Simulated Annealing

1. Simulated Annealing G.Anuradha

2. What is it? • Simulated Annealing is a stochastic optimization method that derives its name from the annealing process used to re-crystallize metals • Comes under the category of evolutionary techniques of optimization

3. What is annealing? • Annealing is a heat process whereby a metal is heated to a specific temperature and then allowed to cool slowly. This softens the metal which means it can be cut and shaped more easily

4. What happens during annealing? • Initially when the metal is heated to high temperatures, the atoms have lots of space to move about • Slowly when the temperature is reduced the movement of free atoms are slowly reduced and finally the metals crystallize themselves

5. Relation between annealing and simulated annealing • Simulated annealing is analogous to this annealing process. • Initially the search area is more, there input parameters are searched in more random space and slowly with each iteration this space reduces. • This helps in achieving global optimized value, although it takes much more time for optimizing

6. Analogy between annealing and simulated annealing • Annealing • Energy in thermodynamic system • high-mobility atoms are trying to orient themselves with other nonlocal atoms and the energy state can occasionally go up. • low-mobility atoms can only orient themselves with local atoms and the energy state is not likely to go up again. • Simulated Annealing • Value of objective function • At high temperatures, SA allows fn. evaluations at faraway points and it is likely to accept a new point. • At low temperatures, SA evaluates the objective function only at local points and the likelihood of it accepting a new point with higher energy is much lower.

7. Cooling Schedule • how rapidly the temperature is lowered from high to low values. • This is usually application specific and requires some experimentation by trial-and-error.

8. Fundamental terminologies in SA • Objective function • Generating function • Acceptance function • Annealing schedule

9. Objective function • E = f(x), where each x is viewed as a point in an input space. • The task of SA is to sample the input space effectively to find an x that minimizes E.

10. Generating function • A generating function specifies the probability density function of the difference between the current point and the next point to be visited. • is a random variable with probability density function g(∆x, T), where T is the temperature.

11. Acceptance function • Decides whether to accept/reject a new value of xnew • Where c – system dependent constant, T is temperature, ∆E is –ve SA accepts new point ∆E is +ve SA accepts with higher energy state Initially SA goes uphill and downhill

12. Annealing schedule • decrease the temperature T by a certain percentage at each iteration.

13. Steps involved in general SA method

14. Steps involved in general SA method Gaussian probability density function-Boltzmann machine is used in conventional GA

15. Travelling Salesman Problem • In a typical TSP problem there are ‘n’ cities, and the distance (or cost) between all pairs of these cities is an n x n distance (or cost) matrix D, where the element dij represents the distance (cost) of traveling from city i to city j. • The problem is to find a closed tour in which each city, except for starting one, is visited exactly once, such that the total length (cost) is minimized. • combinatorial optimization; it belongs to a class of problems known as NP-complete

16. TSP • Inversion: Remove two edges from the tour and replace them to make it another legal tour.

17. TSP • Translation Remove a section (8-7) of the tour and then replace it in between two randomly selected consecutive cities 4 and 5).

18. TSP • Switching: Randomly select two cities and switch them in the tour

19. Put together

20. SA(Extracts from Sivanandem) Step 1:Initialize the vector x to a random point in the set φ Step 2:Select an annealing schedule for the parameter T, and initialize T Step 3:Compute xp=x+Δx where x is the proposed change in the system’s state Step 4:Compute the change in the cool Δf=f(xp)-f(x)

21. Algocontd…. • Step 5: by using metropolis algorithm, decide if xp should be used as the new state of the system or the new state of the system or keep the current state x. Where T replaces kbT. When Δf>=0 a random number is selected from a uniform distribution in the range of [0 1]. If (x xp) > n the state xp is used as the new state otherwise the state remains at x.

22. Algocontd…. • Step 6: Repeat steps 3-5 n number of times • Step 7: If an improvement has been made after the n number of iterations, set the centre point of be the best point • Step 8:Reduce the temperature • Step 9: Repeat Steps 3-8 for t number of temperatures

23. Random Search • Explores the parameter space of an objective function sequentially in a random fashion to find the optimal point that maximizes or minimizes objective function • Simple • Optimization process takes a longer time

24. Primitive version (Matyas)

25. Observations in the primitive version Leads to reverse step in the original method Uses bias term as the center for random vector

26. Modified random search

27. Initial bias is chosen as a zero vector • Each component of dx should be a random variable having zero mean and variance proportional to the range of the corresponding parameter • This method is primarily used for continuous optimization problems

28. Downhill Simplex Method (Nelder-Mead) • Keep track of n+1 points in n dimensions • Vertices of a simplex (triangle in 2D tetrahedron in 3D, etc.) • At each iteration: simplex can move,expand, or contract • Sometimes known as amoeba method:simplex “oozes” along the function

29. Downhill Simplex Method (Nelder-Mead) • Basic operation: reflection location probed byreflection step worst point(highest function value)

30. Downhill Simplex Method (Nelder-Mead) • If reflection resulted in best (lowest) value so far,try an expansion • Else, if reflection helped at all, keep it location probed byexpansion step

31. Downhill Simplex Method (Nelder-Mead) • If reflection didn’t help (reflected point still worst) try a contraction location probed bycontration step

32. Downhill Simplex Method (Nelder-Mead) • If all else fails shrink the simplex aroundthe best point

33. Downhill Simplex Method (Nelder-Mead) • Method fairly efficient at each iteration(typically 1-2 function evaluations) • Can take lots of iterations • Somewhat flakey – sometimes needs restart after simplex collapses on itself, etc. • Benefits: simple to implement, doesn’t need derivative, doesn’t care about function smoothness, etc.