Optimization Using The Simulated Annealing Algorithm

In this paper we will briefl y review the simulated annealin g algorithm, an al gorithm with applications in optimization and pattern reco gnition used extensivel y in artificial intelligence. In earlier papers the authors anal yzed a simulation of the annealin g of a solid, a dodecahedron in particular. Our use of this al gorithm, which is based in the field of combinatorial optimization, reflects properties of Boltzman machines a neural network characterized b y massive parallelism. We will demonstrate two implementations of this al gorithm in simulated annealin g. Each of the implementations depends upon a nei ghborhood structure and a transition mechanism. In the first implementation our nei ghborhood structure is a linear transformation of the vector space of all confi gurat ons and the transition probabilit y s deterministic. In this case, we will use techniques from character theor y of finite groups to anal yze simulated annealing. In the second implementation, a special case of which includes the first implementation, our neighborhood structure is a set-valued function and the transition mechanism is stochastic in nature. In this case, we use techniques from matrix anal ysis, in particular properties of doubl y stochastic matrices, to anal yze simulated annealing modeled and based on a class of Boltzman machines. For pattern reco gnition, we use the simulated annealin g algorithm to solve the classic seven-segment display problem. This is a classification problem which we will solve b y choosing an appropriate Boltzmann machine.