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## Simulated Annealing

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**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 metals • 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 slow 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**• 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.**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**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**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)**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.**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**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**Observations in the primitive version**Leads to reverse step in the original method Uses bias term as the center for random vector**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