aco-03-04-2013
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Transcript of aco-03-04-2013
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Ant Colony Optimization: an introductionDaniel Chivilikhin03.04.2013
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OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem
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OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem
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Biological inspiration: from real to artificial ants
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Ant coloniesDistributed systems of social insectsConsist of simple individualsColony intelligence >> Individual intelligence
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Ant CooperationStigmergy indirect communication between individuals (ants)Driven by environment modifications
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Denebourgs double bridge experimentsStudied Argentine ants I. humilisDouble bridge from ants to food source
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Double bridge experiments: equal lengths (1)
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Double bridge experiments: equal lengths (2)Run for a number of trialsAnts choose each branch ~ same number of trials
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Double bridge experiments: different lengths (2)
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Double bridge experiments: different lengths (2) The majority of ants follow the short path
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OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem
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Solving NP-hard combinatorialproblems
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Combinatorial optimizationFind values of discrete variablesOptimizing a given objective function
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Combinatorial optimization = (S, f, ) problem instanceS set of candidate solutionsf objective function set of constraints set of feasible solutions (with respect to )Find globally optimal feasible solution s*
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NP-hard combinatorial problemsCannot be exactly solved in polynomial timeApproximate methods generate near-optimal solutions in reasonable timeNo formal theoretical guaranteesApproximate methods = heuristics
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Approximate methodsConstructive algorithmsLocal search
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Constructive algorithmsAdd components to solution incrementallyExample greedy heuristics: add solution component with best heuristic estimate
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Local searchExplore neighborhoods of complete solutionsImprove current solution by local changesfirst improvementbest improvement
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What is a metaheuristic?A set of algorithmic conceptsCan be used to define heuristic methodsApplicable to a wide set of problems
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Examples of metaheuristicsSimulated annealingTabu searchIterated local searchEvolutionary computationAnt colony optimizationParticle swarm optimizationetc.
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OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem
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The ACO metaheuristic
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ACO metaheuristicA colony of artificial ants cooperate in finding good solutionsEach ant simple agentAnts communicate indirectly using stigmergy
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Combinatorial optimization problem mapping (1)Combinatorial problem (S, f, (t))(t) time-dependent constraintsExample dynamic problemsGoal find globally optimal feasible solution s*Minimization problemMapped on another problem
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Combinatorial optimization problem mapping (2)C = {c1, c2, , cNc} finite set of componentsStates of the problem:X = {x = , |x| < n < +}Set of candidate solutions:
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Combinatorial optimization problem mapping (3)Set of feasible states:
We can complete into a solution satisfying (t)Non-empty set of optimal solutions:
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Combinatorial optimization problem mapping (4) X states S candidate solutions feasible states S* optimal solutions
S*
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Combinatorial optimization problem mapping (5)Cost g(s, t) for each In most cases g(s, t) f(s, t) GC = (C, L) completely connected graphC set of componentsL edges fully connecting the components (connections)GC construction graph
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Combinatorial optimization problem mapping (last )Artificial ants build solutions by performing randomized walks on GC(C, L)
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Construction graphEach component ci or connection lij have associated:heuristic informationpheromone trail
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Heuristic informationA priori information about the problemDoes not depend on the antsOn components ci iOn connections lij ijMeaning: cost of adding a component to the current solution
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Pheromone trailLong-term memory about the entire search processOn components ci iOn connections lij ijUpdated by the ants
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Artificial ant (1)Stochastic constructive procedureBuilds solutions by moving on GCHas finite memory for:Implementing constraints (t)Evaluating solutionsMemorizing its path
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Artificial ant (2)Has a start state xHas termination conditions ekFrom state xr moves to a node from the neighborhood Nk(xr)Stops if some ek are satisfied
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Artificial ant (3)Selects a move with a probabilistic rule depending on:Pheromone trails and heuristic information of neighbor components and connectionsMemoryConstraints
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Artificial ant (4)Can update pheromone on visited components (nodes)and connections (edges)Ants act:ConcurrentlyIndependently
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The ACO metaheuristicWhile not doStop():ConstructAntSolutions() UpdatePheromones() DaemonActions()
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ConstructAntSolutionsA colony of ants build a set of solutionsSolutions are evaluated using the objective function
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UpdatePheromonesTwo opposite mechanisms:Pheromone depositPheromone evaporation
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UpdatePheromones: pheromone depositAnts increase pheromone values on visited components and/or connectionsIncreases probability to select visited components later
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UpdatePheromones: pheromone evaporationDecrease pheromone trails on all components/connections by a same valueForgetting avoid rapid convergence to suboptimal solutions
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DaemonActionsOptional centralized actions, e.g.:Local optimizationAnt elitism (details later)
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ACO applicationsTraveling salesmanQuadratic assignmentGraph coloringMultiple knapsackSet coveringMaximum cliqueBin packing
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OutlineBiological inspiration of ACOSolving NP-hard combinatorial problemsThe ACO metaheuristicACO for the Traveling Salesman Problem
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ACO for the Traveling Salesman Problem
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Traveling salesman problemN set of nodes (cities), |N| = nA set of arcs, fully connecting NWeighted graph G = (N, A)Each arc has a weight dij distanceProblem:Find minimum length Hamiltonian circuit
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TSP: construction graphIdentical to the problem graphC = NL = Astates = set of all possible partial tours
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TSP: constraintsAll cities have to be visitedEach city only onceEnforcing allow ants only to go to new nodes
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TSP: pheromone trailsDesirability of visiting city j directly after i
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TSP: heuristic informationij = 1 / dijUsed in most ACO for TSP
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TSP: solution constructionSelect random start cityAdd unvisited cities iterativelyUntil a complete tour is built
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ACO algorithms for TSPAnt SystemElitist Ant SystemRank-based Ant SystemAnt Colony SystemMAX-MIN Ant System
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Ant System: Pheromone initializationPheromone initializationij = m / Cnn, where:m number of antsCnn path length of nearest-neighbor algorithm
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Ant System: Tour constructionAnt k is located in city i is the neighborhood of city iProbability to go to city :
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Tour construction: comprehension = 0 greedy algorithm = 0 only pheromone is at workquickly leads to stagnation
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Ant System: update pheromone trails evaporationEvaporation for all connections(i, j) L:ij (1 ) ij, [0, 1] evaporation ratePrevents convergence to suboptimal solutions
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Ant System: update pheromone trails depositTk path of ant kCk length of path TkAnts deposit pheromone on visited arcs:
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Elitist Ant SystemBest-so-far ant deposits pheromone on each iteration:
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Rank-based Ant SystemRank all antsEach ant deposits amounts of pheromone proportional to its rank
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MAX-MIN Ant SystemOnly iteration-best or best-so-far ant deposits pheromonePheromone trails are limited to the interval [min, max]
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Ant Colony SystemDiffers from Ant System in three points:More aggressive tour construction ruleOnly best ant evaporates and deposits pheromoneLocal pheromone update
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Ant Colony SystemTour Construction
Local pheromone update:ij (1 )ij + 0,
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Comparing Ant System variants
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State of the art in TSPCONCORDE http://www.tsp.gatech.edu/concorde.html Solved an instance of 85900 citiesComputation took 286-2719 CPU days!
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Current ACO research activityNew applicationsTheoretical proofs
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Further readingM. Dorigo, T. Sttzle. Ant Colony Optimization. MIT Press, 2004.http://iridia.ulb.ac.be/~mdorigo/ACO/
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Next timeSome proofs of ACO convergence
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Thank you!
Any questions?
This presentation is available at:http://rain.ifmo.ru/~chivdan/presentations/Daniel Chivilikhin [mailto: [email protected]]
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Used resourceshttp://teamaltman.com/wp-content/uploads/2011/06/Uncertainty-Ant-Apple-1024x1024.jpg http://myrealestatecoach.files.wordpress.com/2012/04/ant.jpghttp://ars.els-cdn.com/content/image/1-s2.0-S1568494613000264-gr3.jpghttp://www.theorie.physik.uni-goettingen.de/forschung/ha/talks/stuetzle.pdf http://www.buyingandsellingwebsites.com/wp-content/uploads/2011/12/Ant-150.jpghttp://moodle2.gilbertschools.net/moodle/file.php/1040/Event-100_Days/Ant_Hormiga.gifhttp://4.bp.blogspot.com/_SPAe2p8Y-kg/TK3hQ8BUMtI/AAAAAAAAAHA/p75K-GcT_oo/s1600/ant+vision.GIFhttp://1.bp.blogspot.com/-tXKJQ4nSqOY/UQZ3vabiy_I/AAAAAAAAAjA/IK8jtlhElqk/s1600/16590492-illustration-of-an-ant-on-a-white-background.jpg