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== Introduction to Temperature Scheduling ==

== Introduction to Temperature Scheduling ==

<font; color="red"> Add description featured in D. Mitra et al about crystal growing annealing </font>

<font; color="red"> Add description featured in D. Mitra et. al. about crystal growing annealing </font>

Temperature scheduling in simulated annealing refers to the process of controlling the ''temperature'' in a particular run through either geometric or adaptive means.  The effect that temperature has on an annealing run is that it determines the probability that a solution with a higher cost function value will be accepted over one with a lower value, which is known as ''hill climbing''.  There are two main stages of temperature scheduling(warming up and cooling down), which are punctuated by two stopping conditions (equilibrium and frozen criterion respectively).  The warming up stage represents a period during simulated annealing in which the temperature is raised to a high enough point that the probability of the acceptance of a neighboring solution is nearly one.  Such a high probability of acceptance ensures that the final solution does not depend on the initial starting point.  The warming period usually occurs for a predefined number of ''steps'' (known as a chain length).  When the predefined number of steps have occurred, the annealing run has reached the equilibrium point, which simply is the term used to define the point at which warming has ended and cooling will begin.  Continuing the analogy to classical annealing, the cooling stage of the annealing run, corresponds to the period during which the temperature is cooled down to reduce the acceptance ratio of lesser quality solutions.  The cooling phase helps the algorithm to find an ultimate solution.  Depending on the parameters set, the cooling phase is ended by either a set number of steps or a determined acceptance ratio lesser than some value.  This point in the process is known as the freezing point (or frozen criterion).

Temperature scheduling in simulated annealing refers to the process of controlling the ''temperature'' in a particular run through either geometric or adaptive means.  The effect that temperature has on an annealing run is that it determines the probability that a solution with a higher cost function value will be accepted over one with a lower value, which is known as ''hill climbing''.  There are two main stages of temperature scheduling(warming up and cooling down), which are punctuated by two stopping conditions (equilibrium and frozen criterion respectively).  The warming up stage represents a period during simulated annealing in which the temperature is raised to a high enough point that the probability of the acceptance of a neighboring solution is nearly one.  Such a high probability of acceptance ensures that the final solution does not depend on the initial starting point.  The warming period usually occurs for a predefined number of ''steps'' (known as a chain length).  When the predefined number of steps have occurred, the annealing run has reached the equilibrium point, which simply is the term used to define the point at which warming has ended and cooling will begin.  Continuing the analogy to classical annealing, the cooling stage of the annealing run, corresponds to the period during which the temperature is cooled down to reduce the acceptance ratio of lesser quality solutions.  The cooling phase helps the algorithm to find an ultimate solution.  Depending on the parameters set, the cooling phase is ended by either a set number of steps or a determined acceptance ratio lesser than some value.  This point in the process is known as the freezing point (or frozen criterion).

In ''the diamond problem'', a solution consists of values given to two amplitudes and three phases.  The amplitudes represent the intensity of the light from the reference mirror and the diamond {the front and the back only differ by a constant).  The phases represent each of the three surfaces: the reference mirror and the front and back of the diamond.  Each of these quantities is represented by a certain order Legendre polynomial, which is represented in the form of a matrix with the number of rows and columns to match the order with an offset of one.  Since the order Legendre Polynomial for each of the amplitudes and phases is set by the user, the dimensionality of the problem varies depending on the desired order.  Thus, the lowest dimensionality that ''the diamond problem'' could take on would be 11 and the highest would be (cutting the Legendre polynomial order off at five) 125.  However, it is most likely somewhere in between most likely slightly greater than 50, due to wanting to fully exploit the range of the Legendre polynomials to give the greatest accuracy in mapping the diamond surface.

In ''the diamond problem'', a solution consists of values given to two amplitudes and three phases.  The amplitudes represent the intensity of the light from the reference mirror and the diamond {the front and the back only differ by a constant).  The phases represent each of the three surfaces: the reference mirror and the front and back of the diamond.  Each of these quantities is represented by a certain order Legendre polynomial, which is represented in the form of a matrix with the number of rows and columns to match the order with an offset of one.  Since the order Legendre Polynomial for each of the amplitudes and phases is set by the user, the dimensionality of the problem varies depending on the desired order.  Thus, the lowest dimensionality that ''the diamond problem'' could take on would be 11 and the highest would be (cutting the Legendre polynomial order off at five) 125.  However, it is most likely somewhere in between most likely slightly greater than 50, due to wanting to fully exploit the range of the Legendre polynomials to give the greatest accuracy in mapping the diamond surface.

== Convergence and Simulated Annealing ==

==Convergence and Prospective strategies==

*Markov Chains

*Markov Chains

*Random Walk

*Random Walk

*ParSA-->[2]

*ParSA-->[2]

The primary objective for potential cooling strategies lies in the determination of the &alpha; and ''K'' factors given in the ParSA section on improving solution quality in a lesser amount of time.  By tracking both the chain length ''n'' and speed of convergence P(X_n &notin; Cost_min), one can find a linear plot relating all four of the quantities through the following relationship:

The primary objective for potential cooling strategies lies in the determination of the &alpha; and ''K'' factors given in the ParSA section on improving solution quality in a lesser amount of time.  By tracking both the chain length ''n'' and speed of convergence P(X_n &notin; Cost_min), one can find a linear plot relating all four of the quantities through the following relationship:

<math>\ln{P(X_n \not\in Cost_{min})} = \alpha \ln{K} - \alpha \ln{n}</math>

<math>\ln{P(X_n \not\in Cost_{min})} = \alpha \ln{K} - \alpha \ln{n}</math>