PHYCS 6730 Lab Exercise: Nonlinear Optimization: Simulated Annealing

These exercises experiment with simulated annealing using the Metropolis et al algorithm. The files you need are in ~p6730/exercises/nonlinear_optimization.

Exercise 1 Sample cost function

We will use the sample cost function 
   f(x) = (4 - x2)2 + x;
Plot this function with your favorite plotting utility. Where are the minima? Which is the global minimum? What is the barrier height between them?

Please put your answers in the answer file for this exercise.

Exercise 2. Metropolis et al algorithm

The code ~p6730/exercises/nonlinear_optimization/metropolis does a Metropolis et al search for the minimum, based on these parameters:

  1. Random number seed. Needed to initialize the pseudorandom number sequence.
  2. Starting coordinate x.
  3. Number of trial steps to take.
  4. Temperature. For simulated annealing, discussed later.
  5. Step scale in x. Controls the step size.
Run the program metropolis with the values in in2, arranged in the order of the prompts. The x values generated on standard output should give a histogram that resembles the distribution exp(-f(x)/5), suitably normalized. Check this. See hist documentation for a histogramming tool.

In your answer file compare the observed ratio of counts in the histogram at x = 0 and x = 2 with what you should get from the expected probability distribution. A rough estimate is sufficient.

Exercise 3. Simulated annealing

Let's start from the false minimum around x = 2 and see if we end up at the true minimum as we cool the system down slowly. A example of the cooling process is given in in3, but the temperature is not cool enough and the number of trials is too small. Experiment with the cooling process to see if the system ends up in the true global minimum after cooling to a temperature of 0.1. You should adjust the step scale so the acceptance rate is kept around 50%.

In your answer file give your chosen cooling schedule and describe the result.