We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be inexactly solved by randomized algorithms in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an \(\varepsilon \)-optimal and \(\varepsilon \)-feasible solution, both frameworks achieve the best-known oracle complexities.

我们开发了一个不精确的原始对偶一阶平滑框架来解决一类具有原始强凸性的非双线性鞍点问题。与现有方法相比，我们的框架对原始预言机复杂性有了显着的改进，同时具有竞争性的双预言机复杂性。此外，我们还考虑了原对偶耦合项具有大量分量函数的情况。为了有效地处理这种情况，我们开发了平滑框架的随机版本，它允许每次迭代中的原始子问题和对偶子问题通过预期的随机算法得到不精确的解决。对该框架的收敛性进行了预期和高概率的分析。就原始和双重预言复杂性而言，该框架比其确定性对应框架有了显着改进。作为一个重要的应用，我们采用这两个框架来解决具有许多功能约束的凸优化问题。为了获得\(\varepsilon \)最优且\(\varepsilon \)可行的解决方案，这两个框架都实现了最著名的预言机复杂性。