  Nonconvex and nonsmooth optimization problems are frequently encountered in
much of statistics, business, science and engineering, but they are not yet
widely recognized as a technology in the sense of scalability. 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 and nonsmooth 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 without
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 apply 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.
