Nonconvex optimization problems are frequently encountered in much of statistics, business, science and engineering, but they are not yet widely recognized as a technology. A reason for this relatively low degree of popularity is the lack of a well developed system of theory and algorithms to support the applications, as is the case for its convex counterpart. This paper aims to take one step in the direction of disciplined nonconvex optimization. In particular, we consider in this paper some constrained nonconvex optimization models in block decision variables, with or without coupled affine constraints. In the case of no coupled constraints, we show a sublinear rate of convergence to an $\epsilon$-stationary solution in the form of variational inequality for a generalized conditional gradient method, where the convergence rate is shown to be dependent on the H\"olderian continuity of the gradient of the smooth part of the objective. For the model with coupled affine constraints, we introduce corresponding $\epsilon$-stationarity conditions, and propose two proximal-type variants of the ADMM to solve such a model, assuming the proximal ADMM updates can be implemented for all the block variables except for the last block, for which either a gradient step or a majorization-minimization step is implemented. We show an iteration complexity bound of $O(1/\epsilon^2)$ to reach an $\epsilon$-stationary solution for both algorithms. Moreover, we show that the same iteration complexity of a proximal BCD method follows immediately. Numerical results are provided to illustrate the efficacy of the proposed algorithms for tensor robust PCA.

本文介绍了一些带有或没有耦合的非凸优化模型，使用了相关的优化算法，如条件梯度和ADMM， 为专门处理非凸和非光滑优化问题的理论和算法的发展提出了一步。通过数值实验，证明了这些算法的高效性，特别是在张量鲁棒PCA的场景下。

结构化非凸非光滑优化：算法和迭代复杂度分析