Advanced Computer Graphics

1. Advanced Computer Graphics Spring 2002 Professor Brogan

2. Simulated Annealing • Monte Carlo approach for minimizing multivariate functions • Monte Carlo = Random = Stochastic • Requires one ‘goodness’ metric – result of evaluation function • Multivariate – selects multiple parameter values to minimize evaluation function

3. Algorithm Outline • Select some initial guess of evaluation function parameters: x0 • Evaluate evaluation function, E(x0)=v • Compute a random displacement, x’0 • The Monte Carlo event • Evaluate E(x’0) = v’ • If v’ < v; set new state, x1 = x’0 • Else set x1 = x’0 with Prob(E,T) • This is the Metropolis step • Repeat with updated state and temp

4. What is Annealing? • Used to treat work-hardened parts made out of low-carbon steels • Heat to a specific temperature, then soak, and then cool slowly • Thermodynamics – molecules can move around when they are at high temps. Slow cooling permits self organization into minimum energy configurations

5. Back to our Situation • We approximate nature’s alignment of molecules by allowing uphill transitions • exp (-E/kT) Boltzmann Probabilty Distribution • Even when T is small, a disruption is possible • exp (-(E2-E1) / kT) Metropolis Step • The rate at which T is decreased and the amount it is decreased is prescribed by an annealing schedule

6. What have we got? • Always move downhill if possible • Sometimes go uphill • Optimality guaranteed with slow annealing schedule • No need for smooth search space • No derivatives • Can be discrete search space • Traveling salesman problem

7. Optimization • Given: • What value of minimizes f()? • Expensive to computer f()? • Expensive to compute partials of f()=Jacobian? • Global vs. local solutions

8. Constrained Optimization • Minimize • Subject to: • There are a priori limitations on the possible values of the independent variables

9. Linear Programming • Special type of constrained optimization • Function f() and constraints are linear