The Indefinite Proximal Gradient Method

We introduce a variant of the proximal gradient method in which the quadratic term is diagonal but may be indefinite, and is safeguarded by a trust region. Our method is a special case of the proximal quasi-Newton trust-region method of arXiv:2103.15993v3. We provide closed-form solution of the step computation in certain cases where the nonsmooth term is separable and the trust region is defined in infinity norm, so that no iterative subproblem solver is required. Our analysis expands upon that of arXiv:2103.15993v3 by generalizing the trust-region approach to problems with bound constraints. We provide an efficient open-source implementation of our method, named TRDH, in the Julia language in which Hessians approximations are given by diagonal quasi-Newton updates. TRDH evaluates one standard proximal operator and one indefinite proximal operator per iteration. We also analyze and implement a variant named iTRDH that performs a single indefinite proximal operator evaluation per iteration. We establish that iTRDH enjoys the same asymptotic worst-case iteration complexity as TRDH. We report numerical experience on unconstrained and bound-constrained problems, where TRDH and iTRDH are used both as standalone and subproblem solvers. Our results illustrate that, as standalone solvers, TRDH and iTRDH improve upon the quadratic regularization method R2 of arXiv:2103.15993v3 but also sometimes upon their quasi-Newton trust-region method, referred to here as TR-R2, in terms of smooth objective value and gradient evaluations. On challenging nonnegative matrix factorization, binary classification and data fitting problems, TRDH and iTRDH used as subproblem solvers inside TR improve upon TR-R2 for at least one choice of diagonal approximation.