Lecture 4: Simulated Annealing. An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved

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Gervais Patterson
5 years ago
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1 Lecture 4: Simulated Annealing Lec04/1

2 Neighborhood based search Step 1: s=s0; Step 2: Stop if terminating condition satisfied; Step 3: Generate N(s); Step 4: s =FindBetterSolution(N(s)); Step 5: s=s ; Step 6: Return to Step 2. In Step 5, the new solution can be worse than the current one! The question is, can we get a better one later? Lec04/2

3 Meta-heuristics used in Tabu Search Short-term memory is used to prevent search from cycling (re-visit the same solution or same region). Medium-term memory is used to intensify the search by investigating some potentially good region in more detail. Long-term memory is used to diversify the search by investigating un-explored regions. These memories are used to control the search process, so that even if we may not find a better solution locally, in a long run we may find the global optimal solution. Lec04/3

4 Simulated annealing Simulated annealing (SA) is another meta-heuristic algorithm for finding the global optimal solutions. Different from TS, SA is memoryless. That is, we do not store the search history to escape from local optimum. The only thing to do is to accept some worse local solution with a certain probability. If we control this probability properly, we can get, theoretically, the global optimal solution. Lec04/4

5 History of SA SA was independently introduced by several persons: S. Kirkpatrick, C. D. Gellatt and M. P. Vecchi in V. Cerny in So far, SA has been applied to Graph partitioning and VLSI design. Solving combinatorial optimization problems. Solving continuous optimization problems. Lec04/5

6 What is annealing? In metallurgy, annealing is a technique to control the cooling process of a material. If the cooling process is properly controlled (slow cooling down), we can crystallize the material well, and obtain materials without defects (e.g. very hard steel). Solid state -> (melt, liquefy) liquid state -> (cool down) -> Another object Lec04/6

7 Why annealing? In fact, high temperature can cause atoms to become unstuck from their initial positions and wander randomly. Cooling down very slowly provides more chances of finding a configuration with lower internal energy (or a new equilibrium state). /press_release/2005/ Lec04/7

8 What is simulated annealing? SA is a computer simulation of the annealing process. We start from a high enough temperature, so that search can have better chance to find the rough position of the global optimal solution. Each potential solution is considered an atom. High temperature allows the solutions to change their positions more easily in the solution space. Cooling down the temperature slowly can obtain a globally good solution. Lec04/8

9 Probability of accepting a solution In SA, we accept a solution s in N(s) un-conditionally if it is better (i.e. f(s ) < f(s) ) than the current solution s. If s is worse (i.e. f(s ) > f(s) ), we accept it with a probability given by p s T = e (f s f s T ) where T is the temperature parameter. It has been proved theoretically (based on Markov chain theory) that SA asymptotically converges to the global optimal solution provided that T is reduced slowly enough. Lec04/9

10 Why accept worse solutions? Go towards the local minimum Escape from the local minimum Escape from global minimum (if T is high) Lec04/10

11 SA algorithm Step 1: Initialize s and T; Step 2: Find N(s); Find a new solution s in N(s) at random. If f(s )<f(s): s=s ; Else s=s with probability p(s T); Step 3: If not equilibrium state, return to Step 2; else, update T; Step 4: If terminating condition satisfied, stop. Lec04/11

12 Initialize the temperature The initial temperature cannot be too high, or search will become random walk. The initial temperature cannot be too low, or search will be done locally. To make a balance, some methods have been proposed in the literature. One method is to select T, such that the acceptance rate of a worse solution is between 0.4 and 0.5. Lec04/12

13 Equilibrium state To see if search is in an equilibrium state at a certain temperature, we need to find a large number of solutions, or transit a large number of times from the starting state. Theoretically, the number of solutions to find for each temperature is exponential to the problem size. In practice, we should make some balance between the computational cost and the solution quality. Lec04/13

14 Equilibrium state The simplest way is to define a number Q before search, and fix it during search. During search, we consider that solution is in equilibrium state after Q transitions. To reduce computational cost, some heuristics can be used. For example, we can cool down the temperature once we get a much better solution, without waiting for the equilibrium state. Lec04/14

15 Cooling control In SA, the temperature is decreased gradually such that T0>T1>T2>..>0 Slow cooling down is necessary to obtain the global optimal solution. In practice, again, we should make some compromise between the computational cost and the solution quality. Lec04/15

16 Strategies for cooling Linear T i = T 0 i x d, i=1,2,.. Geometric T i =at i-1, i=1,2,.. Logarithmic T i =T 0 /log(i), i=1,2,.. In metallurgy, the cooling process cannot be defined so easily like this! Lec04/16

17 Terminating condition The simplest way to stop search is to see if T is already zero or not. In practice, we may stop search if the probability of accepting a worse solution is almost zero. For example, we may stop when T is T min =0.01. Lec04/17

18 Example function y = f(x) y = (4-2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + (-4 + 4*x(2)^2)*x(2)^2; ObjectiveFunction X0 = [ ]; [x,fval,exitflag,output] = simulannealbnd(objectivefunction,x0) x = ( ) fval = Lec04/18

19 Results x = fval = exitflag = 1 output = iterations: 2875 funccount: 2898 message: 'Optimization terminated: change in best function value less than options.tolf ' rngstate: [1x1 struct] problemtype: 'unconstrained' temperature: [2x1 double] totaltime: Lec04/19

20 Homework Tabu search uses several memories to control the search process. SA is memoryless, but may need longer computing time to find good solutions. Do you have some idea to combine TS and SA? That is, to make some balance between memory and time cost? Submit your answer in hard copy before the class of next week. Lec04/20

5. Simulated Annealing 5.1 Basic Concepts. Fall 2010 Instructor: Dr. Masoud Yaghini

5. Simulated Annealing 5.1 Basic Concepts Fall 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Real Annealing and Simulated Annealing Metropolis Algorithm Template of SA A Simple Example References