PROBLEM SOLVING WITH OPTIMIZATION NETWORKS (PHD THESIS)

Andrew H. Gee

July 1993

Combinatorial optimization problems, which are characterized by a discrete set as opposed to a continuum of possible solutions, occur in many areas of engineering, science and management. Such problems have so far resisted efficient, exact solution, despite the attention of many capable researchers over the last few decades. It is not surprising, therefore, that most practical solution algorithms abandon the goal of finding the optimal solution, and instead attempt to find an approximate, useful solution in a reasonable amount of time. A recent approach makes use of highly interconnected networks of simple processing elements, which can be programmed to compute approximate solutions to a variety of difficult problems. When properly implemented in suitable parallel hardware, these `optimization networks' are capable of extremely rapid solutions rates, thereby lending themselves to real-time applications.

This thesis takes a detailed look at problem solving with optimization networks. Three important questions are identified concerning the applicability of optimization networks to general problems, the convergence properties of the networks, and the likely quality of the networks' solutions. These questions are subsequently answered using a combination of rigorous analysis and simple, illustrative examples. The investigation leads to a clearer understanding of the networks' capabilities and shortcomings, confirmed by extensive experiments. It is concluded that optimization networks are not as attractive as they might have previously seemed, since they can be successfully applied to only a limited number of problems exhibiting special, amenable properties.

Key words: Combinatorial optimization, neural networks, mean field annealing.

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