An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

This paper focuses on the minimization of a sum of a twice continuously differentiable function f and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of f involving the $$\varrho $$ ϱ th power of the KKT residual. For $$\varrho =0$$ ϱ = 0 , we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1/2. For $$\varrho \in (0,1)$$ ϱ ∈ ( 0 , 1 ) , by assuming that cluster points satisfy a locally Hölderian error bound of order q on a second-order stationary point set and a local error bound of order $$q>1\!+\!\varrho $$ q > 1 + ϱ on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on q and $$\varrho $$ ϱ . A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on $$\ell _1$$ ℓ 1 -regularized Student’s t-regressions, group penalized Student’s t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method.