  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.
