SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS USING NEURAL NETWORKS

Sreeram V. B. Aiyer (PhD Thesis)

October 1992

Combinatorial optimization problems arise naturally in many areas of science and engineering. Unfortunately, the accurate solution of a large class of these problems requires an amount of computation which increases exponentially with the problem size. The extensive research into ways of avoiding this difficulty has recently been given even further impetus by the widespread occurrence of these problems in the field of Artificial Intelligence, especially in the pattern recognition problems associated with speech and vision processing. However, the only computationally tractable methods that have so far been developed all rely on some form of heuristic: essentially an intelligent guess which offers a reasonable compromise between solution quality and computational complexity.

Neural networks offer a novel and potentially powerful heuristic for solving these problems. They are also intrinsically parallel systems, with significant potential for fast hardware implementation. Out of recent research, two related families of neural network for solving combinatorial optimization problems have emerged: Hopfield networks, and Mean Field Annealing networks. A number of researchers has applied these networks to a wide variety of problems, and there now exists a large body of rather inconclusive experimental results, with several researchers reporting very poor performances. No theoretical framework has yet been developed which can either account for these inconclusive results in a systematic way, or offer a robust method of correcting the root causes of these poor performances. Consequently, some researchers have concluded that these networks are fatally flawed.

This thesis is aimed at developing such a theoretical framework. By doing this, it will be possible to explain and correct the poor network performances that have emerged from the experimental results. This framework also has the further advantage of making it possible to analyze how these networks achieve their solutions, and to characterize which types of problems these networks can be expected to solve accurately. Based on this analysis, a series of modifications will then be described which greatly improve the efficiency and final solution quality of these networks. Finally a mapping of the Viterbi algorithm onto these network will be developed, which will be used to demonstrate that combinatorial optimization problems of practical significance to speech recognition can be solved using neural networks.

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