{"id":"fd5ab641-8b3f-41f6-8db7-05458343ccd8","arxiv_id":"2608.09755","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new QO-free method for constrained Riemannian optimization is proven globally, strongly, and superlinearly convergent, overcoming the Maratos effect for the first time in this setting.","lead":"This paper designs an iterative method for minimizing a function on a curved space, a Riemannian manifold, with equality and inequality constraints. It proves the method converges from any feasible start and, near the solution, accelerates to superlinear speed, avoiding a known failure mode called the Maratos effect.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Superlinear convergence proof requires Assumption 5.4 (second-order retraction), which fails for the retractions used in the numerical experiments, so the Maratos-effect claim is not established for the tested implementation.","rationale":"The reader's weakest assumption identifies exactly Assumption 5.4, and I agree this is the most load-bearing condition. The global convergence analysis in Sections 4 does not use second-order retractions, and Theorem 5.1's strong convergence argument appears sound. The local superlinear claim, however, hinges on the unit-step acceptance proof in Theorem 5.2, and that proof explicitly cancels the second-order retraction term. Since common and widely used retractions on the manifolds tested in Section 6 are not second-order, the numerical evidence does not validate the theorem's hypotheses. This does not make the theorem false\\u2014it is a conditional statement\\u2014but it substantially narrows the practical scope of the 'Maratos effect overcome' claim and should be disclosed prominently. The omitted proofs of Lemma 5.9 and equation (56) are secondary but reinforce the need for a conditional decision: the authors should supply those details or explicitly state the retraction assumption and adjust the experiments accordingly. Overall, the reader's CONDITIONAL verdict remains appropriate; no verdict change is needed.","tokens_in":39737,"tokens_out":18059,"duration_ms":143246,"concrete_test":"Directly compute D^2/dt^2 R_x(t\\xi)|_{t=0} for the retractions used in the experiments: for Manopt's obliquefactory, R_x(t\\xi)=(x+t\\xi)/\\|x+t\\xi\\| gives D^2=-\\|\\xi\\|^2 x\\ne0, and for fixedrankembeddedfactory the SVD-truncation retraction should be checked similarly; if nonzero, Assumption 5.4 is violated and Theorem 5.2 does not cover the reported runs. Additionally, rerun the fixed-rank and oblique experiments with a second-order retraction (e.g., the exponential map where available) and check whether unit-step acceptance is eventually obtained and whether \\|\\hat x_{k+1}-\\hat x_*\\|/\\|\\hat x_k-\\hat x_*\\|\\to0; this would show whether Assumption 5.4 is a real barrier or merely an artifact of the proof technique.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is Assumption 5.4, the requirement that the retraction be second-order. This assumption enters only in Theorem 5.2: the Taylor expansion (39) of F along R_{x_k}(\\eta_k+\\tilde\\eta_k) drops the term \\langle\\mathrm{grad}\\,F,\\,D^2R(0)[\\eta_k,\\eta_k]\\rangle, and the whole cancellation leading to (48) depends on this term being zero. For a general first-order retraction this term is O(\\|\\eta_k\\|^2), the same order as the quantities balanced by \\sigma<1/2 at the end of the proof; with a positive constant it can make (19) fail for t=1. Consequently the eventual unit-step acceptance, and hence Theorem 5.3's Q-superlinear rate, is not proven for retractions that are only first-order. This matters because the experiments in Section 6 use Manopt's standard retractions: on the oblique manifold the elementwise normalization retraction and on the fixed-rank manifold the SVD-truncation projection are both first-order and generically have nonzero second derivative. Thus the numerical demonstration of superlinear convergence is not covered by the theorem as stated. The paper's headline claim of being the first constrained-manifold algorithm that provably overcomes the Maratos effect is therefore contingent on a geometric condition that is not met by the implemented retractions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Euclidean QO-free (quadratic-optimization-free) methods to nonlinearly constrained optimization on Riemannian manifolds. Problem (P) is converted via a semi-penalty scheme into an inequality-constrained problem (P_ρ), and Algorithm 1 generates strictly feasible iterates by solving three linear systems with a common operator, a reduced least-squares correction subproblem, and a Riemannian arc search. The authors prove global convergence (Theorem 4.1), strong convergence (Theorem 5.1), eventual acceptance of the unit step size (Theorem 5.2), and Q-superlinear convergence (Theorem 5.3) under assumptions including LICQ, SOSC, a Dennis–Moré condition on the Hessian approximations, and a second-order retraction assumption. Numerical experiments on nonnegative low-rank matrix completion and nonnegative PCA compare the method with RALM, REPMs, and RSQO, reporting competitive accuracy and speed.","tokens_in":40095,"tokens_out":5637,"duration_ms":52857,"significance":"If the convergence claims hold, this is a meaningful contribution: it provides a feasible QO-free method on manifolds that avoids solving quadratic subproblems at each iteration, achieves global and strong convergence, and addresses the Maratos effect without imposing it as an assumption. The algorithmic device of three linear systems sharing one operator, plus a higher-order correction direction solved by a least-squares subproblem, is a natural and potentially useful extension of the Euclidean theory. The paper is largely rigorous and self-contained for the main theorems, although some proof details are deferred to the companion Euclidean results. The numerical experiments are reproducible in structure and show competitive performance. The main qualification is that the superlinear convergence and Maratos-effect statements are established only under Assumption 5.4, which is not satisfied by the retractions used in the reported experiments; this limits the scope of the headline claim and needs to be addressed.","major_comments":[{"comment":"The proof of Theorem 5.2 relies critically on the second-order retraction assumption: in the Taylor expansion (39), the term involving the second covariant derivative of the retraction is dropped, and the subsequent cancellation leading to (48) uses that this term is zero. For a general first-order retraction, this term is generically O(||η_k||^2) and can be positive, so the unit-step acceptance (19) is not guaranteed. The retractions used in the numerical experiments — the elementwise normalization retraction on the oblique manifold and the SVD-truncation projection on the fixed-rank manifold — are only first-order and generically have nonzero second derivative. Consequently, the superlinear convergence shown in Figure 1 and Tables 1–5 is not covered by Theorem 5.3. Please either prove the result for first-order retractions (e.g., by modifying the correction or arc-search mechanism), run experiments with second-order retractions (such as exponential maps on manifolds where they are tractable), or explicitly state in the abstract and conclusions that the Maratos-effect and superlinear convergence guarantees require second-order retractions and that the experiments are heuristic for that aspect.","section":"Section 5, Assumption 5.4 and Theorem 5.2; Section 6"},{"comment":"The proof of Lemma 4.2 is omitted: the text says the boundedness of {||A_k^{-1}||_op} follows the same lines as Lemma 3.1 in [14]. This lemma is load-bearing for Lemma 4.3 and hence for the global convergence Theorem 4.1, and in the manifold setting one must contend with a sequence of operators acting on different tangent spaces. Please provide a self-contained proof or a precise statement of how the Euclidean proof is adapted to the Riemannian setting, including the role of Assumptions 4.1–4.4 in controlling the spectral norm of the inverse operators.","section":"Section 4, Lemma 4.2"},{"comment":"The key estimate ||η_{k0} + o(||η_{k0}||)||^2 = o(||x̂_k − x̂_*||) is asserted by analogy with [14, Lemma 4.8], without proof. This estimate is the bridge between the master direction and the local error and is essential for the Q-superlinear rate. The analogy is not immediate because of the presence of the normal-coordinate chart, the metric G_{x̂_k}, and the retraction R̂. Please supply the full derivation in this paper, or at least verify explicitly that each step in the Euclidean proof carries over to the current setting.","section":"Section 5, Theorem 5.3, Eq. (56)"}],"minor_comments":[{"comment":"In the sentence following Eq. (11), “It has been shown in [49, 50] that αk k and βk i satisfy” should read “α_k^i and β_k^i”; the current text contains a typo.","section":"Section 3.1, Eq. (11)"},{"comment":"In Lemma 5.9 and Theorem 5.3, the notation “o‖x̂_k − x̂_∗‖” and “o(‖x̂_k − x̂_∗‖” is missing parentheses or has unbalanced delimiters; please write o(‖x̂_k − x̂_∗‖) consistently.","section":"Section 5, Lemma 5.9 and Theorem 5.3"},{"comment":"The text says “ROQ-free” in the description of Figure 1; this should be “RQO-free”.","section":"Section 6.3, Figure 1 caption and text"},{"comment":"The method name is spelled inconsistently as “RSQO” in most places and “RSQP” in one sentence; please standardize the abbreviation.","section":"Section 6.2"},{"comment":"Equation (57) and the following display contain an unmatched closing parenthesis in ‖R̂_{x̂}(η̂_{k0}+o(‖η̂_{k0}‖)) − (x̂_k + η̂_{k0}+o(‖η̂_{k0}‖)‖; the typography should be corrected.","section":"Section 5, Theorem 5.2, Eq. (57)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the main algorithmic idea is sound. The decisive issue is the gap between Assumption 5.4 and the retractions used in the experiments, together with two load-bearing proof details that are currently deferred to the Euclidean reference. If the authors supply the missing proofs or adjust the claims and experiments so that the assumptions match the implementation, the paper is publishable. I see no circularity or novelty-disclosure concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a referee's time, but the headline claim is narrower than it looks. The global convergence part is solid. The superlinear convergence and Maratos-effect claim rest on Assumption 5.4, which requires a second-order retraction, and the numerical experiments use retractions — column normalization on the oblique manifold and SVD truncation on the fixed-rank manifold — that are not second-order. So the theorem does not cover the implementation that actually demonstrates the fast convergence. That is the main soft spot, and it is load-bearing for the superlinear half of the paper.\n\nWhat is genuinely new: this is the first QO-free method on Riemannian manifolds with equality and inequality constraints. The three linear systems share a common operator and are always feasible, which is a real advantage over RSQO. The correction direction via least squares is a sensible extension of the Euclidean trick. The global and strong convergence proofs are mostly detailed and follow the Euclidean counterparts faithfully. The local analysis via normal coordinates, especially Lemma 5.4, is careful and useful.\n\nThe soft spots beyond the retraction issue: Lemma 4.2 is deferred to the Qi-Qi paper without proof, and equation (56) in Theorem 5.3 is asserted by analogy rather than shown. These are probably fixable, but they sit exactly where the superlinear rate is established. On the retraction mismatch, the Taylor expansion (39) drops the D^2R term; for a first-order retraction that term is O(||η||^2) and can break the unit-step acceptance. The authors either need a proof under a weaker geometric condition, or experiments with a genuinely second-order retraction, or an explicit statement that the Maratos-free result is conditional on Assumption 5.4. The code is available, which is good, but the experiments do not close the gap.\n\nFor whom: researchers in constrained Riemannian optimization. The global convergence machinery and the linear-system formulation are worth reading even if the superlinear claim is later tightened. I would send it to peer review, with a request to fix the gap or qualify the claim.","headline":"Solid global convergence for a first Riemannian QO-free method, but the superlinear/Maratos claim rests on a second-order retraction assumption that the paper's own experiments violate.","tokens_in":40577,"tokens_out":2479,"would_cite":false,"duration_ms":24238,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C30","90C48"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes the first quadratic-optimization-free method for constrained optimization on Riemannian manifolds and proves global, strong, and Q-superlinear convergence under standard assumptions.","keywords":["Constrained optimization","Riemannian manifolds","QO-free method","Global convergence","Superlinear convergence","Maratos effect","second-order retraction"],"falsifier":"Run Algorithm 1 on a small constrained problem on the sphere or the oblique manifold using the exponential map as the retraction, initialize near a KKT point satisfying strict complementarity and SOSC, and record the sequence of ratios $\\|x_{k+1}-x_*\\|/\\|x_k-x_*\\|$. The theorem predicts these ratios tend to zero; observing a single run where they are bounded away from zero while the stated assumptions hold would disprove the superlinear claim.","tokens_in":39592,"feed_emoji":"📈","tokens_out":11179,"duration_ms":90920,"temperature":0.7,"pith_summary":"Given a smooth cost and smooth equality and inequality constraints on a complete Riemannian manifold, the paper wants a method that always solves feasible subproblems, is globally convergent, and converges Q-superlinearly near a solution. It extends the Euclidean quadratic-optimization-free (QO-free) approach, which solves linear systems instead of quadratic programs, to the manifold setting. The algorithm starts from a strictly feasible point, builds a master tangent direction from three linear systems with one shared operator, adds a higher-order correction direction, and performs an arc search. The paper proves that the whole sequence converges to a KKT point and that, near a solution satisfying strict complementarity and second-order sufficient conditions, the unit step is eventually accepted, giving superlinear convergence. If correct, this is the first constrained manifold method that provably overcomes the Maratos effect rather than assuming it away.","feed_headline":"Manifold optimizer skips QP subproblems, converges superlinearly","feed_subtitle":"It provably avoids the Maratos effect that stalls earlier manifold methods.","key_machinery":"The carrier of the argument is the linear operator $A_k: T_{x_k}M\\times\\mathbb{R}^{m+\\ell}\\to T_{x_k}M\\times\\mathbb{R}^{m+\\ell}$ defined by\n$$A_k(\\eta,\\$\\lambda$)=\\left(H_k[\\eta]+\\sum_{i\\in L}\\lambda_i\\,\\mathrm{grad}\\,c_i(x_k),\\ \\alpha_i^k\\langle\\mathrm{grad}\\,c_i(x_k),\\eta\\rangle-\\sqrt{2}\\beta_i^k\\lambda_i\\right),$$\nwhose nonsingularity makes the three search-direction systems (10), (14), and (15) uniquely solvable with one shared factorization. The coefficients $\\alpha_i^k$ and $\\beta_i^k$ come from the complementarity function $\\phi(a,b)=\\sqrt{a^2+b^2}-a-b$ and encode the complementarity conditions. Around this, the method uses the semi-penalty reformulation $F_\\rho=f-\\rho\\sum_{i\\in E}c_i$, a $\\theta_k$ combination rule that balances descent and feasibility, and a higher-order correction direction from the linear least-squares subproblem (18). The superlinear proof transfers the iteration to Riemannian normal coordinates and uses a second-order retraction to control the second-order terms in the arc-search expansion.","core_discovery":"The central discovery is that the QO-free mechanism is compatible with Riemannian geometry and with equality constraints, provided the equality constraints are treated by a semi-penalty reformulation and the retraction is sufficiently accurate. Algorithm 1 solves three linear systems that share the same nonsingular operator $A_k$, combines their solutions to obtain a feasible descent direction $\\eta_k$, and then solves a reduced least-squares problem for a correction $\\tilde{\\eta}_k$. The paper proves under Assumptions 4.1--4.4 that every accumulation point of the iterates is a KKT point of the original problem, and under Assumptions 5.1--5.5 that the entire sequence converges and satisfies $\\|\\hat{x}_{k+1} - \\hat{x}_*\\| = o(\\|\\hat{x}_k - \\hat{x}_*\\|)$.","pith_inferences":["A natural stress test is to track the ratio $\\|x_{k+1}-x_*\\|/\\|x_k-x_*\\|$ on a matrix manifold equipped with a first-order projection retraction; if the unit step is still accepted and the ratio tends to zero, then Assumption 5.4 is not numerically essential for the local result.","The same proof architecture should transfer to active-set variants, since the correction subproblem is activated only near the solution; that would reduce per-iteration cost on problems with many constraints.","If the strict-feasibility initialization is replaced by a strongly subfeasible direction phase, the method could be extended to problems whose feasible sets are hard to enter from outside."],"forward_implications":["Every iterate is strictly feasible for the penalized problem $P_\\rho$, so the method never hits an infeasible subproblem; this removes a failure mode present in earlier Riemannian SQP methods.","Solving three linear systems with a common operator costs essentially one factorization, making each outer iteration inexpensive when the manifold dimension is moderate.","The penalty parameter $\\rho_k$ is increased only finitely many times; afterwards the method behaves as a feasible descent method and its accumulation points are KKT points of the original problem.","Near a solution satisfying strict complementarity, SOSC, the Hessian approximation condition, and a second-order retraction, the step size $t_k=1$ is eventually accepted, so the fast local convergence is not blocked by the Maratos effect."],"supporting_citations":[{"why":"Supplies the Euclidean QO-free template: three linear systems with one operator, the master direction, the higher-order correction subproblem, and the local superlinear proof that the manifold version adapts.","marker":"[14]"},{"why":"Provides the $\\theta_k$ combination rule that balances descent and feasibility in the master direction.","marker":"[11]"},{"why":"Supplies the semi-penalty scheme that converts equality constraints into the penalized problem $P_\\rho$.","marker":"[47]"},{"why":"Supplies the finite penalty-increase update rule that guarantees $\\rho_k$ is fixed after finitely many iterations.","marker":"[48]"},{"why":"The existing Riemannian SQP method whose subproblems can be infeasible and which assumes the Maratos effect away; the paper's claims are framed against it and its test problems are reused.","marker":"[40]"},{"why":"Provides the Riemannian LICQ, KKT conditions, and SOSC used as the optimality framework and in the normal-coordinate transfer.","marker":"[36]"},{"why":"Provides the standard definitions of retractions, Riemannian Hessians, and normal coordinates used throughout the algorithm and proofs.","marker":"[32]"},{"why":"Provides the Riemannian augmented Lagrangian and exact penalty comparators and the global-convergence-only context that the new method improves on.","marker":"[38]"}],"fun_headline_variants":["Riemannian solver skips QP steps, hits superlinear convergence","Superlinear manifold method drops quadratic subproblems","QO-free algorithm on manifolds achieves superlinear convergence","New manifold method avoids Maratos, converges superlinearly","Riemannian optimizer: no QP, superlinear convergence proof"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise for the fast-convergence claim is Assumption 5.4: the retraction must be second-order, matching the exponential map up to second order along every tangent direction, because the proof's Taylor expansion near the solution has no other way to control the second-order term. If a practical retraction satisfies only the usual first-order property, the unit-step acceptance and hence the superlinear rate are not guaranteed by this paper.","fun_headline_variants_meta":{"raw":{"variants":["Riemannian solver skips QP steps, hits superlinear convergence","Superlinear manifold method drops quadratic subproblems","QO-free algorithm on manifolds achieves superlinear convergence","New manifold method avoids Maratos, converges superlinearly","Riemannian optimizer: no QP, superlinear convergence proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000457,"raw_usage":{"total_tokens":2269,"prompt_tokens":899,"completion_tokens":1370,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1287}},"tokens_in":515,"tokens_out":1370,"duration_ms":9626,"temperature":1.0,"reasoning_tokens":1287,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:22:47.847437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 on a small constrained problem on the sphere or the oblique manifold using the exponential map as the retraction, initialize near a KKT point satisfying strict complementarity and SOSC, and record the sequence of ratios $\\|x_{k+1}-x_*\\|/\\|x_k-x_*\\|$. The theorem predicts these ratios tend to zero; observing a single run where they are bounded away from zero while the stated assumptions hold would disprove the superlinear claim.","supporting_citations":[{"cited_title":"SIAM Jou rnal on Opti- mization 11(1), 113–132 (2000) https://doi.org/10.1137/S1052623499353935","cited_arxiv_id":null,"evidence_quote":"Supplies the Euclidean QO-free template: three linear systems with one operator, the master direction, the higher-order correction subproblem, and the local superlinear proof that the manifold version adapts."},{"cited_title":"SIAM Journal on Control and Optimization 26(4), 788–811 (1988) https://doi.org/10.1137/0326046","cited_arxiv_id":null,"evidence_quote":"Provides the $\\theta_k$ combination rule that balances descent and feasibility in the master direction."},{"cited_title":"Mathematical Programming 11, 67–80 (1976) https://doi.org/10.1007/BF01580371","cited_arxiv_id":null,"evidence_quote":"Supplies the semi-penalty scheme that converts equality constraints into the penalized problem $P_\\rho$."},{"cited_title":"SIAM Journal on Optimization 14(1), 173–199 (2003) https://doi.org/10.1137/S1052623401392123","cited_arxiv_id":null,"evidence_quote":"Supplies the finite penalty-increase update rule that guarantees $\\rho_k$ is fixed after finitely many iterations."},{"cited_title":"Pacific Journal of Optimiz ation 10(2), 415–434 (2014)","cited_arxiv_id":null,"evidence_quote":"Provides the Riemannian LICQ, KKT conditions, and SOSC used as the optimality framework and in the normal-coordinate transfer."},{"cited_title":"Princeton University Press, Princeton, NJ (2008)","cited_arxiv_id":null,"evidence_quote":"Provides the standard definitions of retractions, Riemannian Hessians, and normal coordinates used throughout the algorithm and proofs."}],"review_version":1}