Iterative Improvement Algorithms (continued)

• Simulated Annealing
  
  • Named after a metal-casting technique called annealing where molten metal is heated and then gradually cooled resulting in an even distribution of the molecules and a desired crystalline structure
  • Attempts to fix the problem with hill-climbing methods where the search gets stuck in a local maximum (in our case where we are searching for a global minimum, so the problem is getting stuck in a local minimum).
  • Basic idea: Instead of picking the best move, pick a random move; if the successor state obtained by this move is an improvement over the current state, then do it. Otherwise, make the move with some probability < 1. The probability decreases exponentially with the badness of the move.
  • Define a temperature function that decreases over time. At each move, compute the current temperature T, and use T to determine the probability with which to allow a move to a worse state. In the limit, T goes to 0 at which point the method is doing hill-climbing. Hence the probability is proportional to T.
  • If temperature is lowered slowly enough, simulated annealing is complete and admissible. Intuitively, this is the case because the temperature can be controlled so that it is large enough to move off a local maximum, but small enough to not move off a global maximum.
  • Algorithm
    Assume we are trying to find a state where some evaluation function f is a global maximum:

    current = Initial-State(problem)
    for t = 1 to infinity do
       T = Schedule(t)   ; T is the current temperature, which
    		     ; is monotonically decreasing with t
       if T=0 then return current  ; halt when temperature = 0
       next = Select-Random-Successor-State(current)
       deltaE = f(next) - f(current)  ; If positive, next is
    				  ; better than current.
    				  ; Otherwise, next is
    				  ; worse than current.
       if deltaE > 0 then current = next  ; always move to
    				      ; a better state
       else current = next with probability p = e^(deltaE / T)
    			  ; as T -> 0, p -> 0
    			  ; as deltaE -> -infinity, p -> 0
    end