A VMiPG method for composite optimization with nonsmooth term having no closed-form proximal mapping

This paper concerns the minimization of the sum of a twice continuously differentiable function $f$ and a nonsmooth convex function $g$ without closed-form proximal mapping. For this class of nonconvex and nonsmooth problems, we propose a line-search based variable metric inexact proximal gradient (VMiPG) method with uniformly bounded positive definite variable metric linear operators. This method computes in each step an inexact minimizer of a strongly convex model such that the difference between its objective value and the optimal value is controlled by its squared distance from the current iterate, and then seeks an appropriate step-size along the obtained direction with an armijo line-search criterion. We prove that the iterate sequence converges to a stationary point when $f$ and $g$ are definable in the same o-minimal structure over the real field $(\mathbb{R},+,\cdot)$, and if addition the objective function $f+g$ is a KL function of exponent $1/2$, the convergence has a local R-linear rate. The proposed VMiPG method with the variable metric linear operator constructed by the Hessian of the function $f$ is applied to the scenario that $f$ and $g$ have common composite structure, and numerical comparison with a state-of-art variable metric line-search algorithm indicates that the Hessian-based VMiPG method has a remarkable advantage in terms of the quality of objective values and the running time for those difficult problems such as high-dimensional fused weighted-lasso regressions.