Monte Carlo methods

Model: Monte-Carlo tries to solve deterministic problems using optimizations based on probabilistic interpretations. It draws samples from a probability distribution. Simulated Annealing is a special case of Monte-Carlo. When trials and errors are scattered in their results, an objective function that can measure the cost or benefit will help with convergence. If the samples are large, a batch analysis mode is recommended. The approach to minimize or maximize the objective function is also possible via gradient descent methods but the use of simulated annealing can overcome local minimum even if the cost is higher because it will accept with a certain probability. In Simulated annealing, the current cost is computed, and the new cost is based on the direction of change. If the cost improves, the temperature decreases.

 

Sample implementation follows:

def annealingoptimize(domain,costf,T=10000.0,cool=0.95,step=1):
     # Initialize the values randomly
     vec=[float(random.randint(domain[i][0],domain[i][1]))
          for i in range(len(domain))]
     while T>0.1:
          # Choose one of the indices
          i=random.randint(0,len(domain)-1)
          # Choose a direction to change it
          dir=random.randint(-step,step)
          # Create a new list with one of the values changed
          vecb=vec[:]
          vecb[i]+=dir
          if vecb[i]<domain[i][0]: vecb[i]=domain[i][0]
          elif vecb[i]>domain[i][1]: vecb[i]=domain[i][1]

          # Calculate the current cost and the new cost
          ea=costf(vec)
          eb=costf(vecb)
          p=pow(math.e,(-eb-ea)/T)
          # Is it better, or does it make the probability
          # cutoff?
          if(eb<ea or random.random( )<p):
               vec=vecb
          # Decrease the temperature
          T=T*cool
     return vec