- Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis Bo Jiang (jiang.bomail.shufe.edu.cn ) Tianyi Lin (darren.august.tygmail.com ) Shiqian Ma (sqmaucdavis.edu) Shuzhong Zhang (zhangsumn.edu) Abstract: 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. Keywords: Structured Nonconvex Optimization, $\epsilon$-Stationary Point, Iteration Complexity, Conditional Gradient Method, Alternating Direction Method of Multipliers, Block Coordinate Descent Method Category 1: Nonlinear Optimization Citation: Download: [PDF]Entry Submitted: 05/08/2016Entry Accepted: 05/09/2016Entry Last Modified: 11/14/2017Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.