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Travelling Salesman Problem: Convergence Properties of Optimization Algorithms Group 2 Zachary Estrada Chandini Jain Jonathan Lai

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Introduction 1.Marcus Peinado and Thomas Lengauer. `go with the winners' generators with applications to molecular modeling. RANDOM, pages 135{149, 1997. A B C D F E Travelling Salesman ProblemSurface Reconstruction

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Hierarchy of Optimization Methods

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Hamiltonian Description Where, r i is the position of particle i V k is the number of vertices particle i is connected to V 0 is the actual number of vertices particle i should be connected to k b = 1, bond constant k v = 1024, vertex constant Penalizes vertices with connections unequal to required connection

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Test Systems Honeycomb Lattice Square Lattice Random Positions Square Lattice Sheared Honeycomb Lattice

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Java Heavylifting Cross-Platform Abstraction TDD - JUnit Threads Python Analysis Tcl (VMD) Analysis Awk Input File Code Implementation

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Simulated Annealing: Controlled Cooling 1."Optimization by Simulated Annealing" S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, Science 13 May 1983: 220 (4598), 671-680.

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

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

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

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

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1.http://en.wikipedia.org/wiki/File:Aco_branches.svg Ant Colony

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Pheromone Matrix Each Ant Constructs a Solution Update Pheromone Matrix Ant Colony - Implementation

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Update pheromones based on energy Ant Colony Implementation (contd..)

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

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Ant Colony Discussion

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Genetic Algorithms: Survival of the Fittest 11011100010100101110 1101110001 110011001110111 Generate an initial random population Evaluate fitness of individuals Select parents for crossover based on fitness Perform crossover to produce children Mutate randomly selected children Introduce children into the population and replace individuals with least fitness 1.A genetic algorithm tutorial, Darrell Whitley, Statistics and Computing, Volume 4, Number 2, 65-85, DOI: 10.1007/BF00175354

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Genetic Algorithm: Generation Rules Selection - Fitness proportionate/roulette-wheel selection: -Area of the wheel assigned to each parent in proportion to fitness Crossover - Matrix Crossover Variant: -Select a column M at random and interchange column data between parents -After interchange, V k > V 0 for any particle, disconnect from farthest neighbor Mutation - 2-Opt Operator Variant: -Connect all particles between two randomly chosen points i 1 and i 2 with a randomly chosen neighbor -After interchange, V k > V 0 for any particle, disconnect from farthest neighbor

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Genetic Algorithm: Energy v/s Iterations Hexagonal Lattice Crossover Mutation

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Genetic Algorithm: Energy v/s Iterations Sheared Hexagonal Lattice Crossover Mutation

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Genetic Algorithm: Energy v/s Iterations Square Lattice Random Positions Crossover Mutation

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Go With The Winners

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GWTW Simulated Annealing with survival of fittest Moves are predetermined Create/destroy bonds Swap bonds to explore phase space faster Survival of the fittest Select single winner of system Kill off lower half of population Repopulate single winner clone

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Honeycomb Lattice: Comparison Simulated Annealing Ant Colony Optimization Genetic Algorithm Go With the Winner Avg Energy 535.967999 703.6449535.967999 Best Energy 535.967999 Avg Run Time (s) 79791131422 Avg Iterations 800000942800400000 All Algorithms Best Solutions

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Square Lattice: Comparison Simulated Annealing Ant Colony Optimization Genetic Algorithm Go With the Winner Avg Energy 1277450 1518 Best Energy 1277450 Avg Run Time (s) 1093241131713 Avg Iterations 8000001282800400000 Simulated AnnealingOther Algorithms Best Solutions

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Ant Colony OptimizationGenetic AlgorithmGo With the Winner Sheared Hexagonal Lattice: Comparison Best Solutions Ant Colony Optimization Genetic Algorithm Go With the Winner Avg Energy 554.928606.89764962.64 Best Energy 554.928 962.64 Avg Run Time (s) 4155940 Avg Iterations 2433100400000

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Simulated Annealing Ant Colony Optimization Genetic Algorithm Go With the Winner Avg Energy 2999.99463.1672.7627082708.94068 Best Energy 2999.99463.1612.1811652571.18136 Avg Run Time (s) 8217151871435 Avg Iterations 80000051014200400000 Square Lattice-Random Positions: Comparison Best Solutions Simulated AnnealingAnt Colony OptimizationGenetic AlgorithmGo With the Winner

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Comparison Between Solutions Square Lattice - Random Positions ACO GA Connections 634 similar 270 unique 138 : 132 Left v/s Right

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Sample Simulation ACO with Honeycomb

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Conclusion ACO outperforms other algorithms in all test cases GA generates multiple solutions search space exploration SA is not very efficient for this problem GWTWs accelerates convergence of SA methods; also yields lower energy solutions Choice of move is essential for efficient computation Must highly tune code to run

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Thank You

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