An extended delayed weighted gradient algorithm for solving strongly convex optimization problems

The recently developed delayed weighted gradient method (DWGM) is competitive with the well-known conjugate gradient (CG) method for the minimization of strictly convex quadratic functions. As well as the CG method, DWGM has some key optimality and orthogonality properties that justify its practical performance. The main difference with the CG method is that, instead of … Read more

An Efficient Retraction Mapping for the Symplectic Stiefel Manifold

This article introduces a new retraction on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a matrix inversion of size $2p$–by–$2p$, which is much less expensive than those required for the available retractions in the literature. Later, with the new retraction, we design a constraint … Read more

Proximal Point Algorithm on the Stiefel Manifold

In this paper, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. To solve this problem, we develop a geodesic-free proximal point algorithm, which does not require the use of the Riemannian distance. The proposed method can be regarded as an iterative fixed-point method, which repeatedly applies a proximal operator … Read more

An Accelerated Minimal Gradient Method with Momentum for Convex Quadratic Optimization

In this article we address the problem of minimizing a strictly convex quadratic function using a novel iterative method. The new algorithm is based on the well–known Nesterov’s accelerated gradient method. At each iteration of our scheme, the new point is computed by performing a line–search scheme using a search direction given by a linear … Read more

Global convergence of Riemannian line search methods with a Zhang-Hager-type condition

In this paper, we analyze the global convergence of a general non–monotone line search method on Riemannian manifolds. For this end, we introduce some properties for the tangent search directions that guarantee the convergence, to a stationary point, of this family of optimization methods under appropriate assumptions. A modified version of the non–monotone line search … Read more

A family of optimal weighted conjugate-gradient-type methods for strictly convex quadratic minimization

We introduce a family of weighted conjugate-gradient-type methods, for strictly convex quadratic functions, whose parameters are determined by a minimization model based on a convex combination of the objective function and its gradient norm. This family includes the classical linear conjugate gradient method and the recently published delayed weighted gradient method as the extreme cases … Read more

Spectral Residual Method for Nonlinear Equations on Riemannian Manifolds

In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real–valued function that adapts this … Read more

Two novel gradient methods with optimal step sizes

In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve approximate information of the Hessian of … Read more

Implicit steepest descent algorithm for optimization with orthogonality constraints

Optimization with orthogonality constraints problems appear widely in applications from science and engineering. We address these types of problems from an numerical approach. Our new framework combines the steepest gradient descent using implicit information with and operator projection in order to construct a feasible sequence of points. In addition, we adopt an adaptive Barzilai–Borwein steplength … Read more

A Hybrid Gradient Method for Strictly Convex Quadratic Programming

In this paper, a reliable hybrid algorithm for solving convex quadratic minimization problems is presented. At each iteration, two points are computed: first, an auxiliary point $\dot{x}_k$ is generated by performing a gradient step equipped with an optimal steplength, then, the next iterate $x_{k+1}$ is obtained through a weighted sum of $\dot{x}_k$ with the penultimate … Read more