Convergence analysis of a proximal-type algorithm for DC programs with applications to variable selection

We consider a minimization problem of the form $P(\phi ,g,h):$ $$\min \{f(x):=\phi (x)+g(x)-h(x)\text{\hspace{0.17em}⁣}:x\in {\mathbb{R}}^n\},$$ where $\phi$ is a differentiable function and $g,$ $h$ are convex functions, and introduce iterative methods to finding a critical point of $f$ when $f$ is differentiable. We show that the point computed by proximal point algorithm at each iteration can be used to determine a descent direction for the objective function at this point. This algorithm can be considered as a combination of proximal point algorithm together with a linesearch step that uses this descent direction. We also study convergence results of these algorithms and the inertial proximal methods proposed by Maing$\acute{e}$ and Moudafi (SIAM J. Optim. {\bf 19}(2008), 397–413) under the main assumption that the objective function satisfies the Kurdika–{\L}ojasiewicz property. The proposed algorithm is then applied to solve the variable selection problem in linear regression.