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Norm convergence conjecture for relaxed Douglas–Rachford methods on Hilbert lattice cones

Let KK be a Hilbert lattice cone and let AA be a finite-codimension closed affine manifold. For parameters 0≤a,b,c<10\leq a,b,c<1, consider the relaxed Douglas–Rachford iteration defined in the paper. Norm convergence conjecture. The iteration converges in norm whenever

(1−a)=(1−b)=12(1−c).(1-a)=(1-b)=\frac{1}{2(1-c)}.

In particular, the corresponding von Neumann and Douglas–Rachford methods should always converge in norm. The conjecture would resolve the unresolved case (c) of the paper's main convergence result, even in codimension two for the von Neumann method; the source presents it as an expected foundation for resolving that case.

Sources & referencesView supporting material

Primary source

Jonathan M. Borwein, Brailey Sims and Matthew K. Tam, “Norm Convergence of Realistic Projection and Reflection Methods”, arXiv:1312.7323 (2014).

convergenceoptimizationprojection-methods

Progress summary

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No public proof or counterexample has been found, so the conjecture remains open despite a related 2024 result for a different method.

The conjecture asserts norm convergence of the relaxed Douglas–Rachford iteration for a Hilbert lattice cone and a finite-codimension closed affine manifold under the parameter relation $$(1-a)=(1-b)=\frac{1}{2(1-c)}.TheoriginalpapersaysthiswouldimplynormconvergenceforboththevonNeumannandDouglas–RachfordmethodsandresolvecaseThe original paper says this would imply norm convergence for both the von Neumann and Douglas–Rachford methods and resolve casecofitsmaintheorem,includingcodimensionof its main theorem, including codimension2$ for the von Neumann method.

2024 alternating-projection result

A 2024 paper gives new norm-convergence conditions for alternating projections involving a lattice cone and a finite-codimension subspace, including codimension N≥2N\geq 2. It does not claim to prove this relaxed Douglas–Rachford conjecture or settle its codimension-two case.

Current status (as of August 2026): The exact relaxed Douglas–Rachford norm-convergence conjecture remains unsettled; no proof, counterexample, verification, or AI-generated claim was found.

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