  We present a new modeling paradigm for optimization that we call random field
optimization. Random fields are a powerful modeling abstraction that aims to
capture the behavior of random variables that live on infinite-dimensional
spaces (e.g., space and time) such as stochastic processes (e.g., time series,
Gaussian processes, and Markov processes), random matrices, and random spatial
fields. This paradigm involves sophisticated mathematical objects (e.g.,
stochastic differential equations and space-time kernel functions) and has been
widely used in neuroscience, geoscience, physics, civil engineering, and
computer graphics. Despite of this, however, random fields have seen limited
use in optimization; specifically, existing optimization paradigms that involve
uncertainty (e.g., stochastic programming and robust optimization) mostly focus
on the use of finite random variables. This trend is rapidly changing with the
advent of statistical optimization (e.g., Bayesian optimization) and
multi-scale optimization (e.g., integration of molecular sciences and process
engineering). Our work extends a recently-proposed abstraction for
infinite-dimensional optimization problems by capturing more general
uncertainty representations. Moreover, we discuss solution paradigms for this
new class of problems based on finite transformations and sampling, and
identify open questions and challenges.
