For regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, ISQA+, that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, ISQA+ possesses superlinear convergence not just only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition more general than that in existing literature. Experiments on real-world problems also confirm that ISQA+ greatly improves the state of the art for regularized optimization.
Mathematical Programming 2023