Broximal Alignment for Global Non-Convex Optimization

TL;DR

This paper introduces a new framework for global non-convex optimization that bypasses gradient-based methods, using the Ball Proximal Point Method and a novel structural condition called Broximal Alignment.

Contribution

It proposes Broximal Alignment, a structural condition ensuring convergence of BPM to global minima without requiring convexity or smoothness assumptions.

Findings

BPM converges to global minima under Broximal Alignment.

Broximal Alignment generalizes several non-convex frameworks.

The approach does not rely on gradient norms or stationarity.

Abstract

Most non-convex optimization theory is built around gradient dynamics, leaving global convergence largely unexplored. The dominant paradigm focuses on stationarity, certifying only that the gradient norm vanishes, which is often a weak proxy for actual optimization success. In practice, gradient norms can stagnate or even increase during training, and stationary points may be far from global solutions. In this work, we propose a new framework for global non-convex optimization that avoids gradient-based reasoning altogether. We revisit the Ball Proximal Point Method (BPM), a trust-region-style algorithm introduced by Gruntkowska et al. (2025), and propose a novel structural condition - Broximal Alignment - under which BPM provably converges to a global minimizer. Our condition requires no convexity, smoothness, or Lipschitz assumptions, and it permits multiple and disconnected global minima as well as non-optimal local minima. We show that this class generalizes standard non-convex frameworks such as quasiconvexity, star convexity, quasar convexity, and the aiming condition. Our results provide a new conceptual foundation for global non-convex optimization beyond stationarity.