  In this paper, we propose first-order feasible methods for
difference-of-convex (DC) programs with smooth inequality and simple geometric
constraints. Our strategy for maintaining feasibility of the iterates is based
on a "retraction" idea adapted from the literature of manifold optimization.
When the constraints are convex, we establish the global subsequential
convergence of the sequence generated by our algorithm under strict feasibility
condition, and analyze its convergence rate when the objective is in addition
convex according to the Kurdyka-Lojasiewicz (KL) exponent of the extended
objective (i.e., sum of the objective and the indicator function of the
constraint set). We also show that the extended objective of a large class of
Euclidean norm (and more generally, group LASSO penalty) regularized convex
optimization problems is a KL function with exponent $\frac12$; consequently,
our algorithm is locally linearly convergent when applied to these problems. We
then extend our method to solve DC programs with a single specially structured
nonconvex constraint. Finally, we discuss how our algorithms can be applied to
solve two concrete optimization problems, namely, group-structured compressed
sensing problems with Gaussian measurement noise and compressed sensing
problems with Cauchy measurement noise, and illustrate the empirical
performance of our algorithms.
