I. Introduction
In this paper, we focus on the nonsmooth and nonconvex optimization problem \begin{equation*}\mathop {\min }\limits_{{\mathbf{x}} \in {\mathbb{R}^n}} f({\mathbf{x}}): = {\left\| {{Y^ \top }{\mathbf{x}}} \right\|_1},{\text{ s}}{\text{.t}}{\text{., }}{\left\| {\mathbf{x}} \right\|_2} = 1,\tag{1}\end{equation*} where Y ∈ ℝn×p is a given matrix and ||•||1 denotes the ℓ1 norm of a vector. The set is known as a sphere constraint and is a special case of the Stiefel manifold. Note that (1) has a nonconvex constraint and a nonsmooth objective, which make it very challenging to solve. Though manifold optimization has been a very active area [1], most existing algorithms require computing the derivative of the objective function and are therefore not applicable to (1). The problem (1), which should be contrasted with the ℓ1-PCA problem considered in [2], has recently received much attention because of its many interesting applications, among which are two representative ones: orthogonal dictionary learning (ODL) and dual principal component pursuit (DPCP). In the following we briefly survey the literature on these two problems.