Convex relaxations for global optimization under uncertainty described by continuous random variables

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, and many other applications. 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. However, branch-and-bound 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 branch-and-bound. Furthermore, these relaxations obey a second-order pointwise convergence property, which is sufficient for finite termination of branch-and-bound under standard assumptions. Empirical results are shown for three simple examples. © 2017 American Institute of Chemical Engineers AIChE J, 2018