Retractions by Alternating Projections

Shixiang Chen; Wen Huang; Yixiao He

arxiv: 2605.17384 · v2 · pith:H3GXZRPLnew · submitted 2026-05-17 · 🧮 math.OC

Retractions by Alternating Projections

Shixiang Chen , Yixiao He , Wen Huang This is my paper

Pith reviewed 2026-05-21 08:30 UTC · model grok-4.3

classification 🧮 math.OC

keywords alternating projectionsretractions on manifoldsintersection of manifoldsclean intersectionsecond-order retractionmanifold optimizationNewtonSLRA

0 comments

The pith

Alternating projections between two clean-intersecting manifolds define a retraction on their common points.

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

The paper establishes that when two sufficiently smooth embedded submanifolds in Euclidean space intersect cleanly, repeated alternating projections between them converge locally to a map on the intersection manifold. This limiting map satisfies the definition of a retraction, allowing direct use in manifold-constrained optimization without explicit charts. When the manifolds have one extra degree of smoothness, the map becomes a second-order retraction. The same construction also recovers the local behavior of the NewtonSLRA method as a second-order retraction on the intersection. This supplies a new family of retraction-based algorithms for optimization problems whose feasible set is an intersection.

Core claim

Under the assumption that two C^{2,1} embedded submanifolds M1, M2 subset R^n intersect cleanly, the associated alternating mapping admits a well-defined local limiting map ψ on the intersection manifold M = M1 ∩ M2, and that ψ is a retraction on M. If, in addition, M1 and M2 are C^{3,1}, then ψ is a second-order retraction. The standard NewtonSLRA scheme can be understood as inducing a second-order retraction on M.

What carries the argument

The local limiting map ψ of the alternating projection operator, which sends nearby points to the intersection and obeys the first-order retraction conditions (identity on M and differential equal to the identity at points of M).

If this is right

This supplies retraction-based optimization methods for any problem whose constraint set is the intersection of two smooth manifolds.
The framework covers both exact and inexact alternating-projection-type iterations in a unified way.
NewtonSLRA on low-rank matrix problems is revealed to be a second-order retraction method on the intersection manifold.
Local convergence analyses for manifold optimization can now be applied directly to alternating-projection iterates on intersections.

Where Pith is reading between the lines

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

The construction may extend to inexact or noisy projections that still converge to the same limiting retraction.
Similar limiting maps could be derived for alternating projections among three or more manifolds whose total intersection is clean.
The retraction property opens the door to using standard Riemannian trust-region or line-search methods on intersection constraints without deriving new formulas.

Load-bearing premise

The two submanifolds must cross each other in a way that their tangent spaces together fill the whole space at every common point.

What would settle it

Take two concrete C^{2,1} surfaces in R^3 that intersect cleanly, run many steps of alternating projection from a nearby test point, and check whether the limit lies exactly on the intersection while the map's derivative at intersection points equals the identity.

Figures

Figures reproduced from arXiv: 2605.17384 by Shixiang Chen, Wen Huang, Yixiao He.

**Figure 1.** Figure 1: Tangential retraction error ∥PTR0Mr RetrR0 (tη) − (R0 + tη) ∥F versus the step size t in log–log scale. For very small t, the error curves plateau around 10−15 due to floating-point roundoff in double precision. The numerical results are reported in [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗

read the original abstract

Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two $C^{2,1}$ embedded submanifolds $\mathcal{M}_1, \mathcal{M}_2 \subset \mathbb{R}^n$ intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map $\psi$ on the intersection manifold $\mathcal{M}=\mathcal{M}_1\cap \mathcal{M}_2$, and that $\psi$ is a retraction on $\mathcal{M}$. If, in addition, $\mathcal{M}_1$ and $\mathcal{M}_2$ are $C^{3,1}$, then $\psi$ is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on $\M$. This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold.

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.

Desk Editor's Note private letter to a colleague

Alternating projections induce a retraction on clean intersections of two manifolds, giving a practical new tool for optimization on those sets.

read the letter

The paper shows that for two C^{2,1} embedded submanifolds intersecting cleanly, the alternating projection map has a local limiting map ψ that is a retraction on the intersection manifold M. With C^{3,1} regularity it becomes second-order. They also frame the NewtonSLRA method as inducing such a retraction. This is the main contribution: a direct link between classical alternating projections and the retraction machinery used in manifold optimization, including for inexact variants.

Referee Report

0 major / 3 minor

Summary. The paper develops a unified framework showing that, for two C^{2,1} embedded submanifolds M1, M2 subset R^n that intersect cleanly, the alternating-projection mapping admits a well-defined local limiting map ψ on the intersection manifold M = M1 ∩ M2, and that ψ is a retraction on M. Under the additional assumption that M1 and M2 are C^{3,1}, ψ is a second-order retraction. The work further interprets the NewtonSLRA scheme as inducing a second-order retraction on M, thereby supplying new retraction-based tools for optimization on intersection manifolds.

Significance. If the local convergence and retraction properties hold under the stated hypotheses, the manuscript supplies a concrete bridge between classical alternating-projection methods and the retraction formalism used in manifold optimization. The clean-intersection hypothesis together with the explicit regularity classes (C^{2,1} and C^{3,1}) yields a precise setting in which the limiting map ψ can be shown to satisfy the first- and second-order retraction axioms, and the NewtonSLRA example demonstrates immediate applicability. This perspective may allow existing manifold-optimization algorithms to be applied directly to intersection-constrained problems without additional transversality assumptions.

minor comments (3)

[Abstract] Abstract: the phrase 'the associated alternating mapping' is introduced without a one-sentence definition or reference to the standard iteration; a parenthetical clarification would improve readability for readers outside the immediate area.
[Main results] The transition from the C^{2,1} case (first-order retraction) to the C^{3,1} case (second-order retraction) is stated clearly, but the precise Taylor-expansion argument that upgrades the order could be highlighted with an explicit equation reference in the main theorem statement.
[Framework for inexact methods] The discussion of inexact projections is mentioned in the abstract but receives limited space in the provided text; a short paragraph summarizing how the error terms are controlled in the limit would strengthen the claim of robustness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our work and for the positive assessment of its significance in bridging alternating projections with retraction-based manifold optimization. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under clean-intersection hypothesis

full rationale

The central result establishes existence of a local limiting map ψ from the alternating projection operator that satisfies the retraction axioms on M = M1 ∩ M2 when M1, M2 are C^{2,1} and intersect cleanly. This follows directly from standard local analysis of projections (differentiability of the projection operators near the intersection, fixed-point properties, and Taylor expansion of the composite map) once the tangent-space identity T_x M = T_x M1 ∩ T_x M2 is granted by the clean-intersection assumption. The upgrade to second-order retraction under C^{3,1} regularity is likewise a direct consequence of one extra derivative in the expansion. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation whose content is unverified outside the present work. The derivation therefore remains independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the clean-intersection assumption for C^{2,1} manifolds and the existence of the limiting map ψ; no numerical free parameters are introduced.

axioms (1)

domain assumption Two C^{2,1} embedded submanifolds intersect cleanly
Invoked to ensure the alternating mapping has a well-defined local limit on the intersection manifold.

invented entities (1)

local limiting map ψ no independent evidence
purpose: Defines the retraction induced by the alternating-projection iteration on the intersection
Constructed as the limit of the alternating mapping; no independent falsifiable evidence supplied beyond the derivation itself.

pith-pipeline@v0.9.0 · 5736 in / 1367 out tokens · 37971 ms · 2026-05-21T08:30:43.184553+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

under the assumption that two C^{2,1} embedded submanifolds M1, M2 ⊂ R^n intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map ψ on the intersection manifold M = M1 ∩ M2, and that ψ is a retraction on M

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

[1]

¨Uber einige abbildungsaufgaben.Gesammelte Mathematische Abhandlungen 11, pages 65–83, 1869

HA Schwarz. ¨Uber einige abbildungsaufgaben.Gesammelte Mathematische Abhandlungen 11, pages 65–83, 1869

work page
[2]

John von Neumann.Functional Operators. II. The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, 22. Princeton University Press, Princeton, NJ, 1950

work page 1950
[3]

The method of successive projections for finding a common point of convex sets

Lev M Bregman. The method of successive projections for finding a common point of convex sets. InSoviet Math. Dokl., volume 6, pages 688–692, 1965

work page 1965
[4]

The method of projections for finding the common point of convex sets.USSR Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967

LG Gubin, Boris T Polyak, and EV Raik. The method of projections for finding the common point of convex sets.USSR Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967

work page 1967
[5]

On the convergence of von neumann’s alternating projection algorithm for two sets.Set-Valued Analysis, 1(2):185–212, 1993

Heinz H Bauschke and Jonathan M Borwein. On the convergence of von neumann’s alternating projection algorithm for two sets.Set-Valued Analysis, 1(2):185–212, 1993

work page 1993
[6]

On projection algorithms for solving convex feasibility problems.SIAM review, 38(3):367–426, 1996

Heinz H Bauschke and Jonathan M Borwein. On projection algorithms for solving convex feasibility problems.SIAM review, 38(3):367–426, 1996

work page 1996
[7]

Local linear convergence for alternating and averaged nonconvex projections.Foundations of Computational Mathematics, 9(4):485– 513, 2009

Adrian S Lewis, D Russell Luke, and J´ erˆ ome Malick. Local linear convergence for alternating and averaged nonconvex projections.Foundations of Computational Mathematics, 9(4):485– 513, 2009

work page 2009
[8]

Alternating projections on manifolds.Mathematics of Operations Research, 33(1):216–234, 2008

Adrian S Lewis and J´ erˆ ome Malick. Alternating projections on manifolds.Mathematics of Operations Research, 33(1):216–234, 2008

work page 2008
[9]

Transversality and alternating projections for nonconvex sets.Foundations of Computational Mathematics, 15(6):1637–1651, 2015

Dmitriy Drusvyatskiy, Alexander D Ioffe, and Adrian S Lewis. Transversality and alternating projections for nonconvex sets.Foundations of Computational Mathematics, 15(6):1637–1651, 2015

work page 2015
[10]

Alternating projections on nontangential manifolds

Fredrik Andersson and Marcus Carlsson. Alternating projections on nontangential manifolds. Constructive approximation, 38(3):489–525, 2013

work page 2013
[11]

Restricted normal cones and the method of alternating projections: applications.Set-Valued and Variational Analysis, 21(3):475–501, 2013

Heinz H Bauschke, D Russell Luke, Hung M Phan, and Xianfu Wang. Restricted normal cones and the method of alternating projections: applications.Set-Valued and Variational Analysis, 21(3):475–501, 2013

work page 2013
[12]

Restricted normal cones and the method of alternating projections: theory.Set-Valued and Variational Analysis, 21(3):431–473, 2013

Heinz H Bauschke, D Russell Luke, Hung M Phan, and Xianfu Wang. Restricted normal cones and the method of alternating projections: theory.Set-Valued and Variational Analysis, 21(3):431–473, 2013

work page 2013
[13]

On local convergence of the method of alternating projections.Foundations of Computational Mathematics, 16:425–455, 2016

Dominikus Noll and Aude Rondepierre. On local convergence of the method of alternating projections.Foundations of Computational Mathematics, 16:425–455, 2016

work page 2016
[14]

Regularity of collections of sets and convergence of inexact alternating projections.Journal of Convex Analysis, 23(3):823–847, 2016

Alexander Y Kruger and Nguyen H Thao. Regularity of collections of sets and convergence of inexact alternating projections.Journal of Convex Analysis, 23(3):823–847, 2016

work page 2016
[15]

Local linear convergence for inexact alternating projections on nonconvex sets.Vietnam Journal of Mathematics, 47:669–681, 2019

Dmitriy Drusvyatskiy and Adrian S Lewis. Local linear convergence for inexact alternating projections on nonconvex sets.Vietnam Journal of Mathematics, 47:669–681, 2019

work page 2019
[16]

A quadratically convergent algorithm for structured low-rank approximation.Foundations of Computational Mathematics, 16(2):457–492, 2016

´Eric Schost and Pierre-Jean Spaenlehauer. A quadratically convergent algorithm for structured low-rank approximation.Foundations of Computational Mathematics, 16(2):457–492, 2016. 48

work page 2016
[17]

Relaxed newtonslra for approximate gcd

Kosaku Nagasaka. Relaxed newtonslra for approximate gcd. InInternational Workshop on Computer Algebra in Scientific Computing, pages 272–292. Springer, 2021

work page 2021
[18]

A quadratically convergent alternating projection method for nonconvex sets.arXiv preprint arXiv:2511.22916, 2025

Nachuan Xiao, Shiwei Wang, Tianyun Tang, and Kim-Chuan Toh. A quadratically convergent alternating projection method for nonconvex sets.arXiv preprint arXiv:2511.22916, 2025

work page arXiv 2025
[19]

Absil and J´ erˆ ome Malick

P.-A. Absil and J´ erˆ ome Malick. Projection-like retractions on matrix manifolds.SIAM Journal on Optimization, 22(1):135–158, 2012

work page 2012
[20]

Optimization techniques on riemannian manifolds.Fields Institute Commu- nications, 3, 1994

Steven T Smith. Optimization techniques on riemannian manifolds.Fields Institute Commu- nications, 3, 1994

work page 1994
[21]

Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre.Optimization algorithms on matrix manifolds. Princeton University Press, 2009

work page 2009
[22]

Absil, Christopher G Baker, and Kyle A Gallivan

P.-A. Absil, Christopher G Baker, and Kyle A Gallivan. Trust-region methods on riemannian manifolds.Foundations of Computational Mathematics, 7:303–330, 2007

work page 2007
[23]

Cambridge University Press, 2023

Nicolas Boumal.An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

work page 2023
[24]

An adaptive regularized proximal newton-type methods for composite optimization over the stiefel manifold.Computational Optimization and Applica- tions, 89(2):419–457, 2024

Qinsi Wang and Wei Hong Yang. An adaptive regularized proximal newton-type methods for composite optimization over the stiefel manifold.Computational Optimization and Applica- tions, 89(2):419–457, 2024

work page 2024
[25]

A low-rank augmented lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs.Operations Research, 2025

Di Hou, Tianyun Tang, and Kim-Chuan Toh. A low-rank augmented lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs.Operations Research, 2025

work page 2025
[26]

New York: Springer, 2012

John M Lee.Introduction to smooth manifolds. New York: Springer, 2012

work page 2012
[27]

Variational analysis of determinantal varieties.arXiv preprint arXiv:2511.22613, 2025

Yan Yang, Bin Gao, and Ya-xiang Yuan. Variational analysis of determinantal varieties.arXiv preprint arXiv:2511.22613, 2025

work page arXiv 2025
[28]

Local differentiability of distance functions

Ren´ e Poliquin, R Rockafellar, and Lionel Thibault. Local differentiability of distance functions. Transactions of the American mathematical Society, 352(11):5231–5249, 2000

work page 2000
[29]

Rockafellar and R

R.T. Rockafellar and R. J-B Wets.Variational analysis, volume 317. Springer Science & Business Media, 2009

work page 2009
[30]

Springer, 2001

Frank Deutsch and F Deutsch.Best approximation in inner product spaces, volume 7. Springer, 2001

work page 2001
[31]

Global rates of convergence for nonconvex optimization on manifolds.IMA Journal of Numerical Analysis, 39(1):1–33, 2019

Nicolas Boumal, Pierre-Antoine Absil, and Coralia Cartis. Global rates of convergence for nonconvex optimization on manifolds.IMA Journal of Numerical Analysis, 39(1):1–33, 2019

work page 2019
[32]

Proximal gradient method for nonsmooth optimization over the Stiefel manifold.SIAM Journal on Optimization, 30(1):210–239, 2020

Shixiang Chen, Shiqian Ma, Anthony Man-Cho So, and Tong Zhang. Proximal gradient method for nonsmooth optimization over the Stiefel manifold.SIAM Journal on Optimization, 30(1):210–239, 2020

work page 2020
[33]

Stochastic optimization over proximally smooth sets.SIAM Journal on Optimization, 35(1):157–179, 2025

Damek Davis, Dmitriy Drusvyatskiy, and Zhan Shi. Stochastic optimization over proximally smooth sets.SIAM Journal on Optimization, 35(1):157–179, 2025. 49

work page 2025
[34]

Qaplib–a quadratic assign- ment problem library.Avialable at http://www

Peter Hahn, Miguel Anjos, RE Burkard, SE Karisch, and F Rendl. Qaplib–a quadratic assign- ment problem library.Avialable at http://www. seas. upenn. edu/qaplib, 2006

work page 2006
[35]

Absil and I

P.-A. Absil and I. V Oseledets. Low-rank retractions: a survey and new results.Computational Optimization and Applications, 62(1):5–29, 2015. Appendix A Technical lemma Lemma A.1.Let{t k}k≥0 and{a k}k≥0 be two nonnegative sequences satisfying tk+1 ≤(1 +ε k)t k +α k ak +p k, k≥0,(A.1) ak+1 ≤c a k +β k tk +b k, k≥0,(A.2) wherec∈[0,1)andε k, αk, βk, pk, bk ≥...

work page 2015

[1] [1]

¨Uber einige abbildungsaufgaben.Gesammelte Mathematische Abhandlungen 11, pages 65–83, 1869

HA Schwarz. ¨Uber einige abbildungsaufgaben.Gesammelte Mathematische Abhandlungen 11, pages 65–83, 1869

work page

[2] [2]

John von Neumann.Functional Operators. II. The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, 22. Princeton University Press, Princeton, NJ, 1950

work page 1950

[3] [3]

The method of successive projections for finding a common point of convex sets

Lev M Bregman. The method of successive projections for finding a common point of convex sets. InSoviet Math. Dokl., volume 6, pages 688–692, 1965

work page 1965

[4] [4]

The method of projections for finding the common point of convex sets.USSR Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967

LG Gubin, Boris T Polyak, and EV Raik. The method of projections for finding the common point of convex sets.USSR Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967

work page 1967

[5] [5]

On the convergence of von neumann’s alternating projection algorithm for two sets.Set-Valued Analysis, 1(2):185–212, 1993

Heinz H Bauschke and Jonathan M Borwein. On the convergence of von neumann’s alternating projection algorithm for two sets.Set-Valued Analysis, 1(2):185–212, 1993

work page 1993

[6] [6]

On projection algorithms for solving convex feasibility problems.SIAM review, 38(3):367–426, 1996

Heinz H Bauschke and Jonathan M Borwein. On projection algorithms for solving convex feasibility problems.SIAM review, 38(3):367–426, 1996

work page 1996

[7] [7]

Local linear convergence for alternating and averaged nonconvex projections.Foundations of Computational Mathematics, 9(4):485– 513, 2009

Adrian S Lewis, D Russell Luke, and J´ erˆ ome Malick. Local linear convergence for alternating and averaged nonconvex projections.Foundations of Computational Mathematics, 9(4):485– 513, 2009

work page 2009

[8] [8]

Alternating projections on manifolds.Mathematics of Operations Research, 33(1):216–234, 2008

Adrian S Lewis and J´ erˆ ome Malick. Alternating projections on manifolds.Mathematics of Operations Research, 33(1):216–234, 2008

work page 2008

[9] [9]

Transversality and alternating projections for nonconvex sets.Foundations of Computational Mathematics, 15(6):1637–1651, 2015

Dmitriy Drusvyatskiy, Alexander D Ioffe, and Adrian S Lewis. Transversality and alternating projections for nonconvex sets.Foundations of Computational Mathematics, 15(6):1637–1651, 2015

work page 2015

[10] [10]

Alternating projections on nontangential manifolds

Fredrik Andersson and Marcus Carlsson. Alternating projections on nontangential manifolds. Constructive approximation, 38(3):489–525, 2013

work page 2013

[11] [11]

Restricted normal cones and the method of alternating projections: applications.Set-Valued and Variational Analysis, 21(3):475–501, 2013

Heinz H Bauschke, D Russell Luke, Hung M Phan, and Xianfu Wang. Restricted normal cones and the method of alternating projections: applications.Set-Valued and Variational Analysis, 21(3):475–501, 2013

work page 2013

[12] [12]

Restricted normal cones and the method of alternating projections: theory.Set-Valued and Variational Analysis, 21(3):431–473, 2013

Heinz H Bauschke, D Russell Luke, Hung M Phan, and Xianfu Wang. Restricted normal cones and the method of alternating projections: theory.Set-Valued and Variational Analysis, 21(3):431–473, 2013

work page 2013

[13] [13]

On local convergence of the method of alternating projections.Foundations of Computational Mathematics, 16:425–455, 2016

Dominikus Noll and Aude Rondepierre. On local convergence of the method of alternating projections.Foundations of Computational Mathematics, 16:425–455, 2016

work page 2016

[14] [14]

Regularity of collections of sets and convergence of inexact alternating projections.Journal of Convex Analysis, 23(3):823–847, 2016

Alexander Y Kruger and Nguyen H Thao. Regularity of collections of sets and convergence of inexact alternating projections.Journal of Convex Analysis, 23(3):823–847, 2016

work page 2016

[15] [15]

Local linear convergence for inexact alternating projections on nonconvex sets.Vietnam Journal of Mathematics, 47:669–681, 2019

Dmitriy Drusvyatskiy and Adrian S Lewis. Local linear convergence for inexact alternating projections on nonconvex sets.Vietnam Journal of Mathematics, 47:669–681, 2019

work page 2019

[16] [16]

A quadratically convergent algorithm for structured low-rank approximation.Foundations of Computational Mathematics, 16(2):457–492, 2016

´Eric Schost and Pierre-Jean Spaenlehauer. A quadratically convergent algorithm for structured low-rank approximation.Foundations of Computational Mathematics, 16(2):457–492, 2016. 48

work page 2016

[17] [17]

Relaxed newtonslra for approximate gcd

Kosaku Nagasaka. Relaxed newtonslra for approximate gcd. InInternational Workshop on Computer Algebra in Scientific Computing, pages 272–292. Springer, 2021

work page 2021

[18] [18]

A quadratically convergent alternating projection method for nonconvex sets.arXiv preprint arXiv:2511.22916, 2025

Nachuan Xiao, Shiwei Wang, Tianyun Tang, and Kim-Chuan Toh. A quadratically convergent alternating projection method for nonconvex sets.arXiv preprint arXiv:2511.22916, 2025

work page arXiv 2025

[19] [19]

Absil and J´ erˆ ome Malick

P.-A. Absil and J´ erˆ ome Malick. Projection-like retractions on matrix manifolds.SIAM Journal on Optimization, 22(1):135–158, 2012

work page 2012

[20] [20]

Optimization techniques on riemannian manifolds.Fields Institute Commu- nications, 3, 1994

Steven T Smith. Optimization techniques on riemannian manifolds.Fields Institute Commu- nications, 3, 1994

work page 1994

[21] [21]

Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre.Optimization algorithms on matrix manifolds. Princeton University Press, 2009

work page 2009

[22] [22]

Absil, Christopher G Baker, and Kyle A Gallivan

P.-A. Absil, Christopher G Baker, and Kyle A Gallivan. Trust-region methods on riemannian manifolds.Foundations of Computational Mathematics, 7:303–330, 2007

work page 2007

[23] [23]

Cambridge University Press, 2023

Nicolas Boumal.An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

work page 2023

[24] [24]

An adaptive regularized proximal newton-type methods for composite optimization over the stiefel manifold.Computational Optimization and Applica- tions, 89(2):419–457, 2024

Qinsi Wang and Wei Hong Yang. An adaptive regularized proximal newton-type methods for composite optimization over the stiefel manifold.Computational Optimization and Applica- tions, 89(2):419–457, 2024

work page 2024

[25] [25]

A low-rank augmented lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs.Operations Research, 2025

Di Hou, Tianyun Tang, and Kim-Chuan Toh. A low-rank augmented lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs.Operations Research, 2025

work page 2025

[26] [26]

New York: Springer, 2012

John M Lee.Introduction to smooth manifolds. New York: Springer, 2012

work page 2012

[27] [27]

Variational analysis of determinantal varieties.arXiv preprint arXiv:2511.22613, 2025

Yan Yang, Bin Gao, and Ya-xiang Yuan. Variational analysis of determinantal varieties.arXiv preprint arXiv:2511.22613, 2025

work page arXiv 2025

[28] [28]

Local differentiability of distance functions

Ren´ e Poliquin, R Rockafellar, and Lionel Thibault. Local differentiability of distance functions. Transactions of the American mathematical Society, 352(11):5231–5249, 2000

work page 2000

[29] [29]

Rockafellar and R

R.T. Rockafellar and R. J-B Wets.Variational analysis, volume 317. Springer Science & Business Media, 2009

work page 2009

[30] [30]

Springer, 2001

Frank Deutsch and F Deutsch.Best approximation in inner product spaces, volume 7. Springer, 2001

work page 2001

[31] [31]

Global rates of convergence for nonconvex optimization on manifolds.IMA Journal of Numerical Analysis, 39(1):1–33, 2019

Nicolas Boumal, Pierre-Antoine Absil, and Coralia Cartis. Global rates of convergence for nonconvex optimization on manifolds.IMA Journal of Numerical Analysis, 39(1):1–33, 2019

work page 2019

[32] [32]

Proximal gradient method for nonsmooth optimization over the Stiefel manifold.SIAM Journal on Optimization, 30(1):210–239, 2020

Shixiang Chen, Shiqian Ma, Anthony Man-Cho So, and Tong Zhang. Proximal gradient method for nonsmooth optimization over the Stiefel manifold.SIAM Journal on Optimization, 30(1):210–239, 2020

work page 2020

[33] [33]

Stochastic optimization over proximally smooth sets.SIAM Journal on Optimization, 35(1):157–179, 2025

Damek Davis, Dmitriy Drusvyatskiy, and Zhan Shi. Stochastic optimization over proximally smooth sets.SIAM Journal on Optimization, 35(1):157–179, 2025. 49

work page 2025

[34] [34]

Qaplib–a quadratic assign- ment problem library.Avialable at http://www

Peter Hahn, Miguel Anjos, RE Burkard, SE Karisch, and F Rendl. Qaplib–a quadratic assign- ment problem library.Avialable at http://www. seas. upenn. edu/qaplib, 2006

work page 2006

[35] [35]

Absil and I

P.-A. Absil and I. V Oseledets. Low-rank retractions: a survey and new results.Computational Optimization and Applications, 62(1):5–29, 2015. Appendix A Technical lemma Lemma A.1.Let{t k}k≥0 and{a k}k≥0 be two nonnegative sequences satisfying tk+1 ≤(1 +ε k)t k +α k ak +p k, k≥0,(A.1) ak+1 ≤c a k +β k tk +b k, k≥0,(A.2) wherec∈[0,1)andε k, αk, βk, pk, bk ≥...

work page 2015