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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>

''...bringing a fluid into a low-energy state such as growing a crystal, has been considered...to be similar to the process of finding an optimum solution of a combinatorial optimization problem.  Annealing is a well-known process for growing crystals.  It consists of melting the fluid and then lowering the temperature slowing until the crystal is formed.  The rate of the decrease of temperature has to be very low around the freezing temperature.  The Metropolis Monte Carlo method...can be used to simulate the annealing process.  It has been proposed as an effective method for finding global minima of combinatorial optimization problems.''[3]

''In particular, bringing a fluid into a low-energy state such as growing a crystal, has been considered in [11] to be similar to the process of finding an optimum solution of a combinatorial optimization problem.  Annealing is a well-known process for growing crystals.  It consists of melting the fluid and then lowering the temperature slowing until the crystal is formed.  The rate of the decrease of temperature has to be very low around the freezing temperature.  The Metropolis Monte Carlo method [1], [14] can be used to simulate the annealing process.  It has been proposed as an effective method for finding global minima of combinatorial optimization problems.''

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).