Variable metric techniques are a crucial ingredient in many first order optimization algorithms. In practice, they consist in a rule for computing, at each iteration, a suitable symmetric, positive definite scaling matrix to be multiplied to the gradient vector. Besides quasi-Newton BFGS techniques, which represented the state-of-the-art since the 70’s, new approaches have been proposed in the last decade in the framework of imaging problems expressed in variational form. Such recent approaches are appealing since they can be applied to large scale problems without adding significant computational costs and they produce an impressive improvement in the practical performances of first order methods. These scaling strategies are strictly connected to the shape of the specific objective function and constraints of the optimization problem they are applied to; therefore, they are able to effectively capture the problem features. On the other side, this strict problem dependence makes difficult extending the existing techniques to more general problems. Moreover, in spite of the experimental evidence of their practical effectiveness, their theoretical properties are not well understood. The aim of this paper is to investigate these issues; in particular, we develop a unified framework for scaling techniques, multiplicative algorithms and the Majorization–Minimization approach. With this inspiration, we propose a scaling matrix rule for variable metric first order methods applied to nonnegatively constrained problems exploiting a suitable structure of the objective function. Finally, we evaluate the effectiveness of the proposed approach on some image restoration problems.

可变度量技术是许多一阶优化算法中的关键要素。实际上，它们包含一个规则，该规则用于在每次迭代时计算要与梯度向量相乘的合适的对称正定比例矩阵。准牛顿BFGS技术自70年代以来就代表了最先进的技术，近十年来，在以变体形式表示的成像问题的框架内，也提出了新的方法。这样的最新方法之所以吸引人，是因为它们可以应用于大规模问题而无需增加大量的计算成本，并且它们在一阶方法的实际性能方面产生了令人印象深刻的改进。这些缩放策略严格地与特定目标函数的形状以及所应用的优化问题的约束有关；因此，他们能够有效地捕获问题特征。另一方面，这种严格的问题依赖性使得将现有技术扩展到更一般的问题变得困难。而且，尽管有实验证明了它们的实际有效性，但对它们的理论特性却知之甚少。本文的目的是调查这些问题。特别是，我们为缩放技术，乘法算法和主化-最小化方法开发了一个统一的框架。有了这个灵感，我们为可变度量一阶方法提出了一个缩放矩阵规则，该方法适用于利用目标函数的适当结构的非负约束问题。最后，我们评估了该方法在某些图像恢复问题上的有效性。