An uncertainty model for positive-valued parameters with application to robust optimization

TL;DR

This paper introduces a new uncertainty set model for positive parameters in robust optimization, ensuring positivity, computational tractability, and better practical applicability, especially in energy and machine learning problems.

Contribution

The paper proposes a novel convex uncertainty set that maintains positivity and allows for dual reformulation, with theoretical bounds and probabilistic guarantees.

Findings

The new model preserves positivity and is computationally tractable.

Standard models can cause infeasibility, while the proposed set avoids this.

Numerical experiments demonstrate improved robustness in energy and machine learning applications.

Abstract

Many practical optimization problems involve uncertain parameters that are strictly positive. However, the most common uncertainty sets used in robust optimization are the box and the ellipsoidal sets, which may include non-positive values when the level of uncertainty is large. This can lead to overly conservative solutions or make the corresponding robust counterpart infeasible. To overcome this, in this paper, we propose a new uncertainty-set model that not only preserves positivity but is also computationally tractable. The proposed set uses a particular convex function that measures the variation of uncertain parameters from their nominal values. We can also write the dual reformulation of the associated robust problem. For the theoretical results, we show several properties of the proposed model, including analytical bounds that guide the choice of the uncertainty level, as well as a probabilistic guarantee result. To check the validity of our proposal, we consider photovoltaic-battery operation planning problems and support vector machines in the numerical experiments. For these problems, standard uncertainty models may lead to infeasibility of the robust counterpart, while the proposed uncertainty set gives a tractable dual reformulation.