Semidefinite representable reformulations for two variants of the trust-region subproblem

Motivated by encouraging numerical results in Eltved and Burer (2023) [13], in this note we consider two specific variants of the trust-region subproblem and provide exact semidefinite representable reformulations. The first is over the intersection of two balls; the second is over the intersection of a ball and a special second-order conic representable set. Different from the technique developed in Eltved and Burer (2023) [13], the reformulations in this note are based on partitions of the feasible regions into sub-regions with known lifted convex hulls. (c) 2023 Elsevier B.V. All rights reserved.