Design & Analysis of Algorithms Combinatory optimization SCHOOL OF COMPUTING Pasi Fränti

Local search Stochastic variations of local search Genetic algorithms Swarm Intelligence Optimization techniques

Here decision tree picture!!! Optimization techniques in context

Local search

Select one and move Main principle of local search

Structure of local search

Representation of solution Neighborhood function Search strategy Components of local search

Study neighbor solutions Movement in neighborhood

Accept only better solutions Hill climbing

Local and global maxima

Combining local search and hill-climbing

Represent solution as bit string (x 1 x 2,…x n ), where x i  {0,1}. Problem instance: w i = (2,3,5,7,11), S=15. Solution with elements 2,3 and 7 is represented as Local search for knapsack

Single bit change: 0  1 or 1  0 S=15 W=[2, 3, 5, 7, 11] Move in knapsack

Two operations: 0  1 or 1  0 Swap bit location Extended neighborhood S=15 W=[2, 3, 5, 7, 11] 10011

Getting stuck into local maximum S=15 W=[2, 3, 5, 7, 11]

Prevents search to return previously visited solutions Select the next best Tabu! S=15 W=[2, 3, 5, 7, 11] Tabu search

Tabu search (2 nd iteration) S=15 W=[2, 3, 5, 7, 11]

Traveling salesman problem...  p i-1  p i  p i+1   p i-1  p i+1  p i   p i  p i-1  p i+1   p i  p i+1  p i-1   p i+1  p i  p i-1   p i+1  p i-1  p i ... Permute local changes in given route

Local search algorithm for TSP

TSP example E  F  G  H  A4 +   = 2  + 6 E  F  H  G  A = 11 min! E  G  F  H  A  +   = 3  + 3 E  G  H  F  A   = 2  + 5 E  H  G  F  A  +  = 2  + 5 E  H  F  G  A  + 2= 1  + 6

Genetic algorithm

Genetic algorithm (GA) Needs more material!

Generate a set of initial solutions. REPEAT Generate new solutions by crossover. Mutate the new solutions (optional). Evaluate the candidate solutions. Retain best candidates and delete the rest. UNTIL stopping criterion met. Main structure of GA

Permuting pairs for crossover Elitist approach using zigzag scanning among the best solutions

Optimizing chess playing Revise

Tic-tac-toe example

Evaluation function for tic-tac-toe

Minmax example Redraw

Minmax playing: Min’s move

Minimax maximizes the worst-case outcome for max Minmax playing: Max’s move

Chess Game tree

Beyond the horizon

Evaluating Chess position

Positional factors

Initial value range

Result of optimization

Swarm intelligence

Social intelligence: individual behavior maybe naive but joint effect can be intelligent. Decentralized: no central control of the individuals of the colony Self-organized: individual adapts to environment and other members of colony Robust: Task is completed even if some individuals fail Swarm intelligence (SI)

Ant colony optimization (ACO) Main principle: Emitting pheromone between nest and food Joint efforts to carry loads Solving TSP by ants: Sending ants to make randomized tours Short links chosen more often than long ones Good tracks are marked by pheromone Tracks with high pheromone chosen more often

Ant colony optimization Initialization: Generate randomized tours. Smaller links chosen more often than longer ones Simulate pheromone: Subtract cost of the links in best solution (-1) Increase the ones in the worst solution (+1) Tours are evaluated using the original graph.

Ant colony optimization (1 st round)

Ant colony optimization (2 nd round)