{"id":"c82e99dc-1ac6-4ef9-bd14-985fa6184953","arxiv_id":"2512.18348","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A diagonal Barzilai-Borwein quasi-Newton method with one shared scalar parameter converges to Pareto-critical points for smooth nonconvex multiobjective problems and scales to dimension 2000 in tests.","lead":"This paper introduces a cheap quasi-Newton method for optimizing several conflicting functions at once, replacing separate Hessian approximations with a single shared Barzilai-Borwein scale. The authors prove convergence to Pareto-critical points for smooth nonconvex problems and show speedups over a BFGS-based method on large-scale benchmarks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform spectral bound (Assumption 3) is not enforced by BB-DQN's safeguard; omega_k -> 0 as gradients vanish, so alpha_k can approach 0 or infinity and the proof of Theorem 5 has a gap.","rationale":"The reader's verdict and weakest_assumption align with my reading. I considered whether the absence of error bars in the numerical section or the missing BB baseline is more damaging, but those affect the strength of the empirical claim, not the correctness of the central theoretical assertion. The central assertion is that a single shared modified BB scalar matrix preserves global convergence for nonconvex MOPs. The proof of that assertion is Theorem 5, which rests on Assumption 3. The algorithm as written cannot guarantee Assumption 3: the safeguard omega_k shrinks to zero with the gradient norm, so the permitted range for alpha_k is not uniformly bounded away from zero or infinity. Moreover, Step 15 is not well-defined at k=0. These are internal inconsistencies, not disagreement with community consensus. I also note that even the weaker claim of positive definiteness could fail in case 15 if <s_{k-1}, gamma_{k-1}> = 0. Thus the theoretical contribution is conditional on an assumption that the algorithm does not enforce. The numerical experiments do provide some support for practical efficiency, but they cannot substitute for the missing proof. Therefore the appropriate verdict is CONDITIONAL, unchanged from the reader.","tokens_in":14120,"tokens_out":7958,"duration_ms":422126,"concrete_test":"Run Algorithm 1 on JOS1a (n=50) and MOP2 with the specific tie-breaking rule: in case 10-11 choose alpha_k as the lower endpoint of the intersection, and in case 12-13 choose alpha_k = omega_k (both are allowed by Step 13). Record alpha_k and ||d_k^SD|| for all k. Check whether inf_k alpha_k > 0 and sup_k alpha_k < infinity. If alpha_k -> 0 as ||d_k^SD|| -> 0, Assumption 3 is violated for this valid execution, so the proof of Theorem 5 does not apply; the observed sequence may still converge, but it would require a new argument that does not rely on uniform a,b.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 3 is the load-bearing condition for Proposition 4 and Theorem 5: it gives the uniform a>0 and b<infinity used to lower-bound the Wolfe step t_k and to upper-bound 1/alpha_k. In Algorithm 1, B_k is never updated explicitly; the text implies B_{k+1}=alpha_k I, and the only restriction on alpha_k is Steps 10-16. The safeguard interval is [omega_k, omega_k^{-1}] with omega_k = min{c0, c1||grad f(x_k)||^{c2}} (Step 6). Since c2>0, omega_k tends to 0 as ||grad f(x_k)||->0, which is exactly what happens when approaching a Pareto critical point. The allowed interval therefore expands to (0, infinity), and the algorithm explicitly permits alpha_k to be chosen anywhere in it (Step 13), e.g., alpha_k = omega_k -> 0; case 15 is an unguarded quotient that is undefined at k=0 because s_{-1}, gamma_{-1} are never initialized. Consequently no fixed a,b satisfying a <= alpha_k <= b for all k is guaranteed, so Assumption 3 is not a property of the algorithm; it is an extra external condition. Without it, the decrease bound (12) with omega = sigma1(1-sigma2)a/(2bL) collapses, and Theorem 5's proof no longer goes through. The paper's claim that the approximation matrix remains positive definite only asserts alpha_k > 0, which is weaker and not enough to prove convergence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes BB-DQN, a quasi-Newton method for unconstrained nonconvex multiobjective optimization problems (MOPs). The key idea is to use a single shared modified Barzilai-Borwein diagonal matrix, effectively a scalar α_k I, to approximate the Hessians of all objective functions simultaneously, thereby avoiding the cost of maintaining separate approximations. The algorithm computes a descent direction via a dual subproblem, uses a multiobjective Wolfe line search, and adaptively selects α_k with a safeguard. The paper claims global convergence to Pareto critical points without convexity (Theorem 5) and R-linear convergence under strong convexity (Theorem 6), and reports numerical comparisons with M-BFGSMO showing superior performance on several test problems, especially large-scale ones.","tokens_in":14501,"tokens_out":6724,"duration_ms":61426,"significance":"If the theoretical claims are correct, the paper makes a useful practical contribution: a shared diagonal/scalar BB update substantially reduces per-iteration cost compared to methods that maintain separate Hessian approximations, and the numerical experiments indeed show striking speedups on large-scale problems such as JOS1e and TOI4f. The convergence framework is standard (Wolfe line search + sufficient decrease), and the paper provides explicit rates. However, the central convergence proof depends on a uniform spectral bound on B_k that the algorithm's safeguard does not guarantee, and the algorithm box is not fully specified (no explicit B_{k+1} update, undefined quantities at k=0). These issues are load-bearing but appear fixable within the manuscript's scope.","major_comments":[{"comment":"The algorithm never explicitly sets B_{k+1}=α_k I. Step 2 uses B(x) from (6)-(7), but B is never updated, so d_k is undefined for k≥1. The proof of Proposition 4 implicitly uses d_k = α_k^{-1} d_k^SD (see the identity -σ1 t_k D(x_k,d_k) = -σ1 t_k α_k^{-1} D(x_k,d_k^SD)), which requires B_k=α_k I. Please state this update explicitly and define the direction accordingly.","section":"Algorithm 1, Steps 2, 9-16"},{"comment":"Assumption 3 asserts uniform bounds aI ⪯ B_k ⪯ bI, but the algorithm does not enforce them. The safeguard permits α_k ∈ [ω_k, ω_k^{-1}] with ω_k = min{c0, c1||∇f(x_k)||^{c2}}, and c2>0 implies ω_k→0 as the gradient tends to 0, which is exactly what happens near a Pareto critical point. The choice 'α_k ∈ [ω_k, ω_k^{-1}]' does not guarantee any fixed a,b>0. Proposition 4's bound ω = σ1(1-σ2)a/(2bL) requires both a≤α_k and α_k≤b; without a uniform bound the decrease inequality (12) collapses, and Theorem 5's proof no longer follows. The paper's claim that B_k 'remains positive definite' only establishes α_k>0, not uniform boundedness. This is the main technical gap.","section":"Assumption 3 and Proposition 4"},{"comment":"The 'else' branch is undefined at k=0: it references s_{-1} and γ_{-1}, which are never initialized. Moreover, the displayed formula computes α_k from s_{k-1},γ_{k-1}, but the definitions of η_k,m_k,γ_k that follow appear to be for the current iteration and are not used in the assignment. This makes the algorithm not well-defined and not reproducible as written. Please repair the update rule and specify the initialization.","section":"Algorithm 1, Step 15"},{"comment":"Remark 3 states that Assumption 4 implies Assumption 3, but this is not justified. Strong convexity bounds the objective Hessians ∇²f_i(x), not the algorithm-generated B_k (or α_k). Even under strong convexity, the safeguard permits α_k→0 or α_k→∞ as gradients vanish, so the uniform bound needed for Proposition 4 is not guaranteed. The R-linear convergence proof therefore inherits the same gap.","section":"Theorem 6 and Remark 3"}],"minor_comments":[{"comment":"The proof is omitted with the note 'similar to [14, Proposition 2]'. For a standalone paper, please provide a proof or a precise statement of the referenced result, since the Wolfe conditions for vector-valued objectives are not entirely trivial.","section":"Section 4, Proposition 2"},{"comment":"The numerical section contains missing values: the termination criterion is given as 'set as , where is...' and the Wolfe parameters as 'chosen as 0.and .'. These need to be filled in. Also, the captions of Figures 7, 10, and 11 repeatedly say 'BK problem' for problems that are not BK1.","section":"Section 5"},{"comment":"The manuscript inconsistently uses [m] and ⟨m⟩ for the index set, and sometimes writes ∇F_i instead of ∇f_i. Please unify notation. Also, in Step 6 the quantity ||∇f(x_k)|| is not defined for vector-valued f; specify which norm or gradient aggregate is meant.","section":"Notation"},{"comment":"The paper title in the full text reads 'Modified Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems with Barzilai-Borwein diagonal matrix', while the arXiv title is 'Barzilai-Borwein Diagonal Quasi-Newton Method...'. In the numerical figures, 'M-BBQNMO' is used in some captions and 'BB-DQN' in others; please standardize the name.","section":"Title and labels"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant and timely problem, and the numerical gains on large-scale instances are impressive. However, the central convergence proof (Proposition 4 and Theorem 5) depends on a uniform spectral bound that the stated safeguard does not enforce, and the algorithm box has several specification gaps. These are fixable in a revision, but they are load-bearing for the main claims, so I could not recommend acceptance in the present form. I would also welcome a brief statement on whether the code is available for reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the algorithm is a genuinely cheap BB-style diagonal quasi-Newton for multiobjective problems, and the numerical results on large-scale cases are impressive. But the global convergence theorem has a gap that the authors should be asked to fix before acceptance.\n\nWhat's new: a single shared modified BB scalar matrix, updated from function and gradient information, to approximate all objective Hessians at once. That is a reasonable simplification of the shared-BFGS idea from Yang et al. and the BB scheme from Chen et al., and it does avoid the storage/cost problem of maintaining m separate approximations. The convergence framework is standard — Wolfe line search, Pareto critical points, R-linear rate under strong convexity.\n\nThe soft spot, and it is load-bearing: Theorem 5 uses Assumption 3, a uniform bound aI <= B_k <= bI for all k. The algorithm never actually writes B_{k+1} = alpha_k I, and the safeguard in Steps 10-16 lets alpha_k range over [omega_k, omega_k^{-1}] with omega_k = min{c0, c1||grad f(x_k)||^{c2}}. Since c2 > 0, omega_k tends to 0 as the gradient tends to 0, which is where any convergent sequence goes. So the allowed interval expands to (0, inf), and there is no uniform a,b. Proposition 4's decrease bound uses that uniform bound to control t_k from below and alpha_k from above; without it the proof of Theorem 5 doesn't go through. Also, Step 15 references s_{k-1} and gamma_{k-1} at k=0, with no initialization, and Proposition 3 cites a [27] that is not in the reference list. These are fixable, but they need to be fixed.\n\nThe numerics: they show a large practical speedup over M-BFGSMO, especially on the JOS1 and TOI4 series. That is a real selling point. The comparison is narrow, though — no direct BB baseline, no error bars, and the exactly-3-iterations results on JOS1 makes one suspect a loose tolerance or a very easy test family. A performance profile would help.\n\nBottom line: this is a conditional paper. If the authors replace the gradient-dependent safeguard with one that enforces fixed positive bounds on alpha_k (or otherwise prove Assumption 3 plus fix the initialization), the core result probably holds. I would send it to a serious referee, not desk reject it, but I would not cite the convergence theorem in its current form.","headline":"Cheap and promising BB-diagonal quasi-Newton for multiobjective problems, but the global convergence proof assumes a uniform spectral bound the algorithm does not enforce.","tokens_in":14979,"tokens_out":5938,"would_cite":false,"duration_ms":51646,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C29","90C53"],"pacs":[],"model":"deepseek-v4-flash","headline":"A shared BB diagonal matrix can approximate all objective Hessians at once in multiobjective optimization, giving global convergence without convexity.","keywords":["Barzilai-Borwein method","multiobjective optimization","quasi-Newton method","diagonal Hessian approximation","Pareto critical point","global convergence","R-linear convergence","nonconvex optimization"],"falsifier":"Record the generated α_k values when running Algorithm 1 on a nonconvex test problem. If on any run α_k drops below some fixed positive a while the iterate x_k is not yet Pareto-critical, then Assumption 3 would fail for that sequence, meaning the experiment would expose a running case outside the theorem's conditions.","tokens_in":13979,"feed_emoji":"⚙️","tokens_out":3954,"duration_ms":36820,"temperature":0.7,"pith_summary":"The paper introduces BB-DQN, a quasi-Newton method for unconstrained multiobjective problems that replaces the usual collection of Hessian approximations with a single modified Barzilai-Borwein scalar matrix. It claims this shared matrix is enough to approximate all objective Hessians simultaneously, keeping the method cheap while still converging globally to Pareto-critical points even for nonconvex objectives. Under strong convexity, the paper further claims R-linear convergence. Because the method stores and inverts only one scalar matrix, its per-iteration cost approaches that of steepest descent, which the authors argue makes quasi-Newton-type methods practical for large-scale multiobjective problems. Numerical comparisons on test problems with up to 2000 variables show large reductions in CPU time and iterations against an existing quasi-Newton baseline.","feed_headline":"Shared BB matrix replaces m Hessians in multiobjective optimization","feed_subtitle":"One scalar Hessian approximation cuts large-scale multiobjective quasi-Newton cost to a fraction of previous methods.","key_machinery":"The central object is the modified Barzilai-Borwein diagonal matrix, a single scalar α_k multiplied by the identity, chosen so that the resulting B_k approximates the Hessians of all objectives at once. The safeguard interval [ω_k, ω_k^{-1}] with ω_k = min{c0, c1||∇f(x_k)||^{c2}} keeps the parameter in a practical range, while the fallback step reuses previous information when the curvature quotient is negative. This one-scalar approximation does the work that traditionally required m separate full Hessian approximations, and its positivity is what the convergence theory leans on.","core_discovery":"The central claim is that one modified Barzilai-Borwein diagonal matrix B_k = α_k I, updated from function and gradient information, suffices to compute effective quasi-Newton directions for all objectives simultaneously. The direction is d_k = -B_k^{-1} Σ_i λ_i ∇f_i(x_k), with weights from the dual subproblem. The paper proves, as Theorem 5, that under bounded level sets, Lipschitz gradients, and a uniform positive-definiteness assumption on the B_k, the sequence satisfies lim ||d_k^SD|| = 0 and every accumulation point is Pareto-critical. Theorem 6 adds R-linear convergence to a Pareto optimal point when each objective is strongly convex with Lipschitz gradients. The innovation is the safe","pith_inferences":["The proof depends on the safeguard keeping α_k uniformly bounded below, but the safeguard itself shrinks to zero as gradients vanish; if in practice α_k drifts below any fixed positive threshold before reaching a Pareto-critical region, the theoretical guarantee would not cover the observed run.","A natural extension, not explored in the paper, is to use a small number of scalar BB parameters (one per cluster of objectives) rather than a single scalar, which could capture more curvature variation while keeping the cost low.","The algorithm's reliance on a single scalar means it cannot capture direction-dependent curvature, so on strongly anisotropic problems it may behave closer to a preconditioned gradient method than to a full quasi-Newton method; this is a testable hypothesis about where the method might underperform.","The fallback step in Step 15 references s_{k-1} and γ_{k-1} without initialization at k=0; a robust implementation would need to handle the first iteration separately, and convergence proofs should include that case explicitly."],"forward_implications":["If BB-DQN works as claimed, large-scale multiobjective problems with thousands of variables become tractable for quasi-Newton methods at near-steepest-descent cost per iteration, since only one scalar needs to be stored and inverted.","The global convergence result extends quasi-Newton methodology to nonconvex multiobjective settings, where maintaining separate curvature approximations was previously prohibitive.","R-linear convergence under strong convexity means the method does not merely converge but does so at a predictable geometric rate, which matters for practical stopping criteria.","Numerical experiments on the JOS1 and TOI4 families show iteration counts that stay nearly constant as dimension grows, and CPU time reductions of tens to more than fifty times on the largest instances compared with M-BFGSMO.","The same algorithmic template could be applied to multiobjective minimax or bilevel problems where shared curvature models are desirable."],"fun_headline_variants":["One shared BB matrix cuts multiobjective quasi-Newton cost","BB-DQN: one scalar Hessian for all objectives at once","Single shared Hessian approximation rivals M-BFGSMO","Multiobjective optimization with one Barzilai-Borwein matrix","Shared BB matrix proves convergence for nonconvex MOPs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theory assumes the shared BB scalar stays uniformly bounded away from zero (aI ⪯ B_k ⪯ bI for all k), while the safeguard in Step 6 lets α_k shrink to zero as gradients vanish, and Step 15 uses s_{k-1} and γ_{k-1} without initializing them at k=0.","fun_headline_variants_meta":{"raw":{"variants":["One shared BB matrix cuts multiobjective quasi-Newton cost","BB-DQN: one scalar Hessian for all objectives at once","Single shared Hessian approximation rivals M-BFGSMO","Multiobjective optimization with one Barzilai-Borwein matrix","Shared BB matrix proves convergence for nonconvex MOPs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001127,"raw_usage":{"total_tokens":4519,"prompt_tokens":735,"completion_tokens":3784,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":3697}},"tokens_in":479,"tokens_out":3784,"duration_ms":24648,"temperature":1.0,"reasoning_tokens":3697,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:29:45.633449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Record the generated α_k values when running Algorithm 1 on a nonconvex test problem. If on any run α_k drops below some fixed positive a while the iterate x_k is not yet Pareto-critical, then Assumption 3 would fail for that sequence, meaning the experiment would expose a running case outside the theorem's conditions.","supporting_citations":[],"review_version":1}