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

TL;DR

By exploiting the structure of the constraints, it is shown 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.

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.

Figures (8)

• Figure 1: Visualization of the gap between $V(\hat{x}_N^\star)$ and $V'(\xi_N^\star)$ when $q=1$. The continuous horizontal line represents $\xi_{N}^\star$, defining the feasible region $g(x) \leq \xi_{N}^\star$; the solution $\hat{x}_N^\star$ is such that $g(\hat{x}_N^\star) < \xi_{N}^\star$ (red dot). The dashed horizontal line is the value of $b(\delta)$ for some $\delta \in \Delta$. While ${b(\delta) < \xi_{N}^\star}$, it holds that ${g(\hat{x}_N^\star) < b(\delta)}$, meaning that $x_N^\star$ is appropriate for this value of $\delta \in \Delta$.
• Figure 2: Data-set size $N$ computed with \ref{['eq:oneshot']} (top plot) and increment $\Delta N$ of the data-set size when computed with \ref{['eq:Ncheck']} (bottom plot) as a function of $\bar{\epsilon}$ for $q = 100$ when $\beta = 10^{-6}$.
• Figure 3: Weekday power demand of peninsular Spain between January 1st 2014 and December 31st 2024, obtained from Red Eléctrica.
• Figure 4: Comparison of $s_N^{\star,k}$ (circles) with $\varsigma_N^k$ (crosses), for $k = 1,\dots,12$, with the central month indicated on the horizontal axis.
• Figure 5: Comparison of $\sum_{j = 1}^{n_\mathrm{p}} P_{j,t}^{\star}$ (black dashed line) with $P_{i,t}^\mathrm{d}$, $i \in \mathcal{I}_N$ (solid colored lines), for $k=6$.

Theorems & Definitions (16)

• Definition 1: Support list
• Definition 2: Complexity
• Theorem 1: Theorem 6 in garatti2024non-convex
• Theorem 2
• proof
• Proposition 1: a posteriori robustness certificate
• proof
• Remark 1: comparison with previous works
• Remark 2: computational considerations
• Proposition 2: a priori robustness certificate