Random Field Optimization

TL;DR

This paper introduces random field optimization, a new paradigm for modeling and solving optimization problems involving infinite-dimensional stochastic variables, expanding the scope of uncertainty modeling beyond finite variables.

Contribution

It extends existing infinite-dimensional optimization frameworks to incorporate more general uncertainty representations using random fields, with solution methods based on sampling and transformations.

Findings

Introduces a new optimization paradigm using random fields.

Discusses solution methods involving sampling and finite transformations.

Identifies open challenges in applying random field models to optimization.

Abstract

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.