# Splitting forward-backward penalty scheme for constrained variational problems

arXiv: https://arxiv.org/abs/1408.0974 · HTML: https://mlsanity.com/paper/1408.0974/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2014.

## Built on by (most cited first, 1 of 1)

- [Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization](https://mlsanity.com/paper/2505.11434/) (2025)

## Abstract (arXiv)

We study a forward backward splitting algorithm that solves the variational inequality beginequation* A x +nabla Phi(x)+ NC (x) ni 0 endequation* where H is a real Hilbert space, A: Hrightrightarrows H is a maximal monotone operator, Phi: H→ℝ is a smooth convex function, and NC is the outward normal cone to a closed convex set Csubset H. The constraint set C is represented as the intersection of the sets of minima of two convex penalization function Psi1:H→ℝ and Psi2: H→ℝcup +∞. The function Psi1 is smooth, the function Psi2 is proper and lower semicontinuous. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn), the algorithm  leftbeginarrayrcl x1 & ∈ & H, xn+1 & = & (I+λn A+λnβnpartialPsi2)-1(xn-λnnablaPhi(xn)-λnβnnablaPsi1(xn)), n≥ 1. endarrayright.  performs forward steps on the smooth parts and backward steps on the other parts. Under suitable assumptions, we obtain weak ergodic convergence of the sequence (xn) to a solution of the variational inequality. Convergence is strong when either A is strongly monotone or Phi is strongly convex. We also obtain weak convergence of the whole sequence (xn) when A is the subdifferential of a proper lower-semicontinuous convex function. This provides a unified setting for several classical and more recent results, in the line of historical research on continuous and discrete gradient-like systems.