**1.1. Annealing**

Annealing is a mechanical process in which material is slowly cooled allowing the molecules to arrange themselves in such a way that the material is less strained thereby making it more stable.

If materials such as glass or metal are cooled too quickly its constituent molecules will be under high stress lending it to failure (breaking) if further thermal or physical shocks are encountered. Slowing the cooling of the material allows each molecule to move into a place it feels most comfortable, i.e., less stress.

© 2012 Karimi and Verki, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Karimi and Verki, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

High Temperature Movements Thermal Equilibrium

Mean Field Annealing Based Techniques for Resolving VLSI Automatic Design Problems 5

<sup>→</sup> ) ���

(2)

,�� <sup>=</sup> ���(��

satisfaction of the problem constraints; this process is termed 'mapping' the problem onto

Where *E* is energy function in term of �� and �� is *Hopfield term* of energy function. Totally Eq. 2.is motion (updating) equation of state of neurons and its output is ��.Usually a simple nondecreasing monatomic output function in term of �� like �(��) is applied torelate �� to the states. Typically this function is a step function or a hyperbolic tangent function. � is a constant number as the weighting factor of ��.Thereforea Hopfield Neural Network minimizes a cost function that is encoded with its weights by implementation of *gradient* 

As it mentioned before, MFA merges collective computation and annealing properties of Hopfield neural Networks and SA, respectively, to obtain a general algorithm for solving combinatorial optimization problems. MFA can be used for solving a combinatorial optimization problem by choosing a representation scheme in which the final states of the discrete variables (spins or neurons) can be decoded as a solution to the problem. In fact the space of problem is mapped to the space of MFA variables (spins) and there will be a one-toone relation between two spaces. This is called *encoding*. Then, an energy function is formulated in term of spins with a structure that is based on essence of problem whose global minimum value corresponds to an optimum solution of the problem. MFA is expected to compute the optimum solution to the target problem, starting from a randomly chosen initial state, by minimizing this energy function. Steps of applying MFA technique to

1. Choose a representation plan which encodes the configuration space of the target optimization problem using spins. In order to get a good performance, number of possible configurations in the problem domain and the spin domain must be equal. That means there must be a one-to-one mapping between the configurations of spins

2. Formulate the cost function of the problem in terms of spins to derive the energy function of the system. Global minimum of the energy function should correspond to

3. Derive the mean field theory equations using formulated energy function. Derive

5. Set suitable parameters of the energy function and the cooling schedule to obtain

These main steps are same for various types of optimization problems and are explained at

equations are used for updating averages (expected values) of spins.

� <sup>−</sup> ��(�� <sup>→</sup> ) ���

the network. Hopfield gives the motion equation of the *ith* neuron:

��� �� <sup>=</sup> ��

*descent.* For more details see [16]

a problem can be summarized as follows:

the global minimum of the cost function.

and the problem.

4. cooling schedule

the following sections.

efficient algorithm.

**2. MFA technique** 

**Figure 1.** Molecules movement at the cooling process

As the material is kept at a high temperature the molecules are able to move around quite freely thus reducing stress on a large scale, indeed if the material is made too hot it will move into the liquid state allowing free movement of the molecules. As the material is cooled the molecules are not able to move around as freely but still move limited distances reducing stress in regional areas. The result is a material with significantly less internal stress and resistant to failure due to external shock.

The statistic mechanic is a domain in physics that describes the process of slow cooling of *Hamiltonian Ising* for particles or spins with high degree of freedom until they accede on their equilibrium states. The particles that are cooling, on solid state, provide a framework to characteristics improvisation of intricate and large systems. Now this idea is stated inside optimization algorithms to resolve various cases of problems.
