Recovering optimal solutions via SOC-SDP relaxation of trust region subproblem with nonintersecting linear constraints

Abstract

In this paper, we study an extended trust region subproblem (eTRS) in which the unit ball intersects with \begin{document}$m$\end{document} linear inequality constraints. In the literature, Burer et al. proved that an SOC-SDP relaxation (SOCSDPr) of eTRS is exact, under the condition that the nonredundant constraints do not intersect each other in the unit ball. Furthermore, Yuan et al. gave a necessary and sufficient condition for the corresponding SOCSDPr to be a tight relaxation when \begin{document}$m = 2$\end{document} . However, there lacks a recovering algorithm to generate an optimal solution of eTRS from an optimal solution \begin{document}$X^*$\end{document} of SOCSDPr when rank \begin{document}$(X^*)≥ 2$\end{document} and \begin{document}$m≥ 3$\end{document} . This paper provides such a recovering algorithm to complement those known works.