Convex Relaxations for Global Optimization Under Uncertainty Described by Continuous Random Variables

TL;DR

This paper introduces a new method for computing convex and concave relaxations of nonconvex expected-value functions, enabling global optimization under uncertainty with finite termination guarantees.

Contribution

It presents a novel approach to relax nonconvex expected-value functions, facilitating the use of spatial branch-and-bound for global optimization under uncertainty.

Findings

Relaxations obey second-order convergence, ensuring finite termination.

Empirical results demonstrate effectiveness on example problems.

Method enables rigorous bounds for stochastic optimization.

Abstract

This article considers nonconvex global optimization problems subject to uncertainties described by continuous random variables. Such problems arise in chemical process design, renewable energy systems, stochastic model predictive control, etc. Here, we restrict our attention to problems with expected-value objectives and no recourse decisions. In principle, such problems can be solved globally using spatial branch-and-bound (B&B). However, B&B requires the ability to bound the optimal objective value on subintervals of the search space, and existing techniques are not generally applicable because expected-value objectives often cannot be written in closed-form. To address this, this article presents a new method for computing convex and concave relaxations of nonconvex expected-value functions, which can be used to obtain rigorous bounds for use in B&B. Furthermore, these relaxations obey a second-order pointwise convergence property, which is sufficient for finite termination of B&B under standard assumptions. Empirical results are shown for three simple examples.