We present a numerical iterative optimization algorithm for the minimization of a cost function consisting of a linear combination of three convex terms, one of which is differentiable, a second one is prox-simple and the third one is the composition of a linear map and a prox-simple function. The algorithm's special feature lies in its ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function. A proof of convergence of the algorithm, based on an inexact variable metric approach, is also provided. As a special case, one recovers a generalization of the primal-dual algorithm of Chambolle and Pock, and also of the proximal-gradient algorithm. Finally, we show how it is related to a primal-dual iterative algorithm based on inexact proximal evaluations of the non-smooth terms of the cost function.

翻译：本文提出了一种数值迭代优化算法，用于最小化由三个凸项线性组合构成的代价函数，其中一项可微，第二项为邻近简单函数，第三项为线性映射与邻近简单函数的复合。该算法的独特之处在于能够通过单次迭代运行，逼近代价函数在决定三项相对权重的参数取多种不同值时的极小化点。基于非精确变度量方法，本文同时提供了算法的收敛性证明。作为特例，该算法可推广为Chambolle和Pock的原始-对偶算法以及近端梯度算法的广义形式。最后，我们阐明了该算法与基于代价函数中非光滑项的非精确邻近计算的原始-对偶迭代算法之间的关联。