提出新方法优化结构化非凸问题，比传统弗兰克-沃尔夫更高效。
Revisiting Frank-Wolfe for Structured Nonconvex Optimization
- 基于凸函数差分解，设计无投影优化新算法。
- 迭代次数达 $O(1/ε^2)$，部分分解可降至 $O(1/ε)$ 梯度调用。
- 适用于结构化非凸优化，适合追求高效求解的科研与工程场景。
我们提出一种针对可表示为两个凸函数之差的结构化非凸函数的新投影自由（弗兰克-沃尔夫）优化方法。该问题类别包含光滑非凸最小化，使本方法成为经典弗兰克-沃尔夫算法的有力替代。由于DC分解不唯一，通过精心选择分解方式，可更好利用问题结构，提升计算效率，并适应底层几何特性以获得更优局部解。我们证明所提方法在 $O(1/ε^2)$ 次迭代内达到一阶驻点，与标准弗兰克-沃尔夫算法在一般光滑非凸最小化中的复杂度一致。特定分解可实现梯度高效的变体，仅需 $O(1/ε)$ 次梯度查询。最后，数值实验表明该方法优于其他投影自由算法。
原文摘要 · Abstract (English)
We introduce a new projection-free (Frank-Wolfe) method for optimizing structured nonconvex functions that are expressed as a difference of two convex functions. This problem class subsumes smooth nonconvex minimization, positioning our method as a promising alternative to the classical Frank-Wolfe algorithm. DC decompositions are not unique; by carefully selecting a decomposition, we can better exploit the problem structure, improve computational efficiency, and adapt to the underlying problem geometry to find better local solutions. We prove that the proposed method achieves a first-order stationary point in $O(1/ε^2)$ iterations, matching the complexity of the standard Frank-Wolfe algorithm for smooth nonconvex minimization in general. Specific decompositions can, for instance, yield a gradient-efficient variant that requires only $O(1/ε)$ calls to the gradient oracle. Finally, we present numerical experiments demonstrating the effectiveness of the proposed method compared to other projection-free algorithms.
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