The proximal point method and its two variants for monotone vector fields in Hadamard spaces

Fumiaki Kohsaka; Parin Chaipunya

arxiv: 2605.01354 · v1 · submitted 2026-05-02 · 🧮 math.FA · math.MG· math.OC

The proximal point method and its two variants for monotone vector fields in Hadamard spaces

Parin Chaipunya , Fumiaki Kohsaka This is my paper

Pith reviewed 2026-05-08 19:36 UTC · model grok-4.3

classification 🧮 math.FA math.MGmath.OC

keywords spacesfieldsmonotonevectorhadamardmethodpointproximal

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The pith

The proximal point method and its variants converge for monotone vector fields in Hadamard spaces.

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

Hadamard spaces are complete metric spaces with non-positive curvature, generalizing Euclidean space in a way that allows for unique geodesics. Monotone vector fields behave like gradients that do not point in opposing directions along lines. The proximal point method is an iterative procedure that finds points where such a field equals zero by solving a regularized subproblem at each step. The paper first studies basic tools like tangent spaces and resolvents in these spaces, then shows that the standard method and two modified versions produce sequences that exist and converge to a zero of the field.

Core claim

We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces.

Load-bearing premise

The underlying space is Hadamard (complete CAT(0)) and the vector field is monotone, allowing resolvents to be single-valued and the iteration to be well-defined.

read the original abstract

We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces. Before obtaining our results, we investigate some fundamental properties of tangent spaces, resolvents, and monotone vector fields in such spaces.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of Hadamard spaces and monotonicity; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Hadamard spaces are complete CAT(0) metric spaces with unique geodesics
Invoked to define tangent spaces and resolvents
domain assumption Monotone vector fields admit well-defined resolvents in these spaces
Central to the proximal iteration

pith-pipeline@v0.9.0 · 5342 in / 1138 out tokens · 75756 ms · 2026-05-08T19:36:26.147893+00:00 · methodology

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.

Cost.FunctionalEquation; Foundation.LogicAsFunctionalEquation washburn_uniqueness_aczel unclear

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

ρ((1−α)x⊕αy, z)^2 ≤ (1−α)ρ(x,z)^2 + αρ(y,z)^2 − α(1−α)ρ(x,y)^2 (CAT(0) inequality 2.4); quasilinearization ⟨xy, zw⟩ = ½(ρ(x,w)^2 + ρ(y,z)^2 − ρ(x,z)^2 − ρ(y,w)^2).

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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

22 extracted references · 22 canonical work pages

[1]

Ahmadi Kakavandi and M

B. Ahmadi Kakavandi and M. Amini,Duality and subdifferential for convex functions on completeCAT(0)metric spaces, Nonlinear Anal.73(2010), no. 10, 3450–3455

work page 2010
[2]

Aoyama, Y

K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda,Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67(2007), 2350–2360

work page 2007
[3]

Aoyama, F

K. Aoyama, F. Kohsaka, and W. Takahashi,Proximal point methods for monotone operators in Banach spaces, Taiwanese J. Math.15(2011), 259–281

work page 2011
[4]

Ariza-Ruiz, L

D. Ariza-Ruiz, L. Leu¸ stean, and G. L´ opez-Acedo,Firmly nonexpansive mappings in classes of geodesic spaces, Trans. Amer. Math. Soc.366(2014), 4299–4322

work page 2014
[5]

Baˇ c´ ak,The proximal point algorithm in metric spaces, Israel J

M. Baˇ c´ ak,The proximal point algorithm in metric spaces, Israel J. Math.194(2013), 689– 701

work page 2013
[6]

,Convex analysis and optimization in Hadamard spaces, De Gruyter, Berlin, 2014

work page 2014
[7]

I. D. Berg and I. G. Nikolaev,Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata133(2008), 195–218

work page 2008
[8]

Br´ ezis and P.-L

H. Br´ ezis and P.-L. Lions,Produits infinis de r´ esolvantes, Israel J. Math.29(1978), 329–345 (French, with English summary)

work page 1978
[9]

M. R. Bridson and A. Haefliger,Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999. 30 P. CHAIPUNYA AND F. KOHSAKA

work page 1999
[10]

Burago, Y

D. Burago, Y. Burago, and S. Ivanov,A course in metric geometry, American Mathematical Society, Providence, RI, 2001

work page 2001
[11]

Chaipunya, F

P. Chaipunya, F. Kohsaka, and P. Kumam,Monotone vector fields and generation of non- expansive semigroups in completeCAT(0)spaces, Numer. Funct. Anal. Optim.42(2021), 989–1018

work page 2021
[12]

Chaipunya and P

P. Chaipunya and P. Kumam,On the proximal point method in Hadamard spaces, Optimiza- tion66(2017), 1647–1665

work page 2017
[13]

Jost,Convex functionals and generalized harmonic maps into spaces of nonpositive cur- vature, Comment

J. Jost,Convex functionals and generalized harmonic maps into spaces of nonpositive cur- vature, Comment. Math. Helv.70(1995), 659–673

work page 1995
[14]

Khatibzadeh and S

H. Khatibzadeh and S. Ranjbar,Monotone operators and the proximal point algorithm in completeCat(0)metric spaces, J. Aust. Math. Soc.103(2017), 70–90

work page 2017
[15]

Kimura and F

Y. Kimura and F. Kohsaka,Two modified proximal point algorithms for convex functions in Hadamard spaces, Linear Nonlinear Anal.2(2016), 69–86

work page 2016
[16]

Kimura and S

Y. Kimura and S. Saejung,Strong convergence for a common fixed point of two different generalizations of cutter operators, Linear Nonlinear Anal.1(2015), 53–65

work page 2015
[17]

W. A. Kirk and B. Panyanak,A concept of convergence in geodesic spaces, Nonlinear Anal. 68(2008), 3689–3696

work page 2008
[18]

Kohsaka,Fixed points of metrically nonspreading mappings in Hadamard spaces, Appl

F. Kohsaka,Fixed points of metrically nonspreading mappings in Hadamard spaces, Appl. Anal. Optim.3(2019), 213–230

work page 2019
[19]

Martinet,R´ egularisation d’in´ equations variationnelles par approximations successives, Rev

B. Martinet,R´ egularisation d’in´ equations variationnelles par approximations successives, Rev. Fran¸ caise Informat. Recherche Op´ erationnelle4(1970), no. S´ er. R-3, 154–158 (French)

work page 1970
[20]

U. F. Mayer,Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom.6(1998), 199–253

work page 1998
[21]

R. T. Rockafellar,Monotone operators and the proximal point algorithm, SIAM J. Control Optimization14(1976), 877–898

work page 1976
[22]

Saejung and P

S. Saejung and P. Yotkaew,Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal.75(2012), 742–750. (P. Chaipunya)Department of Mathematics, F aculty of Science, King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand Email address:parin.cha@kmutt.ac.th (...

work page 2012

[1] [1]

Ahmadi Kakavandi and M

B. Ahmadi Kakavandi and M. Amini,Duality and subdifferential for convex functions on completeCAT(0)metric spaces, Nonlinear Anal.73(2010), no. 10, 3450–3455

work page 2010

[2] [2]

Aoyama, Y

K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda,Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67(2007), 2350–2360

work page 2007

[3] [3]

Aoyama, F

K. Aoyama, F. Kohsaka, and W. Takahashi,Proximal point methods for monotone operators in Banach spaces, Taiwanese J. Math.15(2011), 259–281

work page 2011

[4] [4]

Ariza-Ruiz, L

D. Ariza-Ruiz, L. Leu¸ stean, and G. L´ opez-Acedo,Firmly nonexpansive mappings in classes of geodesic spaces, Trans. Amer. Math. Soc.366(2014), 4299–4322

work page 2014

[5] [5]

Baˇ c´ ak,The proximal point algorithm in metric spaces, Israel J

M. Baˇ c´ ak,The proximal point algorithm in metric spaces, Israel J. Math.194(2013), 689– 701

work page 2013

[6] [6]

,Convex analysis and optimization in Hadamard spaces, De Gruyter, Berlin, 2014

work page 2014

[7] [7]

I. D. Berg and I. G. Nikolaev,Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata133(2008), 195–218

work page 2008

[8] [8]

Br´ ezis and P.-L

H. Br´ ezis and P.-L. Lions,Produits infinis de r´ esolvantes, Israel J. Math.29(1978), 329–345 (French, with English summary)

work page 1978

[9] [9]

M. R. Bridson and A. Haefliger,Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999. 30 P. CHAIPUNYA AND F. KOHSAKA

work page 1999

[10] [10]

Burago, Y

D. Burago, Y. Burago, and S. Ivanov,A course in metric geometry, American Mathematical Society, Providence, RI, 2001

work page 2001

[11] [11]

Chaipunya, F

P. Chaipunya, F. Kohsaka, and P. Kumam,Monotone vector fields and generation of non- expansive semigroups in completeCAT(0)spaces, Numer. Funct. Anal. Optim.42(2021), 989–1018

work page 2021

[12] [12]

Chaipunya and P

P. Chaipunya and P. Kumam,On the proximal point method in Hadamard spaces, Optimiza- tion66(2017), 1647–1665

work page 2017

[13] [13]

Jost,Convex functionals and generalized harmonic maps into spaces of nonpositive cur- vature, Comment

J. Jost,Convex functionals and generalized harmonic maps into spaces of nonpositive cur- vature, Comment. Math. Helv.70(1995), 659–673

work page 1995

[14] [14]

Khatibzadeh and S

H. Khatibzadeh and S. Ranjbar,Monotone operators and the proximal point algorithm in completeCat(0)metric spaces, J. Aust. Math. Soc.103(2017), 70–90

work page 2017

[15] [15]

Kimura and F

Y. Kimura and F. Kohsaka,Two modified proximal point algorithms for convex functions in Hadamard spaces, Linear Nonlinear Anal.2(2016), 69–86

work page 2016

[16] [16]

Kimura and S

Y. Kimura and S. Saejung,Strong convergence for a common fixed point of two different generalizations of cutter operators, Linear Nonlinear Anal.1(2015), 53–65

work page 2015

[17] [17]

W. A. Kirk and B. Panyanak,A concept of convergence in geodesic spaces, Nonlinear Anal. 68(2008), 3689–3696

work page 2008

[18] [18]

Kohsaka,Fixed points of metrically nonspreading mappings in Hadamard spaces, Appl

F. Kohsaka,Fixed points of metrically nonspreading mappings in Hadamard spaces, Appl. Anal. Optim.3(2019), 213–230

work page 2019

[19] [19]

Martinet,R´ egularisation d’in´ equations variationnelles par approximations successives, Rev

B. Martinet,R´ egularisation d’in´ equations variationnelles par approximations successives, Rev. Fran¸ caise Informat. Recherche Op´ erationnelle4(1970), no. S´ er. R-3, 154–158 (French)

work page 1970

[20] [20]

U. F. Mayer,Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom.6(1998), 199–253

work page 1998

[21] [21]

R. T. Rockafellar,Monotone operators and the proximal point algorithm, SIAM J. Control Optimization14(1976), 877–898

work page 1976

[22] [22]

Saejung and P

S. Saejung and P. Yotkaew,Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal.75(2012), 742–750. (P. Chaipunya)Department of Mathematics, F aculty of Science, King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand Email address:parin.cha@kmutt.ac.th (...

work page 2012