A Proximal Modified Quasi-Newton Method for Nonsmooth Regularized Optimization

Abstract. We develop R2N, a modified quasi-Newton method for minimizing the sum of a [Formula: see text] function [Formula: see text] and a lower semicontinuous prox-bounded [Formula: see text]. Both [Formula: see text] and [Formula: see text] may be nonconvex. At each iteration, our method computes a step by minimizing the sum of a quadratic model of [Formula: see text], a model of [Formula: see text], and an adaptive quadratic regularization term. A step may be computed by way of a variant of the proximal-gradient method. An advantage of R2N over competing trust-region methods is that proximal operators do not involve an extra trust-region indicator. We also develop the variant R2DH, in which the model Hessian is diagonal, which allows us to compute a step without relying on a subproblem solver when [Formula: see text] is separable. R2DH can be used as a standalone solver, but also as a subproblem solver inside R2N. We describe nonmonotone variants of both R2N and R2DH. Global convergence of a first-order stationarity measure to zero holds without relying on local Lipschitz continuity of [Formula: see text], while allowing model Hessians to grow unbounded, an assumption particularly relevant to quasi-Newton models. Under Lipschitz-continuity of [Formula: see text], we establish a tight worst-case evaluation complexity bound of [Formula: see text] to bring said measure below [Formula: see text], where [Formula: see text] controls the growth of model Hessians. Specifically, the latter must not diverge faster than [Formula: see text], where [Formula: see text] is the set of successful iterations up to iteration [Formula: see text]. When [Formula: see text], we establish the tight exponential complexity bound [Formula: see text], where [Formula: see text] is a constant. We describe our Julia implementation and report numerical experience on a classic basis-pursuit problem, an image denoising problem, a minimum-rank matrix completion problem, a nonlinear support vector machine, and an inverse nonlinear problem.