DUALITY IN DYNAMIC PROGRAMMING
Abstract
Duality is one of the most elegant and useful concepts in mathematics. However, there have been very limited duality results and no unified theory for dynamic programming. This dissertation develops duality for dynamic programming from two points of view. For dynamic programming problems with linear transition functions a duality is developed through the use of conjugate functions and convolution. This procedure is shown to mitigate the "curse of dimensionality" for primal problems with multidimensional state spaces. Numerical examples illustrate the computational advantages of the dual solution procedure. Secondly, a dual method is derived for the shortest route problem, which is the prototype dynamic programming problem. This duality is developed through the use of submodular functions and polymatroids. The duality is dependent on polymatroid intersection and the ability to write the functional equation for the shortest route problem as a submodular convolution function. The submodular method is shown to be polynomial for both the dual and the primal problems. Suggestions for extensions of this work and topics for future research in related areas are also discussed.
Degree
Ph.D.
Subject Area
Operations research
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