On a class of robust nonconvex quadratic optimization problems

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.}~ & α\leq\dfrac{1}{2}x^\top (B_1+μB_2)x+(b_1+δb_2)^\top x \leqβ,~ \forall~ μ\in [μ_1,μ_2],\forall~δ\in[δ_1,δ_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 ${\mu }_1\le {\mu }_2$, ${\delta }_1\le {\delta }_2$ and $\alpha <\beta$. We establish the robust alternative result; the robust S-lemma and the robust optimality for the above nonconvex problem.