MonteCarlo Optimization (Simulated Annealing)

1. MonteCarlo Optimization(Simulated Annealing) Mathematical Biology Lecture 6 James A. Glazier

2. Optimization • Other Major Application of Monte Carlo Methods is to Find the Optimal (or a nearly Optimal) Solution of an Algorithmically Hard Problem. • Given want to find that minimizes f. • Definition: then is a Local Minimum of f. • Definition: then is the Global Minimum of f. • Definition: then f has Multiple Degenerate Global Minima,

3. Energy Surfaces • The Number and Shape of Local Minima of f Determine the Texture of the ‘Energy Surface,’ also called the Energy Landscape, Penalty Landscape or Optimization Landscape. • Definition: The Basin of Attraction of a Local Minimum, is: • The Depth of the Basin of Attraction is: • The Radius or Size of the Basin of Attraction is: • If no local minima except global minimum, then optimization is easy and energy surface is Smooth.

4. Energy Surfaces • If multiple local minima with large basins of attraction, need to pick an In each basin, find the corresponding and pick the best. Corresponds to enumerating all states if is a finite set, e.g. TSP. • If many local minima or minima have small basins of attraction, then the energy surface is Rough and Optimization is Difficult. • In these cases cannot find global minimum. However, often, only need a ‘pretty good’ solution.

5. Monte Carlo Optimization • Deterministic Methods, e.g. Newton-Rabson ( ) Only Move Towards BetterSolutions and Trap in Basins of Attraction. Need to Move the Wrong Way Sometimes to Escape Basins of Attraction (also Called Traps). • Algorithm: • Choose a • Start at Propose a Move to • If • If Where,

6. Monte Carlo Optimization—Issues • Given an infinite time, the pseudo-random walk Will explore all of Phase Space. • However, you never know when you have reached the global minimum! So don’t know when to Stop. • Can also take a very long time to escape from Deep Local Basins of Attraction. • Optimal Choice of g(x) and dwill depend on the particular • If g(x)1 forx<x0, then will algorithm will not see minima with depths less thanx0 . • A standard Choice is the Boltzmann Distribution, g(x)=e-x/T, Where T is the Fluctuation Temperature. The Boltzmann Distribution has right equilibrium thermodynamics, but is NOT an essential choice in this application).

7. Temperature and d • Bigger T results in more frequent unfavorable moves. • In general, the time spent in a Basin of Attraction is ~ exp(Depth of Basin/ T). • An algorithm with these Kinetics is Called an Activated Process. • Bigger T are Good for Moving rapidly Between Large and Deep Basins of Attraction but Ignore Subtle (less than T) Changes in • Similarly, large d move faster, but can miss deep minima with small diameter basins of attraction. • A strategy for picking T is called an “Annealing Schedule.”

8. Annealing Schedules • Ideally, want time in all local minimum basins to be small and time in global minimum basin to be nearly infinite. • A fixed value of T Works if depth of the basin of attraction of global minimum>>depth of the basin of attraction of all local minima and radius of the basin of attraction of global minimum~radius of the largest basin of attraction among all local minima. • If so, pick T between these two depths. • If multiple local minima almost degenerate with global minimum, then can’t distinguish, but answer is almost optimal. • If have a deep global minimum with very small basin of attraction (golf-course energy). Then no method helps!

9. Annealing Schedules • If Energy Landscape is Hierarchical or Fractal, then start with large T and gradually reduce t. • Selects first among large, deep basins, then successively smaller and shallower ones until it freezes in one. • Called “Simulated Annealing.” • No optimal choice of Ts. • Generally Good Strategy: • Start with T~ Df/2, if you know typical values of Df for a fixed stepsize d, or T~ typical f, if you do not. • Run until typical Df <<T. Then set T=T/2. Repeat. • Repeat for many initial conditions. • Take best solution.

10. Example—The Traveling Salesman Problem • Simulated Annealing Method Works for Algorithmically Hard (NP Complete) problems like the Traveling Salesman problem. • Put down N points in some space: • Define an Itinerary: • The Penalty Function or Energy or Hamiltonian is the Total Path Length for a Given Itinerary:

11. Example—The TSP (Contd.) • Pick any Initial Itinerary. • At each Monte Carlo Step, pick: • If the initial Itinerary is: • Then the Trial Itinerary is the Permutation: • Then • Apply the Metropolis Algorithm. • A Good Initial Choice of T is: • This Algorithm Works Well, Giving a Permutation with H within a Percent or Better if the Global Optimum in a Reasonable Amount of Time.