Implicit Regularity and Linear Convergence Rates for the Generalized Trust-Region Subproblem.

In this paper we develop efficient first-order algorithms for the generalized trust -region subproblem (GTRS), which has applications in signal processing, compressed sensing, and engineering. Although the GTRS, as stated, is nonlinear and nonconvex, it is well known that objective value exactness holds for its semidefinite programming (SDP) relaxation under a Slater condition. While polynomial-time SDP-based algorithms exist for the GTRS, their relatively large computational complexity has motivated and spurred the development of custom approaches for solving the GTRS. In particular, recent work in this direction has developed first-order methods for the GTRS whose running times are linear in the sparsity (the number of nonzero entries) of the input data. In contrast to these algorithms, in this paper we develop algorithms for computing \epsilon-approximate solutions to the GTRS whose running times are linear in both the input sparsity and the precision log(1/\epsilon) whenever a regularity parameter is positive. We complement our theoretical guarantees with numerical experiments comparing our approach against algorithms from the liter-ature. Our numerical experiments highlight that our new algorithms significantly outperform prior state-of-the-art algorithms on sparse large-scale instances.