In this paper, we propose a general algorithmic framework for the first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities (VIs). This framework allows obtaining many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in VIs. Besides reproducing known results, our framework allows constructing new methods, which we illustrate by constructing a universal conditional gradient method and a universal method for VIs with a composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem's smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operators. We also generalize our framework for relatively strongly convex objectives and strongly monotone VIs.

在本文中，我们提出了一个广义的一阶优化算法框架，包括最小化问题、鞍点问题和变分不等式（VI）。该框架允许获得许多已知方法作为特例，包括加速梯度方法、复合优化方法、水平集方法、Bregman 近似方法。该框架的思想基于构建主要问题组件的不精确模型，即优化中的目标函数或VI中的运算符。除了重现已知结果外，我们的框架还允许构建新方法，我们通过构建通用条件梯度方法和具有复合结构的 VI 的通用方法来说明这些方法。该方法适用于具有最优复杂度的平滑和非平滑问题，而无需先验知道问题的平滑度。作为我们一般框架的一个特例，我们介绍了算子的相对平滑度，并提出了一种用于具有此类算子的 VI 的算法。我们还将我们的框架推广到相对强凸目标和强单调 VI。