# Further properties of the forward-backward envelope with applications to difference-of-convex programming

arXiv: https://arxiv.org/abs/1605.00201 · HTML: https://mlsanity.com/paper/1605.00201/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2016.

## Built on by (most cited first, 3 of 3)

- [Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence](https://mlsanity.com/paper/2402.17722/) (2024)
- [Accelerated forward-backward and Douglas-Rachford splitting dynamics](https://mlsanity.com/paper/2407.20620/) (2024)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)

## Abstract (arXiv)

In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Lojasiewicz function with an exponent of frac12; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale ℓ1-2 regularized least squares problems [37] illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in [35].