A Proximal Gradient Framework for Composite Multiobjective Optimization on Riemannian Manifolds

Kangming Chen

arxiv: 2605.16731 · v1 · pith:6TWU7XGDnew · submitted 2026-05-16 · 🧮 math.OC

A Proximal Gradient Framework for Composite Multiobjective Optimization on Riemannian Manifolds

Kangming Chen This is my paper

Pith reviewed 2026-05-19 21:34 UTC · model grok-4.3

classification 🧮 math.OC

keywords multiobjective optimizationRiemannian manifoldsproximal gradient methodsPareto stationary pointscomposite optimizationmanifold optimizationconvergence rates

0 comments

The pith

A proximal gradient method converges composite multiobjective optimization problems on Riemannian manifolds to Pareto stationary points at an O(1/k) rate.

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

The paper proposes a Riemannian Multiobjective Proximal Gradient Method for composite multiobjective optimization on manifolds. It directly addresses vector-valued objectives rather than using scalarization and proves global convergence to Pareto stationary points with an O(1/k) convergence rate. Two variants are introduced for practicality: one allowing inexact subproblem solutions and another using trust regions for adaptive penalties with O(ε^{-2}) complexity. This matters because it provides a direct way to handle trade-off problems in geometric settings where traditional methods may not apply. Numerical results indicate superior performance over subgradient methods.

Core claim

The Riemannian Multiobjective Proximal Gradient Method (RMPGM) is proposed for composite optimization problems on Riemannian manifolds. Unlike scalarization approaches, it directly handles vector-valued objectives and establishes global convergence to Pareto stationary points with an O(1/k) convergence rate. An inexact variant allows controlled inexactness in subproblems, and a trust-region variant adaptively adjusts the penalty parameter to achieve O(ε^{-2}) iteration complexity.

What carries the argument

The Riemannian Multiobjective Proximal Gradient Method (RMPGM) that performs proximal gradient updates via retractions to manage composite vector objectives on the manifold.

If this is right

Global convergence to Pareto stationary points is guaranteed for the proposed method.
The iterates converge at an O(1/k) rate in a stationarity measure.
The inexact variant maintains the convergence guarantees under bounded errors in subproblem solutions.
The trust-region variant reaches ε-accuracy in O(ε^{-2}) iterations by adapting the penalty.
Numerical experiments show consistent outperformance over subgradient-based methods.

Where Pith is reading between the lines

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

Direct vector treatment may reduce the need to select scalarization weights in applied multiobjective settings.
The retraction-based proximal steps could be adapted to stochastic or online versions of the same problems.
Similar ideas might apply to multiobjective problems with manifold constraints in Euclidean space.

Load-bearing premise

The proximal mappings of the composite functions and the retraction operations on the manifold must be well-defined and maintain descent properties for the vector objectives.

What would settle it

A numerical test on a known multiobjective problem on the sphere manifold where the method does not approach a Pareto stationary point or fails to achieve the stated rate would disprove the convergence claims.

read the original abstract

This paper proposes a Riemannian Multiobjective Proximal Gradient Method (RMPGM) for composite optimization problems on manifolds. Unlike scalarization-based approaches, the proposed framework directly handles vector-valued objectives and establishes global convergence to Pareto stationary points, together with an $\mathcal{O}(1/k)$ convergence rate. We further develop two variants to enhance practicality and performance: an inexact RMPGM that allows controlled inexactness in solving subproblems, and a trust-region RMPGM that adaptively adjusts the penalty parameter and achieves an $\mathcal{O}(\epsilon^{-2}) $iteration complexity. Numerical experiments demonstrate that the proposed methods are consistently outperform subgradient-based baselines.

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

Paper develops a direct proximal method for multiobjective Riemannian optimization but convergence to Pareto points likely needs unstated assumptions on manifold curvature for the proximal steps to work.

read the letter

This paper introduces a Riemannian Multiobjective Proximal Gradient Method for composite problems on manifolds. It directly optimizes vector-valued objectives instead of using scalarization and claims global convergence to Pareto stationary points along with an O(1/k) rate. Two variants are added for inexact solves and trust-region adjustments that adapt the penalty parameter and give O(epsilon^{-2}) complexity in one case. The numerical tests show consistent gains over subgradient baselines. The new element is the direct vector-objective handling with proximal steps on the manifold, plus the two practical variants. This extends scalar proximal ideas without forcing a single weighted sum. The experiments provide some concrete evidence that the approach runs and beats the obvious baselines in the tested cases. The soft spot sits in the convergence claims. Global convergence and the rate rest on the proximal subproblem having well-defined, unique solutions under the retraction, plus a vector descent property. On manifolds with positive sectional curvature the squared-distance term loses geodesic convexity, so existence or uniqueness can fail when the composite term is not coercive or convex. The abstract and framework description do not list curvature or convexity restrictions, which leaves the Euclidean-style argument exposed. If the full proofs add those conditions explicitly, the issue is minor; if they do not, it is central. The citation pattern looks standard and draws on existing Riemannian proximal and multiobjective work without obvious circularity. This is for researchers in Riemannian optimization who need multiobjective tools, especially those applying them to manifold-valued machine learning problems. A reader who wants to see how proximal methods extend beyond single-objective scalar cases will find the variants and rates useful to examine. I would send it for serious peer review. The core framework is specific enough that referees can verify the manifold assumptions and the rate derivations directly.

Referee Report

2 major / 2 minor

Summary. The paper proposes a Riemannian Multiobjective Proximal Gradient Method (RMPGM) for composite multiobjective optimization on Riemannian manifolds. It establishes global convergence to Pareto stationary points with an O(1/k) rate, introduces an inexact variant allowing controlled subproblem inexactness and a trust-region variant with adaptive penalty parameter achieving O(ε^{-2}) iteration complexity, and reports numerical experiments where the methods outperform subgradient baselines.

Significance. If the convergence analysis holds under appropriate manifold assumptions, the direct handling of vector-valued objectives without scalarization would represent a useful extension of proximal-gradient ideas to the Riemannian multiobjective setting, with potential relevance to applications involving manifold constraints. The inclusion of rates, inexact/trust-region variants, and reproducible numerical comparisons adds practical value.

major comments (2)

[§3.2 and Theorem 4.1] §3.2 (proximal mapping definition) and Theorem 4.1: the global convergence and O(1/k) rate rest on the proximal subproblem admitting a unique solution via the retraction and on the vector-valued descent property being preserved; however, no curvature or coercivity conditions are stated to guarantee this on general Riemannian manifolds (e.g., positive sectional curvature on the sphere), leaving the adaptation of Euclidean proximal arguments vulnerable.
[§4.3] §4.3 (trust-region variant): the O(ε^{-2}) complexity claim assumes the adaptive penalty update maintains the necessary monotonicity and stationarity measure decrease; the proof sketch does not explicitly verify that the retraction-based update preserves the required inequality when the composite term is non-convex.

minor comments (2)

[Abstract and §5] The abstract states the methods 'consistently outperform' baselines; the numerical section should report the precise stationarity measure and manifold-specific implementation details used for all compared methods.
[§2] Notation for the vector-valued objective and the Pareto stationarity condition should be introduced with a short reminder of the definition in §2 to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to clarify the assumptions and strengthen the proofs.

read point-by-point responses

Referee: [§3.2 and Theorem 4.1] §3.2 (proximal mapping definition) and Theorem 4.1: the global convergence and O(1/k) rate rest on the proximal subproblem admitting a unique solution via the retraction and on the vector-valued descent property being preserved; however, no curvature or coercivity conditions are stated to guarantee this on general Riemannian manifolds (e.g., positive sectional curvature on the sphere), leaving the adaptation of Euclidean proximal arguments vulnerable.

Authors: We agree that the current presentation would benefit from explicit conditions to ensure the proximal subproblem has a unique solution and that the vector-valued descent property holds. In the revised manuscript we will add Assumption 3.2, which requires the manifold to admit a retraction that is a local diffeomorphism and the composite objective to satisfy a coercivity condition (ensuring the subproblem is strongly convex in a neighborhood of the current point). Under this assumption the uniqueness follows from standard arguments, and the descent property is preserved because the retraction approximates the exponential map to first order. We will update the statement of Theorem 4.1 to reference this assumption explicitly. revision: yes
Referee: [§4.3] §4.3 (trust-region variant): the O(ε^{-2}) complexity claim assumes the adaptive penalty update maintains the necessary monotonicity and stationarity measure decrease; the proof sketch does not explicitly verify that the retraction-based update preserves the required inequality when the composite term is non-convex.

Authors: We acknowledge that the proof sketch in §4.3 is concise and should be expanded. The adaptive penalty update is chosen large enough to dominate the non-convexity of the composite term, following the standard strategy in Euclidean non-convex trust-region methods. In the revision we will insert a detailed lemma (Lemma 4.5) that explicitly verifies the monotonicity and stationarity-measure decrease after the retraction step. The argument uses the first-order agreement between the retraction and the geodesic together with a uniform bound on the second-order remainder term (which exists locally on any smooth manifold). This establishes the O(ε^{-2}) complexity without requiring convexity of the composite term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; convergence claims derived from standard proximal analysis

full rationale

The paper adapts proximal gradient methods to Riemannian manifolds for multiobjective composite problems. Global convergence to Pareto points and the O(1/k) rate follow from descent properties of the proximal mapping and retraction, which are standard adaptations rather than self-referential definitions or fitted inputs presented as predictions. No load-bearing self-citations or uniqueness theorems imported from the same authors reduce the central results to tautologies. The framework is self-contained with independent analytical content against external benchmarks like Euclidean proximal gradient theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the method implicitly relies on standard Riemannian geometry and proximal operator existence but these are not detailed.

pith-pipeline@v0.9.0 · 5628 in / 1209 out tokens · 32537 ms · 2026-05-19T21:34:48.109882+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

η_k = argmin_η∈T_xk M [ψ_xk(η) + (L̃/2)‖η‖²_xk] with ψ_xk(η) := max_i ⟨grad f_i(x_k), η⟩ + g_i(R_xk(η)) − g_i(x_k)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

global convergence to Pareto stationary points … O(1/k) convergence rate under retraction-convexity

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

31 extracted references · 31 canonical work pages

[1]

Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds, p. 224. Princeton University Press, Princeton, NJ (2008)

work page 2008
[2]

Cambridge University Press, Cambridge, UK (2023)

Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge, UK (2023). https://doi.org/10.1017/9781009166164 .https://www.nicolasboumal.net/book

work page doi:10.1017/9781009166164 2023
[3]

Foundations of Computational Mathematics7, 303–330 (2007)

Absil, P.-A., Baker, C.G., Gallivan, K.A.: Trust-region methods on riemannian manifolds. Foundations of Computational Mathematics7, 303–330 (2007)

work page 2007
[4]

Computational and Applied Mathematics45(4), 151 (2026)

Chen, K., Fukuda, E.H.: Riemannian conditional gradient methods for compos- ite optimization problems. Computational and Applied Mathematics45(4), 151 (2026)

work page 2026
[5]

SIAM Journal on Optimization32(4), 2690–2717 (2022)

Sato, H.: Riemannian conjugate gradient methods: General framework and spe- cific algorithms with convergence analyses. SIAM Journal on Optimization32(4), 2690–2717 (2022)

work page 2022
[6]

Mathe- matical Programming199(1), 525–556 (2023) 28

Weber, M., Sra, S.: Riemannian optimization via frank-wolfe methods. Mathe- matical Programming199(1), 525–556 (2023) 28

work page 2023
[7]

Mathematical Methods of Operations Research51(3), 479–494 (2000)

Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Mathematical Methods of Operations Research51(3), 479–494 (2000)

work page 2000
[8]

Gra˜ na Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization175(2), 395–414

work page
[9]

Fukuda, E.H., Gra˜ na Drummond, L.M.: Inexact projected gradient method for vector optimization54(3), 473–493

work page
[10]

1007/s10589-018-0043-x

Tanabe, H., Fukuda, E.H., Yamashita, N.: Proximal gradient methods for multi- objective optimization and their applications72(2), 339–361 https://doi.org/10. 1007/s10589-018-0043-x

work page
[11]

Tanabe, H., Fukuda, E.H., Yamashita, N.: Convergence rates analysis of a mul- tiobjective proximal gradient method17(2), 333–350 https://doi.org/10.1007/ s11590-022-01877-7

work page
[12]

Journal of Machine Learning Research 11(2) (2010)

Journ´ ee, M., Nesterov, Y., Richt´ arik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. Journal of Machine Learning Research 11(2) (2010)

work page 2010
[13]

SIAM Journal on Optimization23(2), 1214–1236 (2013)

Vandereycken, B.: Low-rank matrix completion by riemannian optimization. SIAM Journal on Optimization23(2), 1214–1236 (2013)

work page 2013
[14]

Journal of Optimization Theory and Applications154, 88–107 (2012)

Bento, G., Ferreira, O.P., Oliveira, P.R.: Unconstrained steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications154, 88–107 (2012)

work page 2012
[15]

Journal of Optimization Theory and Applications159, 108–124 (2013)

Bento, G., Cruz Neto, J.X., Santos, P.: An inexact steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications159, 108–124 (2013)

work page 2013
[16]

Bento, G.C., Cruz Neto, J.X.: A subgradient method for multiobjective opti- mization on riemannian manifolds159(1), 125–137 https://doi.org/10.1007/ s10957-013-0307-7

work page
[17]

Applied Mathematics and Computation486, 129001 (2025)

Chen, K., Fukuda, E.H., Sato, H.: Nonlinear conjugate gradient method for vec- tor optimization on riemannian manifolds with retraction and vector transport. Applied Mathematics and Computation486, 129001 (2025)

work page 2025
[18]

Accessed 2023-05-31

Najafi, S., Hajarian, M.: Multiobjective conjugate gradient methods on rieman- nian manifolds197(3), 1229–1248 https://doi.org/10.1007/s10957-023-02224-1 . Accessed 2023-05-31

work page doi:10.1007/s10957-023-02224-1 2023
[19]

Bento, G.d.C., Ferreira, O.P., Pereira, Y.R.L.: Proximal point method for vector optimization on hadamard manifolds46(1), 13–18 29

work page
[20]

Journal of Optimization Theory and Applications196(1), 212–239 (2023)

Eslami, N., Najafi, B., Vaezpour, S.M.: A trust region method for solving multi- criteria optimization problems on riemannian manifolds. Journal of Optimization Theory and Applications196(1), 212–239 (2023)

work page 2023
[21]

Journal of Optimization Theory and Applications179(1), 37–52 (2018) https://doi.org/ 10.1007/s10957-018-1330-5

Bento, G.d.C., Cruz Neto, J.X., Meireles, L.V.: Proximal point method for locally lipschitz functions in multiobjective optimization of hadamard manifolds. Journal of Optimization Theory and Applications179(1), 37–52 (2018) https://doi.org/ 10.1007/s10957-018-1330-5

work page doi:10.1007/s10957-018-1330-5 2018
[22]

Numerical Algorithms, 1–51 (2025) https://doi.org/10

Upadhyay, B.B., Poddar, S., Ferreira, O.P., Yao, J.-C.: An inexact proximal point method with quasi-distance for quasiconvex multiobjective optimization problems on riemannian manifolds. Numerical Algorithms, 1–51 (2025) https://doi.org/10. 1007/s11075-025-02177-8

work page 2025
[23]

SIAM Journal on Optimization 30(1), 210–239 (2020)

Chen, S., Ma, S., Man-Cho So, A., Zhang, T.: Proximal gradient method for nonsmooth optimization over the stiefel manifold. SIAM Journal on Optimization 30(1), 210–239 (2020)

work page 2020
[24]

Huang, W., Wei, K.: Riemannian proximal gradient methods194(1), 371–413 https://doi.org/10.1007/s10107-021-01632-3

work page doi:10.1007/s10107-021-01632-3
[25]

Huang, W., Wei, K.: An inexact riemannian proximal gradient method85(1), 1–32 https://doi.org/10.1007/s10589-023-00451-w

work page doi:10.1007/s10589-023-00451-w
[26]

Accessed 2023-09-21

Zhao, S., Yan, T., Zhu, Y.: Proximal gradient algorithm with trust region scheme on riemannian manifold https://doi.org/10.1007/s10898-023-01326-4 . Accessed 2023-09-21

work page doi:10.1007/s10898-023-01326-4 2023
[27]

Publisher: Society for Industrial and Applied Mathematics

Hosseini, S., Huang, W., Yousefpour, R.: Line search algorithms for locally lips- chitz functions on riemannian manifolds28(1), 596–619 https://doi.org/10.1137/ 16M1108145 . Publisher: Society for Industrial and Applied Mathematics

work page
[28]

Classics in Applied Mathe- matics, vol

Clarke, F.H.: Optimization and Nonsmooth Analysis. Classics in Applied Mathe- matics, vol. 5. Society for Industrial and Applied Mathematics, Philadelphia, PA (1990). https://doi.org/10.1137/1.9781611971309

work page doi:10.1137/1.9781611971309 1990
[29]

Nonlinear Analysis: Theory, Methods & Applications74(12), 3884–3895 (2011)

Hosseini, S., Pouryayevali, M.: Generalized gradients and characterization of epi- lipschitz sets in riemannian manifolds. Nonlinear Analysis: Theory, Methods & Applications74(12), 3884–3895 (2011)

work page 2011
[30]

Journal of Mathematical Analysis and Applica- tions457(2), 1333–1352 (2018)

Daniilidis, A., Deville, R., Durand-Cartagena, E., Rifford, L.: Self-contracted curves in riemannian manifolds. Journal of Mathematical Analysis and Applica- tions457(2), 1333–1352 (2018)

work page 2018
[31]

Optimization73(13), 3821–3858 (2024) 30

Tanabe, H., Fukuda, E.H., Yamashita, N.: New merit functions for multiobjective optimization and their properties. Optimization73(13), 3821–3858 (2024) 30

work page 2024

[1] [1]

Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds, p. 224. Princeton University Press, Princeton, NJ (2008)

work page 2008

[2] [2]

Cambridge University Press, Cambridge, UK (2023)

Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge, UK (2023). https://doi.org/10.1017/9781009166164 .https://www.nicolasboumal.net/book

work page doi:10.1017/9781009166164 2023

[3] [3]

Foundations of Computational Mathematics7, 303–330 (2007)

Absil, P.-A., Baker, C.G., Gallivan, K.A.: Trust-region methods on riemannian manifolds. Foundations of Computational Mathematics7, 303–330 (2007)

work page 2007

[4] [4]

Computational and Applied Mathematics45(4), 151 (2026)

Chen, K., Fukuda, E.H.: Riemannian conditional gradient methods for compos- ite optimization problems. Computational and Applied Mathematics45(4), 151 (2026)

work page 2026

[5] [5]

SIAM Journal on Optimization32(4), 2690–2717 (2022)

Sato, H.: Riemannian conjugate gradient methods: General framework and spe- cific algorithms with convergence analyses. SIAM Journal on Optimization32(4), 2690–2717 (2022)

work page 2022

[6] [6]

Mathe- matical Programming199(1), 525–556 (2023) 28

Weber, M., Sra, S.: Riemannian optimization via frank-wolfe methods. Mathe- matical Programming199(1), 525–556 (2023) 28

work page 2023

[7] [7]

Mathematical Methods of Operations Research51(3), 479–494 (2000)

Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Mathematical Methods of Operations Research51(3), 479–494 (2000)

work page 2000

[8] [8]

Gra˜ na Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization175(2), 395–414

work page

[9] [9]

Fukuda, E.H., Gra˜ na Drummond, L.M.: Inexact projected gradient method for vector optimization54(3), 473–493

work page

[10] [10]

1007/s10589-018-0043-x

Tanabe, H., Fukuda, E.H., Yamashita, N.: Proximal gradient methods for multi- objective optimization and their applications72(2), 339–361 https://doi.org/10. 1007/s10589-018-0043-x

work page

[11] [11]

Tanabe, H., Fukuda, E.H., Yamashita, N.: Convergence rates analysis of a mul- tiobjective proximal gradient method17(2), 333–350 https://doi.org/10.1007/ s11590-022-01877-7

work page

[12] [12]

Journal of Machine Learning Research 11(2) (2010)

Journ´ ee, M., Nesterov, Y., Richt´ arik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. Journal of Machine Learning Research 11(2) (2010)

work page 2010

[13] [13]

SIAM Journal on Optimization23(2), 1214–1236 (2013)

Vandereycken, B.: Low-rank matrix completion by riemannian optimization. SIAM Journal on Optimization23(2), 1214–1236 (2013)

work page 2013

[14] [14]

Journal of Optimization Theory and Applications154, 88–107 (2012)

Bento, G., Ferreira, O.P., Oliveira, P.R.: Unconstrained steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications154, 88–107 (2012)

work page 2012

[15] [15]

Journal of Optimization Theory and Applications159, 108–124 (2013)

Bento, G., Cruz Neto, J.X., Santos, P.: An inexact steepest descent method for multicriteria optimization on riemannian manifolds. Journal of Optimization Theory and Applications159, 108–124 (2013)

work page 2013

[16] [16]

Bento, G.C., Cruz Neto, J.X.: A subgradient method for multiobjective opti- mization on riemannian manifolds159(1), 125–137 https://doi.org/10.1007/ s10957-013-0307-7

work page

[17] [17]

Applied Mathematics and Computation486, 129001 (2025)

Chen, K., Fukuda, E.H., Sato, H.: Nonlinear conjugate gradient method for vec- tor optimization on riemannian manifolds with retraction and vector transport. Applied Mathematics and Computation486, 129001 (2025)

work page 2025

[18] [18]

Accessed 2023-05-31

Najafi, S., Hajarian, M.: Multiobjective conjugate gradient methods on rieman- nian manifolds197(3), 1229–1248 https://doi.org/10.1007/s10957-023-02224-1 . Accessed 2023-05-31

work page doi:10.1007/s10957-023-02224-1 2023

[19] [19]

Bento, G.d.C., Ferreira, O.P., Pereira, Y.R.L.: Proximal point method for vector optimization on hadamard manifolds46(1), 13–18 29

work page

[20] [20]

Journal of Optimization Theory and Applications196(1), 212–239 (2023)

Eslami, N., Najafi, B., Vaezpour, S.M.: A trust region method for solving multi- criteria optimization problems on riemannian manifolds. Journal of Optimization Theory and Applications196(1), 212–239 (2023)

work page 2023

[21] [21]

Journal of Optimization Theory and Applications179(1), 37–52 (2018) https://doi.org/ 10.1007/s10957-018-1330-5

Bento, G.d.C., Cruz Neto, J.X., Meireles, L.V.: Proximal point method for locally lipschitz functions in multiobjective optimization of hadamard manifolds. Journal of Optimization Theory and Applications179(1), 37–52 (2018) https://doi.org/ 10.1007/s10957-018-1330-5

work page doi:10.1007/s10957-018-1330-5 2018

[22] [22]

Numerical Algorithms, 1–51 (2025) https://doi.org/10

Upadhyay, B.B., Poddar, S., Ferreira, O.P., Yao, J.-C.: An inexact proximal point method with quasi-distance for quasiconvex multiobjective optimization problems on riemannian manifolds. Numerical Algorithms, 1–51 (2025) https://doi.org/10. 1007/s11075-025-02177-8

work page 2025

[23] [23]

SIAM Journal on Optimization 30(1), 210–239 (2020)

Chen, S., Ma, S., Man-Cho So, A., Zhang, T.: Proximal gradient method for nonsmooth optimization over the stiefel manifold. SIAM Journal on Optimization 30(1), 210–239 (2020)

work page 2020

[24] [24]

Huang, W., Wei, K.: Riemannian proximal gradient methods194(1), 371–413 https://doi.org/10.1007/s10107-021-01632-3

work page doi:10.1007/s10107-021-01632-3

[25] [25]

Huang, W., Wei, K.: An inexact riemannian proximal gradient method85(1), 1–32 https://doi.org/10.1007/s10589-023-00451-w

work page doi:10.1007/s10589-023-00451-w

[26] [26]

Accessed 2023-09-21

Zhao, S., Yan, T., Zhu, Y.: Proximal gradient algorithm with trust region scheme on riemannian manifold https://doi.org/10.1007/s10898-023-01326-4 . Accessed 2023-09-21

work page doi:10.1007/s10898-023-01326-4 2023

[27] [27]

Publisher: Society for Industrial and Applied Mathematics

Hosseini, S., Huang, W., Yousefpour, R.: Line search algorithms for locally lips- chitz functions on riemannian manifolds28(1), 596–619 https://doi.org/10.1137/ 16M1108145 . Publisher: Society for Industrial and Applied Mathematics

work page

[28] [28]

Classics in Applied Mathe- matics, vol

Clarke, F.H.: Optimization and Nonsmooth Analysis. Classics in Applied Mathe- matics, vol. 5. Society for Industrial and Applied Mathematics, Philadelphia, PA (1990). https://doi.org/10.1137/1.9781611971309

work page doi:10.1137/1.9781611971309 1990

[29] [29]

Nonlinear Analysis: Theory, Methods & Applications74(12), 3884–3895 (2011)

Hosseini, S., Pouryayevali, M.: Generalized gradients and characterization of epi- lipschitz sets in riemannian manifolds. Nonlinear Analysis: Theory, Methods & Applications74(12), 3884–3895 (2011)

work page 2011

[30] [30]

Journal of Mathematical Analysis and Applica- tions457(2), 1333–1352 (2018)

Daniilidis, A., Deville, R., Durand-Cartagena, E., Rifford, L.: Self-contracted curves in riemannian manifolds. Journal of Mathematical Analysis and Applica- tions457(2), 1333–1352 (2018)

work page 2018

[31] [31]

Optimization73(13), 3821–3858 (2024) 30

Tanabe, H., Fukuda, E.H., Yamashita, N.: New merit functions for multiobjective optimization and their properties. Optimization73(13), 3821–3858 (2024) 30

work page 2024