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Borwein–Sims basin-of-attraction conjecture for the sphere-line Douglas–Rachford iteration

Let XX be an NN-dimensional Hilbert space, let S:={x∈X ⁣:∥x∥=1}S:=\{x\in X\colon\|x\|=1\} be the unit sphere, and let L:={x=λa+αb ⁣:λ∈R}L:=\{x=\lambda a+\alpha b\colon\lambda\in\mathbb{R}\}, where a,ba,b are orthonormal and 0<α<10<\alpha<1. Then S∩LS\cap L consists of the two points ±1−α2a+αb\pm\sqrt{1-\alpha^2}a+\alpha b, and the Douglas–Rachford iteration is defined by applying the associated Douglas–Rachford operator to an initial point x∈Xx\in X. Borwein–Sims' conjecture. In this simple example, the basin of attraction is the two open half-spaces forming the complement of the singular manifold

⟨x,a⟩=0.\langle x,a\rangle=0.

The cited local convergence theorem establishes convergence near each intersection point, while this conjecture asserts the corresponding global basin structure away from the singular manifold.

Sources & referencesView supporting material

Primary source

Francisco J. Aragón Artacho and Jonathan M. Borwein, “Global convergence of a non-convex Douglas-Rachford iteration”, arXiv:1203.2392 (2014).

douglas-rachfordfixed-point-iterationsnonconvex-optimization

Progress summary

Refreshed
Solved

A 2015 proof establishes that every starting point off the exceptional hyperplane converges to one of the two intersection points, exactly as conjectured.

Borwein and Sims conjectured that the sphere–line Douglas–Rachford iteration has two basins separated by the singular hyperplane {x:⟨x,a⟩=0}\{x:\langle x,a\rangle=0\}. A 2012 preprint recorded the conjecture and proved only a restricted planar case.

Known results

  • Borwein and Sims established local convergence near each feasible point.
  • A 2012 analysis proved a restricted case: N=2N=2 and α=1/2\alpha=1/\sqrt{2}.
  • Borwein and Aragón Artacho obtained sizable attraction domains before the global result.

2015 global convergence theorem

Benoist constructed a Lyapunov function proving global norm convergence for every α∈(0,1)\alpha\in(0,1) when the initial point lies in either complementary open half-space. The singular hyperplane is excluded, and later expositions explicitly identify this as the affirmative resolution of the Borwein–Sims conjecture.

Current status (as of August 2026): The stated sphere–line basin conjecture is settled by Benoist’s 2015 global convergence result; points off ⟨x,a⟩=0\langle x,a\rangle=0 converge to the corresponding intersection point, while behavior on the singular manifold is outside the conjecture.

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