相关论文: An increasing rank Riemannian method for generaliz…

This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$, as they arise from discretized partial differential…

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the…

In this paper a new Riemannian rank adaptive method (RRAM) is proposed for the low-rank tensor completion problem (LRTCP) formulated as a least-squares optimization problem on the algebraic variety of tensors of bounded tensor-train (TT)…

We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for…

We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret…

We consider a nonsmooth optimization problem on Riemannian manifold, whose objective function is the sum of a differentiable component and a nonsmooth convex function. We propose a manifold inexact augmented Lagrangian method (MIALM) for…

We propose a new algorithm to solve optimization problems of the form $\min f(X)$ for a smooth function $f$ under the constraints that $X$ is positive semidefinite and the diagonal blocks of $X$ are small identity matrices. Such problems…

Recently, a Riemannian proximal Newton method has been developed for optimizing problems in the form of $\min_{x\in\mathcal{M}} f(x) + \mu \|x\|_1$, where $\mathcal{M}$ is a compact embedded submanifold and $f(x)$ is smooth. Although this…

We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for…

This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method. One of the existing approaches for solving nonconvex QCQP problems relaxes the rank one constraint on the unknown…

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate…

We propose a Riemannian optimization approach for computing low-rank solutions of the algebraic Riccati equation. The scheme alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is on the set of…

Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over…

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank…

Low-rank matrix completion (LRMC) has demonstrated remarkable success in a wide range of applications. To address the NP-hard nature of the rank minimization problem, the nuclear norm is commonly used as a convex and computationally…

Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of…

Doubly nonnegative (DNN) relaxation usually provides a tight lower bound for a mixed-binary quadratic program (MBQP). However, solving DNN problems is challenging because: (1) the problem size is $\Omega((n+l)^2)$ for an MBQP with $n$…

We consider the problem of minimizing the sum of a smooth function and a composition of a zero-one loss function with a linear operator, namely zero-one composite optimization problem (0/1-COP). It is a versatile model including the support…

We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear…