Some results on a class of functional optimization problems

TL;DR

This paper explores a broad class of optimization problems linked to natural, economic, and statistical systems, emphasizing the importance of correct probability distributions and introducing methods for Bayesian updating and density estimation, with applications to networks.

Contribution

It introduces a unified framework for functional optimization problems, including Bayesian updates and kernel density estimation, with extensions to network systems and practical risk propagation models.

Findings

Conserved quantities exist in transformed coordinate systems of these problems.

Incorrect probability assumptions can lead to catastrophic outcomes like infinite costs.

The proposed KDE method converges and aids in distribution prediction over paths.

Abstract

We first describe a general class of optimization problems that describe many natural, 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 principle, it is vital to understand the nature of the governing probability distribution. We show that optimiziation for the incorrect probability distribution can have catastrophic 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.