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 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 literature. Our numerical experiments highlight that our new algorithms significantly outperform prior state-of-the-art algorithms on sparse large-scale instances.
翻译:在本文中,我们为普遍信任区域子问题开发了高效的一阶算法(GTRS),该算法在信号处理、压缩遥感和工程方面都有应用。尽管GTRS(GTRS)指出其运行时间在输入数据中线性(非零条目数量),但众所周知,在斯later条件下,其SDP放松的客观价值准确性为SDP所持有。虽然GTRS存在基于GTRS的多元-时间SDP算法,但其相对较大的计算复杂性激励和刺激了解决GTRS的定制方法的发展。特别是,最近朝这个方向开展的工作为GTRS制定了一阶方法,而GTRS的运行时间在输入数据中是线性(非零条目数量),但与这些算法相反,我们在本文中为GTRS计算$\epslon$-直线性解决方案制定了算法,而GTRS的运行时间在输入空间空间和精确的 $\log(1/\ epsilon) 只要定期参数是肯定的。我们理论上的理论保证与数字性实验相辅相成,我们用我们从以前的大规模数字学角度比较了我们的方法,在前的细数级文献中显示。