Global Optimization: Visualizing Heuristic Strategies

1. Global Optimization: Visualizing Heuristic Strategies Rob Dimeo IDL/DAVE Lunchtime Seminar December 14, 2004

2. Conventional Optimization Algorithms • Minimization methods require you choose between: • one that requires only function evaluations… • or one that requires function evaluations and derivatives For functions that have multiple minima these algorithms can get caught in one of the local minima

3. Heuristic Global Optimization Algorithms Many algorithms borrow from a natural paradigm • Simulated Annealing • Genetic Algorithm • Particle Swarm Optimization • Ant Colony Optimization Some are artificial constructs or ad-hoc • Stochastic tree optimization • Stochastic downhill simplex A common problem with global optimization algorithms: Exploration vs. exploitation

4. The Simple Genetic Algorithm Creation of initial population Based on Darwinian survival-of-the-fittest • Search space encoded as chromosomes made of bits • Solutions are bred using rules for reproduction, crossover, and mutation • Population of solutions evolve for some number of generations (undergoing reproduction, crossover, and mutation) and the best fit solution is determined in the last generation • Example: Solution of the 1-d Ising model Fitness (function) evaluation Termination criteria met? yes Done no Selection Crossover Mutation Determination of new population

5. Stochastic Downhill Simplex • Standard downhill simplex (reflections, expansions, and contractions) augmented with a random restart • At conclusion of a series of downhill simplex moves, the simplex is “exploded” and another series of downhill simplex moves proceeds • Example: function minimization

6. Stochastic Tree Search • Start at the original node • Create branches to 2 new nodes • Evaluate the function at each new node (save the minimum). Assign branch probabilities based on function evaluation • In the next iteration begin at the original node • Choose branch based on branch probability until you reach a terminal node • At the terminal node create 2 new nodes • Repeat until some depth has been reached • Enhancement: Specify a branch decay that reduces the probability based on the number of times that node has been visited • One point in search space defines a node • From starting node branches are constructed to two new nodes • Function (c2) evaluated and stored at each new node • Thickness of branch is related to the measure of probability for taking that branch in subsequent iterations Example: Function minimization