Inexact spectral projected gradient methods on convex sets

Abstract

A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

Spectral projected gradient and variable metric methods for optimization with linear inequalities

“…In this section the Inexact Variable Metric (IVM) method is reviewed and the global convergence results proved in [9] are recalled. Let L ∈ IR ++ .…”

“…The test problems were solved with IVM. As in [9], the interior initial guess for the primal variables is generated together with the problem data. The initial guess for the vector of Lagrange multipliers (variables of the nonnegatively constrained quadratic minimization problems) was the null vector.…”

“…The features of the SPG method that make it computationally efficient are the spectral (Barzilai-Borwein) choice of the steplength, introduced in [2] and analyzed by several authors (see, among others, [15,16,23,24,36,37]) and the use of nonmonotone linesearch strategies [30]. The difficulty in computing projections in critical cases motivated the introduction of the Inexact Spectral Projected Gradient (ISPG) method in [9]. The idea is that the search direction does not need to be computed exactly for obtaining global convergence and efficiency.…”

“…So, iterative methods with premature stopping criteria can be used. In [9] the ISPG method was also generalized giving rise to a general Inexact Variable Metric method (IVM) for convex constrained optimization. In [9] the authors implemented the ISPG method using a projection method [11,20,31,38] for approximate computation of projections on a family of matricial optimization problems with linear constraints.…”

“…This paper is organized as follows. The inexact variable metric method proposed in [9] is briefly reviewed in Section 2. Local convergence of Newton-like instances is discussed in Section 3.…”