Travelling Salesman Problem: Convergence Properties of Optimization Algorithms

Travelling Salesman Problem: Convergence Properties of Optimization AlgorithmsGroup 2

Zachary EstradaChandini JainJonathan LaiIntroductionMarcus Peinado and Thomas Lengauer. `go with the winners' generators with applications to molecular modeling. RANDOM, pages 135{149, 1997.ABCDFETravelling Salesman ProblemSurface ReconstructionHierarchy of Optimization Methods

Hamiltonian DescriptionWhere,ri is the position of particleiVk is the number of vertices particlei is connected toV0 is the actual number of vertices particlei should be connected tokb = 1, bond constantkv = 1024, vertex constant

Penalizes vertices with connections unequal to required connection Test Systems

Honeycomb LatticeSquare Lattice Random Positions

Square LatticeShearedHoneycomb LatticeJava HeavyliftingCross-PlatformAbstractionTDD - JUnitThreadsPython AnalysisTcl (VMD) AnalysisAwk Input File

Code ImplementationSimulated Annealing: Controlled Cooling

"Optimization by Simulated Annealing" S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, Science 13 May 1983: 220 (4598), 671-680.

Simulated Annealing Moves http://mathbits.com/mathbits/studentresources/graphpaper/graphpaper.htm

Simulated Annealing Moves http://mathbits.com/mathbits/studentresources/graphpaper/graphpaper.htm

Simulated Annealing Moves http://mathbits.com/mathbits/studentresources/graphpaper/graphpaper.htm

Simulated Annealing Moves http://mathbits.com/mathbits/studentresources/graphpaper/graphpaper.htm

http://en.wikipedia.org/wiki/File:Aco_branches.svgAnt Colony

Pheromone Matrix

Each Ant Constructs a Solution

Update Pheromone Matrix

Ant Colony - Implementation

Update pheromones based on energyAnt Colony Implementation (contd..)

Update pheromones based on energy:Accept moves during walk with probability:Ant Colony Implementation (contd..)

Ant Colony Discussion

Ant Colony Discussion

Ant Colony Discussion

Ant Colony DiscussionGenetic Algorithms: Survival of the Fittest110111000101001011101101110001110011001110111Generate an initial random populationEvaluate fitness of individualsSelect parents for crossover based on fitnessPerform crossover to produce childrenMutate randomly selected childrenIntroduce children into the population and replace individuals with least fitnessA genetic algorithm tutorial, Darrell Whitley , Statistics and Computing, Volume 4, Number 2, 65-85, DOI: 10.1007/BF00175354Genetic Algorithm: Generation RulesSelection - Fitness proportionate/roulette-wheel selection:Area of the wheel assigned to each parent in proportion to fitnessCrossover - Matrix Crossover Variant: Select a column M at random and interchange column data between parentsAfter interchange, Vk> V0 for any particle, disconnect from farthest neighborMutation - 2-Opt Operator Variant: Connect all particles between two randomly chosen points i1 and i2 with a randomly chosen neighborAfter interchange, Vk> V0 for any particle, disconnect from farthest neighborGenetic Algorithm: Energy v/s Iterations

Hexagonal Lattice

CrossoverMutationGenetic Algorithm: Energy v/s Iterations

Sheared Hexagonal Lattice

CrossoverMutationGenetic Algorithm: Energy v/s Iterations

Square Lattice Random Positions

CrossoverMutationGo With The Winners

Go With The WinnersGWTW Simulated Annealing with survival of fittestMoves are predeterminedCreate/destroy bondsSwap bonds to explore phase space fasterSurvival of the fittestSelect single winner of systemKill off lower half of populationRepopulate single winner cloneHoneycomb Lattice: ComparisonSimulated AnnealingAnt ColonyOptimizationGenetic AlgorithmGo With theWinnerAvg Energy535.967999535.967999703.6449535.967999Best Energy535.967999535.967999535.967999535.967999Avg Run Time (s)79791131422Avg Iterations800000942800400000

All AlgorithmsBest SolutionsSquare Lattice: Comparison

Simulated AnnealingAnt ColonyOptimizationGenetic AlgorithmGo With theWinnerAvg Energy12774504501518Best Energy1277450450450Avg Run Time (s)1093241131713Avg Iterations8000001282800400000

Simulated AnnealingOther AlgorithmsBest SolutionsAnt Colony OptimizationGenetic AlgorithmGo With the Winner

Sheared Hexagonal Lattice: ComparisonBest SolutionsAnt ColonyOptimizationGenetic AlgorithmGo With theWinnerAvg Energy554.928606.89764962.64Best Energy554.928554.928962.64Avg Run Time (s)4155940Avg Iterations2433100400000Simulated AnnealingAnt ColonyOptimizationGenetic AlgorithmGo With theWinnerAvg Energy2999.99463.1672.7627082708.94068Best Energy2999.99463.1612.1811652571.18136Avg Run Time (s)8217151871435Avg Iterations80000051014200400000Square Lattice-Random Positions: ComparisonBest Solutions

Simulated AnnealingAnt Colony OptimizationGenetic AlgorithmGo With the Winner

Comparison Between Solutions

Square Lattice - Random PositionsACOGA

Connections634 similar 270 unique

138 : 132 Left v/s RightSample Simulation ACO with Honeycomb

ConclusionACO outperforms other algorithms in all test casesGA generates multiple solutions search space explorationSA is not very efficient for this problemGWTWs accelerates convergence of SA methods; also yields lower energy solutionsChoice of move is essential for efficient computationMust highly tune code to runThank You