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arxiv: 2604.13483 · v1 · submitted 2026-04-15 · 🧮 math.OC

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Broximal Alignment for Global Non-Convex Optimization

Kaja Gruntkowska , Hanmin Li , Xun Qian , Peter Richt\'arik
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Pith reviewed 2026-05-10 12:53 UTC · model grok-4.3

classification 🧮 math.OC
keywords non-convex optimizationproximal point methodglobal convergenceBroximal Alignmenttrust-region methodsnon-smooth optimization
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The pith

Broximal Alignment ensures the ball proximal point method reaches a global minimizer in non-convex problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a structural condition called Broximal Alignment that applies to non-convex objective functions. When this condition holds, the Ball Proximal Point Method is guaranteed to generate a sequence that converges to a global minimizer. The condition imposes no requirements on convexity, smoothness, or Lipschitz continuity and explicitly allows multiple disconnected global minima as well as non-global local minima. It is shown to contain several existing non-convex classes as special cases. The result supplies a convergence certificate that does not rely on gradient norms vanishing at stationary points.

Core claim

We define Broximal Alignment as the property that the ball-proximal updates remain aligned with the global solution set in a manner that forces the iterates to approach a global minimizer. Whenever an objective satisfies this property the sequence produced by the Ball Proximal Point Method converges to a global minimizer, even when the function possesses non-optimal local minima and disconnected global solution sets.

What carries the argument

Broximal Alignment, the structural condition on the objective that forces successive ball-proximal updates to stay consistent with global optimality.

If this is right

  • The Ball Proximal Point Method converges to a global minimizer whenever Broximal Alignment holds.
  • The condition encompasses quasiconvexity, star convexity, quasar convexity, and the aiming condition.
  • Functions with several disconnected global minima are still covered.
  • Non-optimal local minima are permitted without blocking global convergence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical checks for Broximal Alignment on concrete problems could turn the convergence guarantee into a usable certificate.
  • Methods that steer iterates toward alignment during execution might broaden the applicability of trust-region style algorithms.
  • The unification of existing non-convex classes under one condition suggests a route to compare their relative strengths.

Load-bearing premise

The objective function satisfies the Broximal Alignment condition.

What would settle it

A function that meets Broximal Alignment yet whose Ball Proximal Point Method iterates fail to approach any global minimizer.

Figures

Figures reproduced from arXiv: 2604.13483 by Hanmin Li, Kaja Gruntkowska, Peter Richt\'arik, Xun Qian.

Figure 1
Figure 1. Figure 1: The inequality in Assumption 3.1 is equivalent to requiring that u lies inside the ball centered at the midpoint m = 1 2 (x + x⋆) with radius t = 1 2 ∥x − x⋆∥X. Equivalently, the angle at u in the triangle formed by x, u and x⋆ is ≥ 90 de￾grees (in the geometry induced by X). instead postulating this inequality directly, leading to the following assumption (illustrated in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 2
Figure 2. Figure 2: Trajectory of BPM with t = 2π starting from x0 = 20 applied to f(x) = |x| + 10 sin(x) (Example 1). ball BX(x, t) always contains at least two local minimizers whose function values differ. It is easy to check that both Assumptions 3.1 and 3.2 hold for this radius, and so f ∈ F BA I (t) for any t ≥ 2π. The example above shows that our setup does not preclude the existence of multiple local minima. This is i… view at source ↗
Figure 3
Figure 3. Figure 3: Hierarchy of assumptions. Solid arrows encode one￾way implications, dashed lines denote mutual non-implications, and red arrows indicate conditions guaranteeing Broximal Align￾ment for any t > 0. For simplicity, we assume f is L-smooth (i.e., ∥∇f(x) − ∇f(y)∥ ≤ L∥x − y∥ ∀x, y ∈ R d ) with domain dom f = R d . See the appendix and Karimi et al. (2016); Hinder et al. (2019); Khanh & Lara (2025) for proofs. Ba… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a structural condition termed Broximal Alignment under which the Ball Proximal Point Method (BPM) is proven to converge globally to a minimizer for non-convex objectives. The condition is claimed to require no convexity, smoothness, or Lipschitz assumptions, to permit non-optimal local minima and disconnected global minima sets, and to generalize quasiconvexity, star convexity, quasar convexity, and the aiming condition. The work positions this as a new framework for global non-convex optimization that avoids gradient-based stationarity arguments.

Significance. If the central result holds, the manuscript supplies a novel sufficient condition for global convergence of a trust-region-style method in settings where standard assumptions fail and local minima are present. This could broaden the theoretical toolkit for non-convex optimization beyond stationarity certificates, particularly if concrete subclasses or verification procedures are later identified.

major comments (2)
  1. [Definition of Broximal Alignment and main convergence theorem] The Broximal Alignment condition (introduced as the key structural hypothesis for the main convergence theorem) is defined abstractly and shown to be sufficient for global convergence of BPM, yet the manuscript supplies neither an algorithm, a testable criterion, nor a non-trivial sufficient subclass that would allow a user to determine whether a given non-convex function satisfies it. This renders the theorem conditional on a premise whose satisfaction cannot be checked for typical problems containing local minima.
  2. [Generalization statements] The claimed generalizations to quasiconvexity, star convexity, quasar convexity, and the aiming condition are asserted without explicit verification that these classes satisfy Broximal Alignment; the manuscript therefore leaves open whether the new condition is strictly larger than the union of the cited frameworks or merely overlaps with them.
minor comments (1)
  1. [Preliminaries] Notation for the ball-proximal operator and the alignment parameter should be introduced with a single consolidated definition rather than scattered across the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's careful reading of the manuscript and the valuable feedback provided. The comments highlight important aspects regarding the presentation of the Broximal Alignment condition and its generalizations. We respond to each major comment in turn and outline the changes we plan to implement.

read point-by-point responses

  1. Referee: [Definition of Broximal Alignment and main convergence theorem] The Broximal Alignment condition (introduced as the key structural hypothesis for the main convergence theorem) is defined abstractly and shown to be sufficient for global convergence of BPM, yet the manuscript supplies neither an algorithm, a testable criterion, nor a non-trivial sufficient subclass that would allow a user to determine whether a given non-convex function satisfies it. This renders the theorem conditional on a premise whose satisfaction cannot be checked for typical problems containing local minima.

    Authors: We acknowledge that Broximal Alignment is introduced as an abstract condition without a general verification algorithm or criterion. This is by design, as the condition is meant to capture a broad class of functions where standard assumptions do not hold. To make the framework more accessible, we will add to the revised manuscript a section with non-trivial examples and sufficient conditions, including functions exhibiting local minima and disconnected global minima. These examples will illustrate how Broximal Alignment can be satisfied and verified for specific problems. We believe this addresses the concern while preserving the abstract nature of the condition as a theoretical tool. revision: partial

  2. Referee: [Generalization statements] The claimed generalizations to quasiconvexity, star convexity, quasar convexity, and the aiming condition are asserted without explicit verification that these classes satisfy Broximal Alignment; the manuscript therefore leaves open whether the new condition is strictly larger than the union of the cited frameworks or merely overlaps with them.

    Authors: The referee correctly notes that the generalization claims lack explicit verification in the current manuscript. We will rectify this by including detailed proofs in the revision showing that quasiconvex, star-convex, quasar-convex functions, and those satisfying the aiming condition all obey Broximal Alignment. Additionally, we will construct an example of a function that satisfies Broximal Alignment but does not satisfy the other conditions, thereby establishing that the new condition is strictly more general. This will be added as a new subsection. revision: yes

Circularity Check

0 steps flagged

Broximal Alignment condition is independently defined; no reduction of convergence claim to inputs or self-citation chain

full rationale

The paper introduces Broximal Alignment as a new structural condition on the objective function, defined separately from the target convergence result for BPM. The algorithm itself is referenced from prior work (Gruntkowska et al. 2025), but the sufficiency proof under the new condition does not reduce by construction to fitted parameters, self-defined quantities, or a load-bearing self-citation chain. The claimed generalizations to quasiconvexity and related classes are presented as consequences of the definition rather than circular renamings. No equations or steps in the provided derivation chain equate the prediction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are explicitly identified; the framework relies on the newly defined Broximal Alignment condition whose verification details are not provided.

pith-pipeline@v0.9.0 · 5496 in / 1117 out tokens · 21751 ms · 2026-05-10T12:53:33.809245+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 6 canonical work pages

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    Sincegis proper, we havedomg̸=∅andg >−∞everywhere. By construction,f(x)≥g inf >−∞for allx, and domf= domg̸=∅. 15 Broximal Alignment for Global Non-Convex Optimization Hencefis proper. We next verify thatfis closed, i.e., lower semicontinuous. Case 1: x /∈ C.Since C is closed, there exists ε >0 such that B(x, ε)∩ C=∅ . Therefore, for any sequence {xk} with...

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    Let x∈domf satisfy dist(x,X f)> t , and let u∈brox t f(x). Then the ball B(x, t) does not intersect Xf , and in particular does not intersectC. Consequently,f=gonB(x, t), which implies broxt f(x) =brox t g(x). Since Xg ⊆ X f , we also have dist(x,X g)> t , so Assumption 3.1 for g applies. Therefore, there exists x⋆ ∈ X g such that ⟨x−u, u−x ⋆⟩ ≥0. This es...

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    if and only if

    Assume that broxt f(x) ={x} for some x∈domf . If x∈ C , then x∈ X f and the claim holds. Now, suppose x /∈ C. IfB(x, t)∩ C ̸=∅, choosez∈ B(x, t)∩ C. Then f(z) =g inf ≤f(x). Since x is the unique minimizer of f over B(x, t), this forces z=x , contradicting x /∈ C. Hence B(x, t)∩ C=∅ . Therefore,f=gonB(x, t)and broxt f(x) =brox t g(x) ={x}. By Assumption 3....

    2019