Hybrid limited memory gradient projection methods for box-constrained optimization problems

Abstract

Gradient projection methods represent effective tools for solving large-scale constrained optimization problems thanks to their simple implementation and low computational cost per iteration. Despite these good properties, a slow convergence rate can affect gradient projection schemes, especially when high accurate solutions are needed. A strategy to mitigate this drawback consists in properly selecting the values for the steplength along the negative gradient. In this paper, we consider the class of gradient projection methods with line search along the projected arc for box-constrained minimization problems and we analyse different strategies to define the steplength. It is well known in the literature that steplength selection rules able to approximate, at each iteration, the eigenvalues of the inverse of a suitable submatrix of the Hessian of the objective function can improve the performance of gradient projection methods. In this perspective, we propose an automatic hybrid steplength selection technique that employs a proper alternation of standard Barzilai–Borwein rules, when the final active set is not well approximated, and a generalized limited memory strategy based on the Ritz-like values of the Hessian matrix restricted to the inactive constraints, when the final active set is reached. Numerical experiments on quadratic and non-quadratic test problems show the effectiveness of the proposed steplength scheme.

Improved Gradient Descent Iterations for Solving Systems of Nonlinear Equations

“…The BB iterations were modified and investigated in a number of publications [65][66][67][68][69][70][71][72][73][74][75][76][77][78][79]. The so-called Scalar Correction (SC) method from [80] proposed the trial steplength in (17) defined by…”

A new proximal heavy ball inexact line-search algorithm

“…Moreover, our approach to the selection of the steplength α k is in the same line of [29,30], where the authors shows that the effectiveness of line-search based FB methods of the form (2) can be significantly improved by devising clever steplength selection rules, in spite of lower theoretical rate estimates.…”

“…According to the founding in [29,30], we then propose to generalize the implementation of Phila described in the previous section with the subgradient (45) in place of the gradient of f .…”

Special issue for SIMAI 2020–2021: large-scale optimization and applications

“…On the same line is the paper by Crisci et al [7] that aims to make effective gradient projection methods for constrained optimization problems. Moving from an approach originally proposed by Fletcher [8] for the unconstrained case, in [7] the authors propose a limited memory strategy based on the Ritz-like values of the Hessian matrix restricted to the inactive constraints, when the final active set is reached. Computational experiments show the nice behaviuor of their approach on large scale problems for which classical gradient descent methods usually yield very slow convergence.…”