# Robust Quadratic Programming with Mixed-Integer Uncertainty

## TL;DR

This paper develops copositive programming reformulations and SDP approximations for robust quadratic programs with mixed-integer uncertainty, providing more effective solutions than existing methods.

## Contribution

It introduces exact copositive reformulations and a conservative SDP approximation for mixed-integer robust quadratic programs, extending previous continuous-only approaches.

## Key findings

- SDP approximation outperforms S-lemma-based methods
- Reformulations applicable to two-stage problems with recourse
- Demonstrated effectiveness on practical optimization problems

## Abstract

We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. These convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that the popular approximate S-lemma method --- which is valid only in the case of continuous uncertainty --- is weaker than our approximation. We also show that all results can be extended to the two-stage robust quadratic optimization setting if the problem has complete recourse. We assess the effectiveness of our proposed SDP reformulations and demonstrate their superiority over the state-of-the-art solution schemes on instances of least squares, project management, and multi-item newsvendor problems.