible regio is fairly large search may stagate if the feasible regio is very small o use of iformatio from ifeasible solutios a variatio of this method assigs zero fitess to ifeasibles Static Pealty pealty factors do ot deped o curret geeratio umber i ay way remai costat durig whole ru it may ot be a good idea to keep the same pealty factors durig the whole ru pealty factors are problem-depedet Static Pealty extictive pealties very high pealties to prevet use of ifeasibles biary pealties d i is biary: if costrait violated, else 0 distace based pealties usually use square of Euclidea distace Dyamic Pealty curret geeratio umber is ivolved i the computatio of the correspodig pealty factors ormally the pealty factors are defied i such a way that they icrease over time, i.e. over geeratios difficult to derive good dyamic pealty fuctios require settig of iitial values Adaptive Pealty pealty fuctio takes feedback from the search process ot depedet o iitial values e.g. pealty for the geeratio (t + ) is decreased if all best idividuals i the last k geeratios were feasible is icreased if they were all ifeasible stays the same if there are some feasible ad ifeasible idividuals tied as best i the populatio Adaptive Pealty settig the parameters of this type of approach may be difficult tries to avoid either a all-feasible or a allifeasible populatio more recet costrait-hadlig approaches pay attetio to this issue. 4

5 Desigig Pealty Fuctios ideal pealty factor caot be kow a priori for a arbitrary problem if pealty too high ad optimum lies at the boudary of the feasible regio, algorithm will be pushed iside the feasible regio very quickly, ad will ot be able to move back towards boudary if pealty too low, too much search time will be spet explorig the ifeasible regio because pealty will be egligible with respect to objective fuctio. Desigig Pealty Fuctios appropriate choice of pealty method may deped o: ratio of feasible space to the whole search space topological properties of feasible search space evaluatio fuctio umber of variables ad costraits promisig results from use of adaptive pealties Repair Fuctios guaratee feasibility of solutio ifeasibles repaired to closest feasible require heuristic repair cost what to do with repaired solutio replace ifeasible (Lamarckia) do ot replace ifeasible (Darwiia / Baldwiia) Special Represetatio ad Operators preserve feasibility due to differet represetatio, special operators eeded e.g. the TSP with EAs Seperatio of Costraits ad Objectives costraits ad objectives hadled seperately multi-objective cocepts co-evolutio superiority of feasibles Mappig Costrait Hadlig usig decoders to map solutios from the ifeasible regio ito the feasible regio i some cases, special operators to produce solutios that lie o the boudary of the feasible regio have bee defied mai idea is to trasform the whole feasible regio ito a differet shape which is easier to explore by the NIH 5

6 0/ Sigle Kapsack Problem Example Problem: Multidimesioal Kapsack Problem with EAs (MKP) defiitio: sigle kapsack of capacity C ad items each object has weight w i profit p i fid a vector x=(x,x 2,...x ) where x i {0,} such that: w x C i i for which P( x) = p x i i is maximized MKP - Defiitio MKP is a geeralizatio of the 0/ kapsack problem m kapsacks of capacities c,...,c m objects with profits p,...,p each object has m possible weights object i has weight w ij whe cosidered for iclusio i the jth kapsack MKP - Defiitio objective: fid a vector x=(x,...,x ) that guaratees o kapsacks are overfilled ad yields maximum profit solutio lies close to boudary of feasible regio max p x i i subject to w x c ij i j for j =,2,..., m Real-World Applicatio Examples diet problem differet food items m differet elemets (vitami A, B,..., magesium,..., calories etc) each elemet has lower ad / or upper limit for itake each food item has cost objective is to miimize cost ad adhere to utritioal requiremets Real-World Applicatio Examples selectig projects to fud differet projects pla for m years budget determied for each year each project provides a profit objective is to maximize profit ad ot exceed yearly budgets 6

7 MKP - Costrait Hadlig direct search i the complete search space pealty fuctios direct search i the feasible search space clever iitializatio repair ad local search idirect search i the feasible search space permutatio represetatio ordial represetatio real valued represetatio radom keys weight codig MKP - Represetatios biary represetatio if ith positio is: : ith item icluded i all kapsacks 0: ith item is ot icluded i ay of the kapsacks a strig may lead to ifeasible solutio cadidates permutatio / ordial represetatio strig shows order to iclude items real valued represetatio shows a heuristic / radom weight for icludig each item radom keys weight codig Pealty Fuctio used with biary represetatio pealty approach proposed by Khuri et al [Khuri,994] allows ifeasible strigs to joi populatio applies pealty to reduce fitess of ifeasible strig pealty term gets higher whe solutio farther away from feasibility Pealty Fuctio ew graded fitess fuctio defied - f(x) pealty depeds o umber of overfilled kapsacks i.e. umber of violated costraits f ( x) = ( pixi ) s.max( pi ) where s is o. of overfilled kapsacks! Does ot guaratee all feasible solutios. Check! Pealty Fuctio better pealty fuctio which guides populatio towards boudary of feasible regio proposed by Gottlieb [Gottlieb, 999] ew graded fitess fuctio defied - f(x) pealty depeds o amout of costrait violatio p f ( x) = ( pi xi ) w max + max{ CV ( x, i) i I} mi Repair ad Local Search ifeasibles repaired by removig items radomly based o profit-weight ratios locally improve foud good solutios Chu-Beasley, 998 7

8 Permutatio Represetatio cosiders permutatios of all items a first-fit algorithm is used to decode a permutatio ito a feasible solutio starts with the feasible solutio x = (0,..., 0) cosiders each item i the order determied by the permutatio each correspodig decisio variable icreased from 0 to if the iclusio of item does ot violate ay capacity costraits all permutatio operators may be used uiform order based crossover ad swap mutatio reported as good Ordial Represetatio solutio represeted by a vector v = (v,..., v) with vkε {,..., k + } for k ε {,..., }. vector mapped to a permutatio of the items {,..., }, which is further decoded to a feasible solutio via a first-fit heuristic example, assume v = (3,, 2, ) iitially, the ordered list L = (, 2, 3, 4) vector v is decoded by successively removig the elemets 3,, 4, 2 from L yieldig the permutatio (3,, 4, 2) classical operators may be used poor performace Radom Key Represetatio based o real-valued vectors w = (w,...,w), where each item j is assiged a weight wjε[0, ] decoder sorts all items accordig to their weights, which yields a permutatio i icreasig order of weights the permutatio is decoded via the first-fit heuristic two poit crossover ad gaussia mutatio have bee used performace reported to be iferior to permutatio represetatio Weight Biased Represetatio solutio represeted by a vector of realvalued weights to obtai pheotype from weight vector: first, the origial problem P is temporarily modified to P by biasig certai problem parameters accordig to the weights secodly, a problem specific heuristic is used to derive a solutio for P solutio to P is iterpreted ad evaluated for the origial (ubiased) problem P Weight Biased Represetatio classical positioal crossover ad mutatio operators ca be used whe a suitable biasig scheme ad decodig heuristic is used, oly feasible cadidate solutios are created weight-codig approach [Raidl 999] 8