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

Retraction-based first-order feasible methods for difference-of-convex programs with smooth inequality and simple geometric constraints