Alternating DC algorithm for partial DC programming problems

DC (Difference of Convex functions) programming and DCA (DC Algorithm) play a key role in nonconvex programming framework. These tools have a rich and successful history of thirty five years of development, and the research in recent years is being increasingly explored to new trends in the development of DCA: design novel DCA variants to improve standard DCA, to deal with the scalability and with broader classes than DC programs. Following these trends, we address in this paper the two wide classes of nonconvex problems, called partial DC programs and generalized partial DC programs, and investigate an alternating approach based on DCA for them. A partial DC program in two variables (x,y)∈ℝ^n×ℝ^m takes the form of a standard DC program in each variable while fixing other variable. A so-named alternating DCA and its inexact/generalized versions are developed. The convergence properties of these algorithms are established: both exact and inexact alternating DCA converge to a weak critical point of the considered problem, in particular, when the Kurdyka–Łojasiewicz inequality property is satisfied, the algorithms furnish a Fréchet/Clarke critical point. The proposed algorithms are implemented on the problem of finding an intersection point of two nonconvex sets. Numerical experiments are performed on an important application that is robust principal component analysis. Numerical results show the efficiency and the superiority of the alternating DCA comparing with the standard DCA as well as a well known alternating projection algorithm.