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The aim of this paper is to deepen the convergence analysis of the scaledgradient projection (SGP) method, proposed by Bonettini et al. in a recentpaper for constrained smooth optimization. The main feature of SGP is thepresence of a variable scaling matrix multiplying the gradient, which maychange at each iteration. In the last few years, an extensive numericalexperimentation showed that SGP equipped with a suitable choice of the scalingmatrix is a very effective tool for solving large scale variational problemsarising in image and signal processing. In spite of the very reliable numericalresults observed, only a weak, though very general, convergence theorem isprovided, establishing that any limit point of the sequence generated by SGP isstationary. Here, under the only assumption that the objective function isconvex and that a solution exists, we prove that the sequence generated by SGPconverges to a minimum point, if the scaling matrices sequence satisfies asimple and implementable condition. Moreover, assuming that the gradient of theobjective function is Lipschitz continuous, we are also able to prove theO(1/k) convergence rate with respect to the objective function values. Finally,we present the results of a numerical experience on some relevant imagerestoration problems, showing that the proposed scaling matrix selection ruleperforms well also from the computational point of view.

New convergence results for the scaled gradient projection method