Simple simulated annealing

Simple Simulated Annealing (SSA) for Single-Objective Problems... 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

The difficulty with this procedure is that simple refinement routines, such as simplex or least squares, lead only to the nearest minimum in the cost function which is unlikely to be the global minimum. The refinement procedure therefore has to be one that randomly samples different parts of configuration space so as to be able to reach different minima, ultimately selecting the global minimum. Two refinement methods have been proposed, simulated annealing and the genetic algorithm. 

These results show clearly the importance of the optimization criterion to clustering. The computationally simple Ward s method performs better than the simulated annealing approach with a simplistic criterion. However, a criterion that more correctly accounts for the hierarchy, by minimizing the sum of squared error at each level, performs much better. As with partitional clustering the application of simulated annealing to hierarchical clustering requires careful selection of the internal clustering criterion. 

Density functional techniques are, in principle, based on minimization of energy as a functional of the electron density p(r). In practice the density is represented in terms of Kohn-Sham orbitals, and therefore the implementation takes the same broad form [Equation (2)] as Schrodinger based methods. We will forgo extensive discussion of specific forms of [ ] until Section 2 below, and consider simple examples to illustrate the use simulated annealing here. 

---