Robustness certificates in data-driven non-convex optimization with additively-uncertain constraints

TL;DR

This paper develops efficient, distribution-free robustness certificates for data-driven non-convex optimization problems with additive uncertainty, enabling reliable decision-making with minimal computational effort.

Contribution

It introduces a novel approach for a priori and a posteriori robustness certification in non-convex problems, avoiding complex scenario program solutions.

Findings

Certificates are computationally efficient and distribution-free.

Incremental data-set sizing guarantees robustness levels.

Application to unit commitment shows reduced conservativeness.

Abstract

We consider decision-making problems that are formulated as non-convex optimization programs where uncertainty enters the constraints through an additive term, independent of the decision variables, and robustness is imposed using a finite data-set, according to the scenario robust optimization paradigm. By exploiting the structure of the constraints, we show that both a priori and a posteriori distribution-free probabilistic robustness certificates for a possibly sub-optimal solution to the resulting data-driven optimization problem can be obtained with minimal computational effort. Building on these results, we also discuss a one-shot and an incremental procedure to determine the size of the data-set so as to guarantee a user-chosen robustness level. Notably, both the a posteriori robustness assessment and incremental data-set sizing do not require to solve the non-convex scenario program. A comparative analysis performed on the unit commitment problem using real data reveals a limited increase in conservativeness with a significant computational saving with respect to the application of scenario theory results for general, non necessarily structured, non-convex problems.