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We study the issue of numerically solving the nonnegative inverse eigenvalue problem (NIEP). At first, we reformulate the NIEP as a convex composite optimization problem on Riemannian manifolds. Then we develop a scheme of the Riemannian linearized proximal algorithm (R-LPA) to solve the NIEP. Under some mild conditions, the local and global convergence results of the R-LPA for the NIEP are established, respectively. Moreover, numerical experiments are presented. Compared with the Riemannian Newton-CG method in Z. Zhao et al. (Numer. Math. 140:827–855, 2018), this R-LPA owns better numerical performances for large scale problems and sparse matrix cases, which is due to the smaller dimension of the Riemannian manifold derived from the problem formulation of the NIEP as a convex composite optimization problem. [ABSTRACT FROM AUTHOR]

We propose an inexact linearized proximal algorithm with an adaptive stepsize, together with its globalized version based on the backtracking line-search, to solve a convex composite optimization problem. Under the assumptions of local weak sharp minima of order p (p ≥ 1) for the outer convex function and a quasi-regularity condition for the inclusion problem associated to the inner function, we establish superlinear/quadratic convergence results for proposed algorithms. Compared to the linearized proximal algorithms with a constant stepsize proposed in Hu et al. (SIAM J Optim 26(2):1207–1235, 2016), our algorithms own broader applications and higher convergence rates, and the idea of analysis used in the present paper deviates significantly from that of Hu et al. (2016). Numerical applications to the nonnegative inverse eigenvalue problem and the wireless sensor network localization problem indicate that the proposed algorithms are more efficient and robust, and outperform the algorithms in Hu et al. (2016) and some popular algorithms for relevant problems. [ABSTRACT FROM AUTHOR]