{ "schemaVersion": "1.6", "slug": "alternating-projections-exact-worst-case", "title": "Exact worst-case convergence of alternating projections with a common interior ball", "shortTitle": "The worst case is a wedge: exact convergence of alternating projections with a common ball", "url": "https://evidencepress.org/releases/alternating-projections-exact-worst-case/", "oneLine": "A proof of the Luner–Grimmer conjecture: when two convex sets share a ball, the slowest possible run of alternating projections is exactly a thin two-half-plane wedge, in every dimension, with exact extensions to unequal balls and residuals.", "abstract": "Alternating projections (von Neumann's method; POCS in signal processing) look for a point in two convex sets by jumping to the nearest point of one set, then of the other. Suppose both sets contain a common ball of radius δ and the start lies within R of its centre. Luner and Grimmer (arXiv:2410.14811, versions 1 and 2, 2024) built two half-planes for which the distance to the intersection after N cycles equals max over 0 ≤ c ≤ κ of c^(2N−1)(√(R²−δ²) − δ√((1+c)/(1−c))), with κ = 1 − 2δ²/R², and conjectured that nothing is worse. This candidate proves the conjecture for every N in every dimension with a short hand-checkable argument: each trajectory is tracked by two numbers, compared with the half-plane trajectory through a monotone value function, and closed with a sharp error bound. The same argument gives the exact worst case for balls of different radii about a common centre, after any number (at least two) of single projections and in real Hilbert space (Theorem 7), and the exact worst residual, which recovers the exact rate of Zamani and Glineur (2024) as δ → 0 (Theorem 8). Consequences include the sharp equivalent R⁴κ^(2N)/(8eδ²√(R²−δ²)N), a limiting profile, a computable stopping test and practical cycle counts: at R/δ = 10 and accuracy 10⁻³ the exact worst case needs 93 cycles, against 212 from an elementary bound with the sharp factor and 833 from the classical interior-point bound. The formula and the extremal half-planes are Luner and Grimmer's; the contribution is the matching upper bound and the extensions. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-03", "dateModified": "2026-11-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23922446", "doiUrl": "https://doi.org/10.5281/zenodo.23922446", "conceptDoi": "10.5281/zenodo.23922445", "pdfUrl": "https://github.com/ipitchford/alternating-projections-exact-worst-case/releases/download/v0.1.0-candidate/alternating-projections-exact-worst-case-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23922446", "repoUrl": "https://github.com/ipitchford/alternating-projections-exact-worst-case", "releaseUrl": "https://github.com/ipitchford/alternating-projections-exact-worst-case/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/alternating-projections-exact-worst-case/index.md", "bibtexUrl": "https://evidencepress.org/releases/alternating-projections-exact-worst-case/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/alternating-projections-exact-worst-case.mp3", "imageUrl": "https://evidencepress.org/assets/og/alternating-projections-exact-worst-case.png?v=86a81ba1ca", "coverArtUrl": "https://evidencepress.org/assets/art/alternating-projections-exact-worst-case.svg?v=737904647e", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/alternating-projections-exact-worst-case.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/alternating-projections-exact-worst-case.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematics candidate. Every result has a written proof; the computations are finite checks that can reveal errors but do not prove the theorems. Internal AI-assisted review in five roles, an adversarial review by a second model, a proof audit, a second-model proof audit of the new proofs and a re-review found no correctness issue. External specialist review, independent reproduction and formal verification have not been established. The prior-art search was bounded." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23922446", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Asset parity was checked against the frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "run_checks.py (88,902 checks) was rerun with Python -I -B and with -O from a fresh copy of the approved package and again from the frozen archive; outputs matched the shipped results apart from timings and the manuscript hash, and the figures were regenerated byte for byte. GitHub CI reruns the checks on Linux in both modes." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Two independently written exact suites check the scalar identities, and internal reviewers ran their own adversarial searches. This is producer-side diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "The checks need only the Python standard library; the figure script pins numpy 2.3.5 and matplotlib 3.10.8 in requirements.txt; not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematics and exact computation", "AI-assisted internal review, second-model adversarial review and proof audit, proof audit, clean replay, re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (approved text of 3 November 2026 with DOI insertion, 19 pages), its check suites and the frozen release archive. The versioned GitHub and Zenodo research archives were checked byte for byte against the frozen release assets on 3 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "alternating projections", "von Neumann method", "POCS", "convex feasibility", "worst-case convergence", "performance estimation", "interior ball", "Hilbert space" ], "keyResults": [ "Exact worst case (Theorem 1, proof): for closed convex sets in any dimension containing a common ball of radius δ, with the start within R of its centre, the distance to the intersection after N ≥ 1 cycles is at most max over 0 ≤ c ≤ κ of c^(2N−1)(√(R²−δ²) − δ√((1+c)/(1−c))), κ = 1 − 2δ²/R², and two half-planes attain it; if R ≤ √2·δ one cycle suffices. This proves Conjecture 6.2 of Luner and Grimmer (2024), whose formula and construction these are.", "Extensions (Theorems 7 and 8, proof): exact worst cases for balls of different radii a and b about a common centre, after any number k ≥ 2 of single projections, in real Hilbert space; and the exact worst residual, which tends to the Zamani–Glineur (2024) rate as δ → 0.", "Sharp asymptotics (Corollaries 5-6, proof): W_N ~ R⁴κ^(2N)/(8eδ²√(R²−δ²)N); a limiting profile when Nδ²/R² is held fixed; after the first projection, near-worst instances lie only Θ(1/N) from the intersection.", "In practice (Section 9): at R/δ = 10 and accuracy 10⁻³ the exact worst case needs 93 cycles, against 212 from an elementary bound with the sharp factor κ and 833 from the classical interior-point bound; Lemma 3 also gives a computable stopping test for any run." ], "reviews": [], "evidencePackage": "Written proofs of every theorem, lemma, corollary and proposition in a 19-page manuscript. Four standard-library check suites run 88,902 explicit checks in exact rational or high-precision decimal arithmetic: the scalar identities and inequalities of Lemmas 2-4 (two independently written suites), exact projections onto 160 polygon pairs against the formula, and the extension suite (two-radius identities, attainment, a randomised search over 720 instances, the Zamani-Glineur limit, asymptotics and the cycle-count tables). Planted errors make every suite fail. The suites were rerun from a fresh copy and from the frozen archive with and without Python -O, and the figures were regenerated byte for byte.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Luner, A., & Grimmer, B. (2024). Performance estimation for smooth and strongly convex sets. arXiv:2410.14811 (versions 1 and 2, preprint). Source of the formula, the attaining half-planes (Proposition 6.1) and Conjecture 6.2, proved here.", "url": "https://arxiv.org/abs/2410.14811v1", "doi": "10.48550/arXiv.2410.14811" }, { "citation": "Luner, A., & Grimmer, B. (2026). Performance estimation for smooth and strongly convex sets. Mathematical Programming. Version of record; it omits the alternating-projections section.", "url": "https://doi.org/10.1007/s10107-026-02373-x", "doi": "10.1007/s10107-026-02373-x" }, { "citation": "Zamani, M., & Glineur, F. (2024). Exact convergence rate of the subgradient method by using Polyak step size. arXiv:2407.15195. Exact worst residual of alternating projections without interiority, recovered here as a limit.", "url": "https://doi.org/10.48550/arXiv.2407.15195", "doi": "10.48550/arXiv.2407.15195" }, { "citation": "Kayalar, S., & Weinert, H. L. (1988). Error bounds for the method of alternating projections. Mathematics of Control, Signals, and Systems. The exact rate for two subspaces.", "url": "https://doi.org/10.1007/BF02551235", "doi": "10.1007/BF02551235" }, { "citation": "Gubin, L. G., Polyak, B. T., & Raik, E. V. (1967). The method of projections for finding the common point of convex sets. USSR Computational Mathematics and Mathematical Physics. The classical linear rate under an interior point.", "url": "https://doi.org/10.1016/0041-5553(67)90113-9", "doi": "10.1016/0041-5553(67)90113-9" }, { "citation": "De Bernardi, C. A., & Miglierina, E. (2021). A variational approach to the alternating projections method. Journal of Global Optimization. A non-sharp radius-loss inequality with telescoping.", "url": "https://doi.org/10.1007/s10898-021-01025-y", "doi": "10.1007/s10898-021-01025-y" }, { "citation": "Bauschke, H. H., & Borwein, J. M. (1993). On the convergence of von Neumann's alternating projection algorithm for two sets. Set-Valued Analysis. Convergence theory of the method.", "url": "https://doi.org/10.1007/BF01027691", "doi": "10.1007/BF01027691" }, { "citation": "Bauschke, H. H., & Borwein, J. M. (1996). On projection algorithms for solving convex feasibility problems. SIAM Review. Survey of projection methods.", "url": "https://doi.org/10.1137/S0036144593251710", "doi": "10.1137/S0036144593251710" }, { "citation": "Bauschke, H. H., Borwein, J. M., & Li, W. (1999). Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization. Mathematical Programming. Error bounds of the kind sharpened by Lemma 3.", "url": "https://doi.org/10.1007/s101070050083", "doi": "10.1007/s101070050083" }, { "citation": "Bauschke, H. H., Deutsch, F., & Hundal, H. (2009). Characterizing arbitrarily slow convergence in the method of alternating projections. International Transactions in Operational Research. Why rates need conditions such as an interior ball.", "url": "https://doi.org/10.1111/j.1475-3995.2008.00682.x", "doi": "10.1111/j.1475-3995.2008.00682.x" }, { "citation": "Deutsch, F. (2001). Best Approximation in Inner Product Spaces. Springer. Standard reference for projections and the subspace case.", "url": "https://doi.org/10.1007/978-1-4684-9298-9", "doi": "10.1007/978-1-4684-9298-9" }, { "citation": "Drori, Y., & Teboulle, M. (2013). Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming. Origin of performance estimation.", "url": "https://doi.org/10.1007/s10107-013-0653-0", "doi": "10.1007/s10107-013-0653-0" }, { "citation": "Taylor, A. B., Hendrickx, J. M., & Glineur, F. (2016). Smooth strongly convex interpolation and exact worst-case performance of first-order methods. Mathematical Programming. Exact worst-case analysis of first-order methods.", "url": "https://doi.org/10.1007/s10107-016-1009-3", "doi": "10.1007/s10107-016-1009-3" }, { "citation": "Ochiai, H., Sekiguchi, Y., & Waki, H. (2023). Exact convergence rates of alternating projections for nontransversal intersections. Japan Journal of Industrial and Applied Mathematics. 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