# Forward-backward quasi-Newton methods for nonsmooth optimization problems

arXiv: https://arxiv.org/abs/1604.08096 · HTML: https://mlsanity.com/paper/1604.08096/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Apr 2016.

## Built on by (most cited first, 9 of 9)

- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence](https://mlsanity.com/paper/2402.17722/) (2024)
- [An L-BFGS-B approach for linear and nonlinear system identification under ℓ1 and group-Lasso regularization](https://mlsanity.com/paper/2403.03827/) (2024)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Self-concordant smoothing in proximal quasi-Newton algorithms for large-scale convex composite optimization](https://mlsanity.com/paper/2309.01781/) (2023)
- [Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives](https://mlsanity.com/paper/2509.03475/) (2025)
- [SPIRAL: A superlinearly convergent incremental proximal algorithm for nonconvex finite sum minimization](https://mlsanity.com/paper/2207.08195/) (2022)
- [A Proximal Modified Quasi-Newton Method for Nonsmooth Regularized Optimization](https://mlsanity.com/paper/2409.19428/) (2024)
- [Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence](https://mlsanity.com/paper/2604.17067/) (2026)

## Abstract (arXiv)

The forward-backward splitting method (FBS) for minimizing a nonsmooth composite function can be interpreted as a (variable-metric) gradient method over a continuously differentiable function which we call forward-backward envelope (FBE). This allows to extend algorithms for smooth unconstrained optimization and apply them to nonsmooth (possibly constrained) problems. Since the FBE and its gradient can be computed by simply evaluating forward-backward steps, the resulting methods rely on the very same black-box oracle as FBS. We propose an algorithmic scheme that enjoys the same global convergence properties of FBS when the problem is convex, or when the objective function possesses the Kurdyka-Lojasiewicz property at its critical points. Moreover, when using quasi-Newton directions the proposed method achieves superlinear convergence provided that usual second-order sufficiency conditions on the FBE hold at the limit point of the generated sequence. Such conditions translate into milder requirements on the original function involving generalized second-order differentiability. We show that BFGS fits our framework and that the limited-memory variant L-BFGS is well suited for large-scale problems, greatly outperforming FBS or its accelerated version in practice. The analysis of superlinear convergence is based on an extension of the Dennis and Mor'e theorem for the proposed algorithmic scheme.