A new steplength selection for scaled gradient methods with application to image deblurring

Gradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the steplength parameter defining the descent direction, as in the case of the well-known Barzilai and Borwein rules. In a recent paper, a strategy for the steplength selection approximating the inverse of some eigenvalues of the Hessian matrix has been proposed for gradient methods applied to unconstrained minimization problems. In the quadratic case, this approach is based on a Lanczos process applied every m iterations to the matrix of the most recent m back gradients but the idea can be extended to a general objective function. In this paper we extend this rule to the case of scaled gradient projection methods applied to non-negatively constrained minimization problems, and we test the effectiveness of the proposed strategy in image deblurring problems in both the presence and the absence of an explicit edge-preserving regularization term

An Inexact Successive Quadratic Approximation Method for Convex L-1 Regularized Optimization

We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model of the objective function. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iteration. Our inexactness conditions are based on a semi-smooth function that represents a (continuous) measure of the optimality conditions of the problem, and that embodies the soft-thresholding iteration. We give careful consideration to the algorithm employed for the inner minimization, and report numerical results on two test sets originating in machine learning

A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportioning, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.Comment: 30 pages, 17 figure

On the filtering effect of iterative regularization algorithms for linear least-squares problems

Many real-world applications are addressed through a linear least-squares problem formulation, whose solution is calculated by means of an iterative approach. A huge amount of studies has been carried out in the optimization field to provide the fastest methods for the reconstruction of the solution, involving choices of adaptive parameters and scaling matrices. However, in presence of an ill-conditioned model and real data, the need of a regularized solution instead of the least-squares one changed the point of view in favour of iterative algorithms able to combine a fast execution with a stable behaviour with respect to the restoration error. In this paper we want to analyze some classical and recent gradient approaches for the linear least-squares problem by looking at their way of filtering the singular values, showing in particular the effects of scaling matrices and non-negative constraints in recovering the correct filters of the solution

Global convergence enhancement of classical linesearch interior point methods for MCPs

AbstractRecent works have shown that a wide class of globally convergent interior point methods may manifest a weakness of convergence. Failures can be ascribed to the procedure of linesearch along the Newton step. In this paper, we introduce a globally convergent interior point method which performs backtracking along a piecewise linear path. Theoretical and computational results show the effectiveness of our proposal