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Data-Driven Adjustable Robust Optimization

Xiaoxing Ren, Alessio Moreschini, Zhongda Chu, Yulong Gao, Thomas Parisini

TL;DR

Data-Driven Adjustable Robust Optimization addresses feasibility under uncertainty by learning an adjustable uncertainty set $\hat{\mathbb{S}}$ within the nominal set $\mathbb{S}$ from data. It introduces a two-stage framework: Stage 1 constructs the largest feasible $\hat{\mathbb{S}}$ using samples to maximize a size or inclusion measure, and Stage 2 optimizes under that set. The work covers both non-stochastic and distributionally robust (Wasserstein-based) settings, providing tractable reformulations for common set shapes and finite-program representations when the safety set is polyhedral. Numerical demonstrations on a simple problem and an optimal power flow case study illustrate the approach's ability to improve feasibility while balancing robustness under data-driven uncertainty.

Abstract

In this paper, we develop a two-stage data-driven approach to address the adjustable robust optimization problem, where the uncertainty set is adjustable to manage infeasibility caused by significant or poorly quantified uncertainties. In the first stage, we synthesize an uncertainty set to ensure the feasibility of the problem as much as possible using the collected uncertainty samples. In the second stage, we find the optimal solution while ensuring that the constraints are satisfied under the new uncertainty set. This approach enlarges the feasible state set, at the expense of the risk of possible constraint violation. We analyze two scenarios: one where the uncertainty is non-stochastic, and another where the uncertainty is stochastic but with unknown probability distribution, leading to a distributionally robust optimization problem. In the first case, we scale the uncertainty set and find the best subset that fits the uncertainty samples. In the second case, we employ the Wasserstein metric to quantify uncertainty based on training data, and for polytope uncertainty sets, we further provide a finite program reformulation of the problem. The effectiveness of the proposed methods is demonstrated through an optimal power flow problem.

Data-Driven Adjustable Robust Optimization

TL;DR

Data-Driven Adjustable Robust Optimization addresses feasibility under uncertainty by learning an adjustable uncertainty set within the nominal set from data. It introduces a two-stage framework: Stage 1 constructs the largest feasible using samples to maximize a size or inclusion measure, and Stage 2 optimizes under that set. The work covers both non-stochastic and distributionally robust (Wasserstein-based) settings, providing tractable reformulations for common set shapes and finite-program representations when the safety set is polyhedral. Numerical demonstrations on a simple problem and an optimal power flow case study illustrate the approach's ability to improve feasibility while balancing robustness under data-driven uncertainty.

Abstract

In this paper, we develop a two-stage data-driven approach to address the adjustable robust optimization problem, where the uncertainty set is adjustable to manage infeasibility caused by significant or poorly quantified uncertainties. In the first stage, we synthesize an uncertainty set to ensure the feasibility of the problem as much as possible using the collected uncertainty samples. In the second stage, we find the optimal solution while ensuring that the constraints are satisfied under the new uncertainty set. This approach enlarges the feasible state set, at the expense of the risk of possible constraint violation. We analyze two scenarios: one where the uncertainty is non-stochastic, and another where the uncertainty is stochastic but with unknown probability distribution, leading to a distributionally robust optimization problem. In the first case, we scale the uncertainty set and find the best subset that fits the uncertainty samples. In the second case, we employ the Wasserstein metric to quantify uncertainty based on training data, and for polytope uncertainty sets, we further provide a finite program reformulation of the problem. The effectiveness of the proposed methods is demonstrated through an optimal power flow problem.

Paper Structure

This paper contains 13 sections, 2 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Data-driven adjustable robust optimization
  3. Data-driven distributionally adjustable robust optimization
  4. Stage 1
  5. Stage 2
  6. Numerical experiments
  7. Numerical example
  8. The case of non-stochastic uncertainty
  9. The case of stochastic uncertainty
  10. Case study in optimal power flow
  11. The case of non-stochastic uncertainty
  12. The case of stochastic uncertainty
  13. Concluding Remarks

Key Result

Lemma 1

Suppose that the uncertainty set is a polytope of the form $\mathbb{S} = \{\xi: C\xi \leq d\}$. If the safety set $\mathbb{\hat{S}} = \{\xi: A\xi < b \in \mathbb{R}^L\}$ is an open polytope and the half-space $\lbrace \xi : \langle a_l, \xi \rangle \geq b_l \rbrace$ has a nonempty intersection with If the safety set $\mathbb{\hat{S}} = \{\xi: A\xi \leq b \in \mathbb{R}^L\}$ is a closed polytope t

Figures (3)

  • Figure 1: Illustration of the feasible and unfeasible regions of the problem addressed in the motivating example. On the left, we depict the problem with its original uncertainty set, while on the right, we show the problem with a reduced uncertainty set. The benefit of the right-hand side problem is the elimination of the unfeasible region.
  • Figure 2: Motivating example for \ref{['eq:stage_1']}. On the left-hand side, we depict the problem with its original uncertainty set and the randomly generated uncertainty samples. On the right-hand side, we show the new uncertainty set that contains the most uncertainty samples while ensuring feasibility.
  • Figure 3: IEEE-5-bus-system

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Proposition 1
  • proof