This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components separately, linearizing only the smooth parts. This provides us with new generalizations of the classical Frank-Wolfe method and the Conditional Gradient Sliding algorithm, that cater to a subclass of non-differentiable problems. Our algorithms rely on a stronger version of the linear minimization oracle, which can be efficiently implemented in several practical applications. We provide the basic version of our method with an affine-invariant analysis and prove global convergence rates for both convex and non-convex objectives. Furthermore, in the convex case, we propose an accelerated method with correspondingly improved complexity. Finally, we provide illustrative experiments to support our theoretical results.

翻译：本文研究在凸紧集上求解全复合优化问题的一阶算法。我们通过分别处理目标函数中的可微与不可微部分，并仅对光滑部分进行线性化，从而充分利用其结构特性。由此得到经典Frank-Wolfe方法与条件梯度滑动算法的新推广形式，适用于一类非可微问题。我们的算法依赖于一种更强版本的线性最小化预言机，该预言机可在多种实际应用中高效实现。我们为方法的基础版本提供了仿射不变性分析，并证明了针对凸与非凸目标的全局收敛率。此外，在凸情形下，我们提出一种加速方法，其复杂度相应降低。最后，我们通过算例实验验证了理论结果。