#### Date of Award

2018

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematics

#### First Advisor

William A. Gibson

#### Second Advisor

Peter S. Dodds

#### Abstract

We first describe a general class of optimization problems that describe many natu- ral, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances of these problems in statistical physics, facility allocation, and machine learning. A dynamic description and statement of a partial inverse problem follow. When attempting to optimize the state of a system governed by the generalized equipartitioning princi- ple, it is vital to understand the nature of the governing probability distribution. We show that optimiziation for the incorrect probability distribution can have catas- trophic results, e.g., infinite expected cost, and describe a method for continuous Bayesian update of the posterior predictive distribution when it is stationary. We also introduce and prove convergence properties of a time-dependent nonparametric kernel density estimate (KDE) for use in predicting distributions over paths. Finally, we extend the theory to the case of networks, in which an event probability density is defined over nodes and edges and a system resource is to be partitioning among the nodes and edges as well. We close by giving an example of the theory’s application by considering a model of risk propagation on a power grid.

#### Language

en

#### Number of Pages

60 p.

#### Recommended Citation

Dewhurst, David Rushing, "Some results on a class of functional optimization problems" (2018). *Graduate College Dissertations and Theses*. 884.

https://scholarworks.uvm.edu/graddis/884

#### Included in

Economic Theory Commons, Mathematics Commons, Physics Commons