Broximal Gradient Descent: A Projection-Free Sister of 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.

arXiv comment121 pages. Substantially expanded revision with a new title, additional convergence results, practical radius rules, applications, and numerical experiments