Simulated Annealing G.Anuradha

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

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

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

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

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.

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.

Fundamental terminologies in SA Objective function Generating function Acceptance function Annealing schedule

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.

Generating function

Acceptance function Decides whether to accept/reject a new value of x new 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

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

Steps involved in general SA method

Gaussian probability density function-Boltzmann machine is used in conventional GA

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

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

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

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

Put together

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)

Algo contd….

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

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

Primitive version (Matyas)

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

Modified random search

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

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

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

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 by expansion step

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

Downhill Simplex Method (Nelder- Mead) If all else fails shrink the simplex around the best point

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.