相关论文: On the globalization of Riemannian Newton method

Anderson acceleration (AA) is a popular method for accelerating fixed-point iterations, but may suffer from instability and stagnation. We propose a globalization method for AA to improve stability and achieve unified global and local…

In this work, by using techniques and results of differential geometry, we propose a new numerical method on complete Riemannian manifolds to find zeros of vector fields. Our algorithm generalizes the classical secant method

We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular…

In this paper, we consider the composite optimization problems over the Stiefel manifold. A successful method to solve this class of problems is the proximal gradient method proposed by Chen et al. Motivated by the proximal Newton-type…

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free…

The conjugate gradient method is a crucial first-order optimization method that generally converges faster than the steepest descent method, and its computational cost is much lower than that of second-order methods. However, while various…

We introduce a generalized Rayleigh-quotient on the tensor product of Grassmannians enabling a unified approach to well-known optimization tasks from different areas of numerical linear algebra, such as best low-rank approximations of…

In recent years, stochastic variance reduction algorithms have attracted considerable attention for minimizing the average of a large but finite number of loss functions. This paper proposes a novel Riemannian extension of the Euclidean…

Finding an $\epsilon$-stationary point of a nonconvex function with a Lipschitz continuous Hessian is a central problem in optimization. Regularized Newton methods are a classical tool and have been studied extensively, yet they still face…

Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values…

Many machine learning models involve solving optimization problems. Thus, it is important to deal with a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention…

The renormalization group method of Goldenfeld, Oono and their collaborators is applied to asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This…

Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost.…

In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory for finding approximate solutions of nonlinear equations in the setting of Banach spaces. First, we discuss the convergence analysis of our…

This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first…

We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the…

We consider the optimization problem with a generally quadratic matrix constraint of the form $X^TAX = J$, where $A$ is a given nonsingular, symmetric $n\times n$ matrix and $J$ is a given $k\times k$ symmetric matrix, with $k\leq n$,…

The nonlinear eigen-problem $ Ax+F(x)=\lambda x$ is studied where $A$ is an $n\times n$ irreducible Stieltjes matrix. Under certain conditions, this problem has a unique positive solution. We show that, starting from a multiple of the…

This paper proposes a novel general framework of Riemannian conjugate gradient methods, that is, conjugate gradient methods on Riemannian manifolds. The conjugate gradient methods are important first-order optimization algorithms both in…

In this paper we study quantitative aspects of Newton method for finding zeros of mappings f: M_n -> R^n and vector fields X: M_x -> TM_n