A robust approach to warped Gaussian process-constrained optimization

Abstract

Optimization problems with uncertain black-box constraints, modeled by warped Gaussian processes, have recently been considered in the Bayesian optimization setting. This work considers optimization problems with aggregated black-box constraints. Each aggregated black-box constraint sums several draws from the same black-box function with different decision variables as arguments in each individual black-box term. Such constraints are important in applications where, e.g., safety-critical measures are aggregated over multiple time periods. Our approach, which uses robust optimization, reformulates these uncertain constraints into deterministic constraints guaranteed to be satisfied with a specified probability, i.e., deterministic approximations to a chance constraint. While robust optimization typically considers parametric uncertainty, our approach considers uncertain functions modeled by warped Gaussian processes. We analyze convexity conditions and propose a custom global optimization strategy for non-convex cases. A case study derived from production planning and an industrially relevant example from oil well drilling show that the approach effectively mitigates uncertainty in the learned curves. For the drill scheduling example, we develop a custom strategy for globally optimizing integer decisions.

ROmodel: modeling robust optimization problems in Pyomo

“…where the decision which facilities to build has to be made under demand uncertainty, while the decision from which facility to supply individual customers can be made once the uncertainty is resolved, 5. A production planning in which the price at which products can be sold depends on the amount produced through an uncertain black-box function modelled by a (warped) Gaussian process [26],…”

“…ROmodel implements standard reformulations for ellipsoidal and polyhedral uncertainty sets and linear uncertain constraints [3,5]. It also implements reformulations for black-box constrained problems [26]. These are discussed in more detail in Section 4, which also dicussed how ROmodel can be extended to include further reformulations.…”

“…Wiebe et al [26] propose a robust optimization-based approximation of a class of chanceconstraints containing uncertain black-box functions g(•):…”

“…4 can be reformulated exactly. For the warped Gaussian process, Wiebe et al [26] propose an approximation based on Wolfe duality.…”

“…ROmodel therefore allows data-based techniques to be integrated seemlesly with robust optimization methods. As an example, we implement (warped) Gaussian process-based uncertainty sets in ROmodel [26]. These sets seemlesly integrate Gaussian processes trained in the Python library GPy [10] into robust optimization problems.…”