Barzilai-Borwein Diagonal Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems

Hua Liu; Liping Tang; Xinmin Yang; Zhuoxin Fan

REVIEW 4 major objections 4 minor 24 references

Barzilai-Borwein Diagonal Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A shared BB diagonal matrix can approximate all objective Hessians at once in multiobjective optimization, giving global convergence without convexity.

desk verdict 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. read the letter →

arxiv 2512.18348 v2 pith:5ZDQJKXI submitted 2025-12-20 math.OC

Hua Liu , Zhuoxin Fan , Liping Tang , Xinmin Yang This is my paper · ORCID

classification math.OC MSC 90C2690C2990C53

keywords Barzilai-Borweinmethodmultiobjectiveoptimizationquasi-NewtondiagonalHessianapproximationParetocriticalpointglobalconvergenceR-linearnonconvex

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

The 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.

The 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.

Bottom 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Algorithm 1, Steps 2, 9-16] 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.
[Assumption 3 and Proposition 4] 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.
[Algorithm 1, Step 15] 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.
[Theorem 6 and Remark 3] 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.

minor comments (4)

[Section 4, Proposition 2] 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 5] 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.
[Notation] 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.
[Title and labels] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the convergence proof is conditional on stated assumptions, and the flagged issues are proof gaps, not circular reductions.

full rationale

After walking the derivation chain, I find no step where a claimed result is equivalent to its inputs by construction. Proposition 4 and Theorems 5-6 are conditional proofs: they assume Lipschitz gradients, bounded level sets, and the uniform bound aI⪯B(x)⪯bI, and derive the decrease bound and rates from those assumptions. The most suspicious ingredient is Assumption 3, because Algorithm 1's safeguard omega_k = min{c0, c1||∇f(x_k)||^{c2}} need not deliver uniform bounds and Step 15 is uninitialized at k=0; however, the paper never claims to derive Assumption 3 from the algorithm. It explicitly claims only that the approximation matrix remains positive definite, which is weaker than the assumed uniform bound. An unenforced or unwarranted assumption is a correctness/completeness gap, not circularity. Self-citations [15,16] are used as motivation and to describe prior methods, not as the load-bearing justification of the convergence theorems; those theorems are proved from standard Wolfe/Lipschitz inequalities. Proposition 2's proof is omitted ('the proof is similar to which in [14, Proposition 2], thus we omit it here') and Proposition 3 relies on a missing reference ('follows directly from the proof given in [27, proposition 2.2]'); I flag these as completeness defects, but they do not make the derivation circular. The numerical section compares against an external algorithm on standard test problems, so no fitted parameter is renamed as a prediction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central algorithm is defined by a scalar α_k; its theoretical convergence relies on the smooth nonconvex problem class, bounded level sets, Lipschitz gradients, and—for the rate—common strong convexity, plus an unverified uniform bound on α_k. Algorithm hyperparameters c0,c1,c2,σ1,σ2 are free hand choices with missing values in the numerics. No new physical or conceptual entities are introduced.

free parameters (5)

c0
Appears in the safeguard ω_k=min{c0, c1||∇f||^{c2}}; hand-chosen algorithmic constant, value not reported in the numerics.
c1
Scales the safeguard with the gradient norm; hand-chosen, value not reported.
c2
Exponent in the safeguard; hand-chosen, value not reported.
σ1, σ2
Wolfe line-search parameters with 0<σ1<σ2<1; the numerical values are truncated in the manuscript.
B0 = I
Initial Hessian approximation set to the identity matrix.

assumptions (6)

domain assumption Assumption 1: level set L={x: f(x)≤f(x0)} is bounded and f is continuous
Used to obtain bounded-below component functions and compactness in Theorem 5; this defines the problem class.
domain assumption Assumption 2: each ∇f_i is Lipschitz continuous on an open set containing L
Used in Proposition 4 to lower-bound the line-search step and derive the per-iteration decrease.
ad hoc to paper Assumption 3: there exist a,b>0 with aI ⪯ B_k ⪯ bI for all k
Uniform boundedness of the approximate Hessian is assumed but not proved from the algorithm's α selection; this is the load-bearing gap in the global convergence theorem.
domain assumption Assumption 4: each f_i is strongly convex and smooth with common 0<U≤1≤L
Used only for the R-linear convergence theorem; not needed for the nonconvex global convergence claim.
domain assumption Proposition 2: existence of t_k satisfying the modified Wolfe conditions (proof omitted, citing [14])
Line-search termination is assumed rather than proved; standard for continuously differentiable objectives but not fully verified.
standard math KKT/dual representation of the direction subproblem and Proposition 3, whose proof references [27]
Justifies d_k=-B_k^{-1}Σλ_i∇f_i and the equivalence between nonzero descent direction and non-Pareto-criticality; the proof is outsourced to [27], which is missing from the bibliography.

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Cite this review

Pith. "Pith review of Barzilai-Borwein Diagonal Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems." pith.science (2026). https://pith.science/paper/5ZDQJKXI

@misc{pith2026251218348,
  author       = {Pith},
  title        = {Pith review of: Barzilai-Borwein Diagonal Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZDQJKXI}},
  note         = {Machine review of arXiv:2512.18348}
}

read the original abstract

This paper addresses the challenge of developing efficient algorithms for large-scale nonconvex multiobjective optimization problems (MOPs). While quasi-Newton methods are effective, their traditional application to MOPs is computationally expensive as they require maintaining and inverting separate Hessian approximations for each objective function. To overcome this limitation, we propose a novel Barzilai-Borwein diagonal-type Quasi-Newton method (BB-DQN). Our key innovation is the use of a single, shared, and modified BB-type matrix, updated iteratively using function and gradient information, to approximate the Hessians of all objectives simultaneously. We theoretically demonstrate that this approximation matrix remains positive definite throughout the iterative process. Furthermore, we establish the global convergence of the BB-DQN method without convexity assumptions and prove its R-linear convergence under mild conditions. Numerical experiments on a diverse set of test problems confirm that BB-DQN outperforms existing methods like M-BFGSMO, achieving superior performance in terms of computational time, iteration count, and reliability, especially for large-scale instances.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Structural and multidisciplinary optimization 26(6), 369–395 (2004)

Marler, R.T., Arora, J.S.: Survey of multi-objective optimization me thods for engineering. Structural and multidisciplinary optimization 26(6), 369–395 (2004)

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Reviews in chemical engineering 16(1), 1–54 (2000)

Bhaskar, V., Gupta, S.K., Ray, A.K.: Applications of multiobjective o ptimization in chemical engineering. Reviews in chemical engineering 16(1), 1–54 (2000)

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In : Invited Keynote Paper, GL2-2, The Third China-Japan-Korea Joint Sympo sium on Opti- mization of Structural and Mechanical Systems, Kanazawa, Japa n, vol

De Weck, O.L.: Multiobjective optimization: History and promise. In : Invited Keynote Paper, GL2-2, The Third China-Japan-Korea Joint Sympo sium on Opti- mization of Structural and Mechanical Systems, Kanazawa, Japa n, vol. 2, p. 34 (2004)

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Annals of Operations Research 154(1), 29–50 (2007) 17

Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjec- tive optimization. Annals of Operations Research 154(1), 29–50 (2007) 17

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In: 2021 IEEE 4th In ternational Conference on Big Data and Artiﬁcial Intelligence (BDAI), pp

Qu, Q., Ma, Z., Clausen, A., Jørgensen, B.N.: A comprehensive revie w of machine learning in multi-objective optimization. In: 2021 IEEE 4th In ternational Conference on Big Data and Artiﬁcial Intelligence (BDAI), pp. 7–14 (2021). IEEE

2021

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Annals of Operations Research 339(3), 1119–1148 (2024)

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[7] [7]

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Numerical Algorithms, 1–27 (2025)

Chen, W., Tang, L., Yang, X.: Improvements to steepest descen t method for multi-objective optimization. Numerical Algorithms, 1–27 (2025)

2025