On a class of robust nonconvex quadratic optimization problems

TL;DR

This paper investigates a class of robust nonconvex quadratic optimization problems with interval uncertainty, establishing key theoretical results including a robust alternative theorem, a robust S-lemma, and conditions for robust optimality.

Contribution

The paper introduces new theoretical foundations for solving robust nonconvex quadratic problems, including a robust alternative theorem, a robust S-lemma, and robust optimality conditions.

Findings

Established a robust alternative result for the problem.

Proved a robust S-lemma applicable to the problem.

Derived conditions for robust optimality.

Abstract

Let us consider the following robust nonconvex quadratic optimization problem: \begin{equation*} \begin{split} \min &~ \dfrac{1}{2} x^\top Ax+a^\top x \\ \text{s.t.}~ & \alpha\leq\dfrac{1}{2}x^\top (B_1+\mu B_2)x+(b_1+\delta b_2)^\top x \leq\beta,~ \forall~ \mu\in [\mu_1,\mu_2],\forall~\delta\in[\delta_1,\delta_2], \end{split} \end{equation*} where $A$, $B_1$, $B_2$ are real symmetric matrices, $\mu_1,\mu_2,\delta_1,\delta_2,\alpha$, $\beta\in\mathbb{R}$ satisfying $\mu_1\leq \mu_2$, $\delta_1\leq\delta_2$ and $\alpha<\beta$. We establish the robust alternative result; the robust S-lemma and the robust optimality for the above nonconvex problem.