Optimization Using The Simulated Annealing Algorithm

Abstract

Abstract NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 2520 Optimization Using the Simulated Annealing Algorithm Edgar N. Reyes, Dennis I. Merino, Carl Steidley Southeastern Louisiana University/Texas A&M University - Corpus Christi Hammond, LA 70402/Corpus Christi, TX 78412 Abstract In this paper we will briefly review the simulated annealing algorithm, an algorithm with applications in optimization and pattern recognition used extensively in artificial intelligence. In earlier papers the authors analyzed a simulation of the annealing of a solid, a dodecahedron in particular. Our use of this algorithm, which is based in the field of combinatorial optimization, reflects properties of Boltzman machines - a neural network characterized by massive parallelism. We will demonstrate two implementations of this algorithm in simulated annealing. Each of the implementations depends upon a neighborhood structure and a transition mechanism. In the first implementation our neighborhood structure is a linear transformation of the vector space of all configurations and the transition probability is deterministic. In this case, we will use techniques from character theory of finite groups to analyze 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 analysis, in particular properties of doubly stochastic matrices, to analyze simulated annealing modeled and based on a class of Boltzman machines. For pattern recognition, we use the simulated annealing algorithm to solve the classic seven-segment display problem. This is a classification problem which we will solve by choosing an appropriate Boltzmann machine. 1. Introduction. Annealing is the physical process of heating up a solid and following it by a specified slow cooling process. We shall use the simulated annealing algorithm, a method based in the field of combinatorial optimization, to describe simulated controlled cooling processes. In the annealing process, one can interpret the states (and free energy) of the solid in the cooling process as solutions (and cost function, respectively) of a combinatorial optimization problem [1]. Our use of the simulated annealing algorithm reflects properties of Boltzman machines, a neural network model belonging to a class of connectionists models and which has massive parallelism as a feature, amongst others. Also, we will use an appropriate Boltzmann machine to solve a pattern recognition problem, namely, the seven-segment display problem. The display of the decimal digits in a hand-watch for instance uses a seven-segment display. In identifying the digit displayed, we will maximize an overall measurement of desirability of the Boltzmann machine. We shall briefly review some aspects of Boltzman machines and the simulated annealing algorithm. Let (U, C) be a network consisting of units, U = {u i : i = 1, ..., n } , and a set of { } { } connections, C, consisting of unordered pairs u i , u j . A connection u i , u j in C is said to join u i to u j . Intrinsic to Boltzman machines are the notions of a connection strength s and a