Broximal Gradient Descent: A Projection-Free Sister of Projected Gradient Descent
TL;DR
BroxGD is proposed, a projection-free sister method to projected gradient descent for constrained optimization for constrained optimization that expects to find many applications and inspire numerous extensions, much like projected gradient descent.
Abstract
We propose Broximal Gradient Descent (BroxGD), a projection-free sister method to projected gradient descent for constrained optimization. Its forward--backward construction replaces the proximal backward operation by the broximal operation of Gruntkowska et al. (2025). Instead of projecting, each step minimizes a linear function over the intersection of the constraint set $\mathcal{X}$ and a ball $\mathbb{B}(x_k,t_k)$ centered at the current iterate $x_k$, of suitable radius $t_k>0$: \[ x_{k+1}\in\arg\min_{z\in\mathcal{X}\cap\mathbb{B}(x_k,t_k)}\langle\nabla f(x_k),z\rangle. \] We develop a comprehensive convergence theory spanning a wide range of optimization regimes and radius rules. We expect BroxGD to find many applications and inspire numerous extensions, much like projected gradient descent. Our contribution is theoretical; potential applications and toy experiments illustrate the method and suggest directions for future work.
