{"id":"a7ce84a8-9c88-4d95-9588-24e6e57dc123","arxiv_id":"2502.08835","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"BALA, a single-loop bundle-based augmented Lagrangian algorithm, achieves sublinear convergence for convex constrained problems and linear convergence for a class of conic programs including semidefinite programs.","lead":"The authors propose BALA, a single-loop augmented Lagrangian method that approximates the feasible set by simple inner sets built from past iterates. It achieves sublinear convergence for convex problems and linear convergence for structured conic programs such as semidefinite programs, while avoiding the nested inner loops of classical inexact ALM.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's claimed O(ε^{-4}) primal rate for ρ=ε is internally inconsistent: it conflates the PPM parameter α=ρ with 1/ρ and ignores Lemma 2's dual-gap requirement δ=O(ρ ε²).","rationale":"I read the paper as a serious attempt to build a single-loop bundle-based augmented Lagrangian method with sublinear and conditional linear convergence. The fixed-ρ sublinear results (O(ε^{-3}) for dual iterates and O(ε^{-6}) for primal last iterates) appear coherent, and the linear-convergence proof can likely be repaired by correcting the ρ/1/ρ inequality in Theorem 6. The reader's weakest assumption about the oracle cost of evaluating g and constructing v_{k+1} is a real and important limitation: for a general compact convex Ω these are convex optimization problems, and the single-loop efficiency claim is only fully justified for structured sets such as orthants, SOCs, and PSD cones with trace bounds. That concern is valid but is partly acknowledged in Appendix A.2, and it does not by itself falsify a stated theorem. My stress-test pass found a sharper, more directly verifiable problem: the improved-rate clause of Theorem 4 is internally inconsistent. Lemma 5 fixes the proximal parameter to α=ρ, so Theorem 2's O(ε^{-2}) dual rate requires ρ=1/ε, not ρ=ε; and Lemma 2 forces the dual target for primal feasibility to be O(ε³) when ρ=ε, leading to a worse instead of better PBM complexity. The proof in Appendix D.5 asserts the O(ε^{-4}) primal bound without the needed calculation, and the calculation contradicts it. Because this is a concrete error in a central complexity theorem, the paper should not be accepted without revision; however, the main algorithmic idea and the fixed-ρ convergence theorems remain defensible, so conditional acceptance is still the appropriate verdict. I therefore leave the reader's verdict unchanged while flagging an additional required correction beyond those already identified.","tokens_in":31732,"tokens_out":22554,"duration_ms":201804,"concrete_test":"Re-derive the 'ρ=ε' case of Theorem 4 from Eq. (17b) and Theorem 2. Concretely: (i) set the dual target δ=ρβ ε²/2, substitute α=ρ=ε into Theorem 2, and compute the step bound for target δ; this gives O(δ^{-3})=O(ε^{-9}), not O(ε^{-4}). (ii) Check the claimed shortcut directly: with g(y_k)-g⋆=ε and ρ=ε, Eq. (17b) gives ||Ax_{k+1}-b||² ≤ 2/β, contradicting the required ||Ax_{k+1}-b|| ≤ ε. Either calculation settles that the asserted O(ε^{-4}) primal rate does not follow from the cited lemmas; the theorem should be amended to state only the fixed-ρ rates, or a genuinely correct parameter choice should be provided.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The improved part of Theorem 4 ('if we choose ρ=ε, then O(ε^{-2}) and O(ε^{-4}) iterations') is not supported by the paper's own lemmas and is, as stated, false. First, Lemma 5 shows z_{k+1} minimizes g_k(y)+(1/(2ρ))||y-y_k||², so in the PBM notation of Theorem 2, Eq. (8), the proximal parameter is α=ρ, not 1/ρ. Theorem 2's O(ε^{-2}) rate is stated only when α=1/ε, i.e. ρ=1/ε. With ρ=ε, the applicable dual bound is the generic O(ε^{-3}), so the advertised dual improvement evaporates. Second, even granting a dual gap of ε, the primal claim fails. Lemma 2, Eq. (17b), gives ρ||Ax_{k+1}-b||²/2 ≤ (g(y_k)-g⋆)/β, so to make ||Ax_{k+1}-b|| ≤ ε one needs g(y_k)-g⋆ ≤ ρβ ε²/2. With ρ=ε this required dual gap is O(ε³), not ε. Applying Theorem 2 to target δ=O(ε³) with α=ρ=ε (which is not the tuned value 1/δ=ε^{-3}) yields at best O(M² δ^{-3})=O(ε^{-9}) steps, not O(ε^{-4}). Appendix D.5 simply asserts the O(ε^{-4}) bound 'follows from Lemmas 2 and 3' without performing this substitution; performing it contradicts the claim. This is not cosmetic: the improved rates are part of the headline complexity statement, and the same reciprocal confusion between ρ and 1/ρ also appears in Theorem 6, where the statement requires ρ ≥ 1/γ but the proof requires 1/ρ ≥ γ. The fixed-ρ rates O(ε^{-3}) and O(ε^{-6}) appear to survive, but the claimed improvements need correction or removal.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a bundle-based augmented Lagrangian algorithm (BALA) for constrained convex problems of the form (P). At each iteration, BALA solves a subproblem over an inner approximation Ω_k ⊂ Ω, computes a dual candidate via a proximal-type update, tests a sufficient descent criterion (15), and updates Ω_k using both primal and dual information (Assumption 3.1). The authors show that BALA is equivalent to a proximal bundle method applied to the dual (Lemmas 5 and Theorem 7), and exploit this connection to prove sublinear convergence of dual iterates, primal feasibility, and primal cost (Theorems 3 and 4), an average-iterate result (Theorem 5), and, under quadratic growth and quadratic closeness, linear convergence with every iteration being a descent step (Theorem 6). The main application is to semidefinite programs with trace-bounded PSD cones, where the subproblems and dual-function evaluations have closed forms.","tokens_in":32185,"tokens_out":10025,"duration_ms":86272,"significance":"If the results are correct, BALA is a potentially valuable single-loop alternative to inexact ALM for structured convex problems. The primal-dual PBM connection is a clean and useful observation, and the paper is generally well written. The fixed-ρ sublinear rates are plausible and the experiments show encouraging performance on SDPs. However, two load-bearing points—the claimed improved rates with ρ=ε in Theorem 4 and the penalty condition in Theorem 6—contain inconsistencies that need correction before the main claims are fully reliable.","major_comments":[{"comment":"The claim that choosing ρ=ε improves the dual and primal iteration complexities to O(ε^-2) and O(ε^-4) is unsupported and, as stated, false. Lemma 5 shows that the dual candidate z_{k+1} is the proximal point of the model g_k with parameter α=ρ (since the quadratic term is (1/(2ρ))||y-y_k||^2). Theorem 2's accelerated bound O(ε^-2) applies only when α=1/ε, i.e., ρ=1/ε, not ρ=ε. Moreover, to turn a dual gap of δ into a primal residual ||Ax_{k+1}-b|| ≤ ε, Lemma 2 (17b) requires δ ≤ ρβ ε^2/2; with ρ=ε this forces δ=O(ε^3), and the generic PBM bound then gives at best O(δ^-3)=O(ε^-9) iterations, not O(ε^-4). The proof in Appendix D.5 simply asserts that the O(ε^-4) bound 'follows from Lemmas 2 and 3' without performing the substitution. The fixed-ρ rates O(ε^-3) and O(ε^-6) appear consistent, but the improved rates should be removed or corrected.","section":"Theorem 4, Appendix D.5"},{"comment":"The condition ρ ≥ 1/γ stated in Theorem 6 contradicts the inequality 1/ρ ≥ γ used in the proof. In Appendix D.7, the descent argument applies Theorem 8, which requires the proximal parameter θ=1/ρ to satisfy θ ≥ α, where α corresponds to the quadratic-closeness constant γ in (19). The proof explicitly states '1/ρ ≥ γ where γ is the constant in (19)'. The theorem statement requires the opposite inequality, so a user who follows the stated condition ρ ≥ 1/γ may choose a penalty for which the proof's key step (descent at every iteration) is no longer guaranteed. The theorem condition must be changed to ρ ≤ 1/γ (or to 1/ρ ≥ γ) or the proof amended.","section":"Theorem 6, Appendix D.7"},{"comment":"The single-loop efficiency claim rests on Assumption 3.1 item 3, which requires at every iteration an exact point v_{k+1} ∈ Ω with g(z_{k+1}) = -L(v_{k+1}, z_{k+1}), plus exact evaluations of g(y_k) and g(z_{k+1}) in the descent test (15). For a general compact convex Ω these are convex optimization problems potentially as hard as the original problem, and the closed-form constructions in Appendix A.2 are available only for the nonnegative orthant, second-order cone, and PSD cone with a trace bound. The paper should state in the main text that the per-iteration cost is low only for such structured sets, and that the iteration-complexity guarantees do not account for the oracle cost when Ω is arbitrary. Otherwise the headline 'single-loop' advantage is overstated.","section":"Assumption 3.1, Appendix A.2"}],"minor_comments":[{"comment":"In Theorem 2, the Lipschitz constant M is not defined in the theorem statement; it is first identified as ‖b‖+‖A‖D in Appendix D.5, which should be stated earlier.","section":"Theorem 2"},{"comment":"In Lemma 4, the bound O(ε^-2) on null steps between consecutive descent steps should specify the dependence of ε (the target accuracy) rather than an undefined 'sub-optimality measure'.","section":"Lemma 4"},{"comment":"In Appendix D.5, the phrase 'viewing the reciprocal relationship between the penalty parameter ρ and the proximal mapping parameter in Lemma 5' should explicitly state that the proximal parameter is α=ρ.","section":"Appendix D.5"},{"comment":"In Appendix D.7, the expression for µ2 contains unclear parentheses in the term involving ρ^4 E^2; the formula should be typeset and grouped carefully.","section":"Appendix D.7"},{"comment":"In Section 5, the text refers to 'Figure 3' when discussing the main experiments, but the first main-text figure is Figure 1; the cross-reference appears to be a typo.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the core idea is promising. The errors in Theorem 4's improved-rate claim and Theorem 6's condition are localized and appear fixable without changing the overall framework. The dependence on the dual-function oracle for general Ω should be made explicit in the main text. No concerns about citation practices; the self-citations to [4,5,22] are relevant prior results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of the paper is a new single-loop bundle-based ALM (BALA) that inner-approximates Ω by a bundle-defined set, and the main equivalence result (Theorem 7) showing BALA on the primal is exactly PBM on the dual. That equivalence is the right lens, and it makes the fixed-ρ sublinear rates believable. Theorem 5 (an average-iterate result for PBM) is a genuine new claim. The algorithm construction with line-segment inner sets and descent test is not present in the earlier literature, so the algorithmic contribution is real.\n\nThe soft spots are real too, and one is load-bearing. Theorem 4's 'if ρ=ε, then O(ε^-2) and O(ε^-4)' appears false. Lemma 5 gives z_{k+1}=argmin g_k(y)+(1/(2ρ))||y-y_k||^2, so in Theorem 2's notation the proximal parameter is α=ρ, not 1/ρ. The O(ε^-2) rate in Theorem 2 requires α=1/ε, i.e., ρ=1/ε. And even granting a dual gap of ε, Lemma 2 requires g(y_k)-g* ≤ ρβ ε^2/2, so with ρ=ε the required dual tolerance is O(ε^3), which yields worse rates; the appendix asserts O(ε^-4) without doing this substitution. The fixed-ρ rates O(ε^-3)/O(ε^-6) appear to survive.\n\nTheorem 6 has the same reciprocal confusion: the statement requires ρ ≥ 1/γ, but the proof explicitly uses 1/ρ ≥ γ. Since linear convergence is a headline result, the condition needs to be flipped and the proof rechecked.\n\nA separate weakness is that Assumption 3.1 requires exact evaluations of g(y_k) and an exact v_{k+1} realizing the dual function value. The paper treats these as cheap oracles, which is true for orthant/SOC/PSD trace-bound sets, but the assumption should be stated as a first-class oracle requirement. The experiments are illustrative only: no error bars, only one competing algorithm (CGAL), no SBM baseline despite the paper's own connection.\n\nWho this is for: people working on ALM/PPM/bundle methods for conic programs. With the parameter issues fixed, this deserves a serious referee. My recommendation: send to review, major revision, and ask the authors to either prove or remove the ρ=ε improvements and fix Theorem 6's condition.","headline":"A genuinely new single-loop ALM-PBM link, but the advertised ρ=ε speedups and Theorem 6's parameter condition are wrong as printed; the corrected fixed-ρ rates are solid.","tokens_in":32739,"tokens_out":2747,"would_cite":true,"duration_ms":25392,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A bundle-based augmented Lagrangian method (BALA) solves a constrained convex problem in a single loop with no inner subroutine, achieving sublinear convergence rates and linear convergence for semidefinite programs with rank-one…","keywords":["bundle method","augmented Lagrangian","convex optimization","semidefinite programming","convergence rates","single-loop algorithm","proximal bundle method","primal-dual algorithm"],"falsifier":"Take an SDP with a rank-one optimal solution that satisfies quadratic growth (18) and quadratic closeness (19) with known constants $\\alpha, \\gamma$, and run BALA with $\\beta = 1/2$ and $\\rho \\ge 1/\\gamma$ from a point satisfying the hypotheses. If any null step occurs after the iteration $T$ where (19) is guaranteed, then the claim that every iteration is a descent step is false. Alternatively, implement BALA with line-segment approximations on a general compact polytope $\\Omega$ in dimension $n \\ge 100$, counting the cost of the exact dual evaluations; if the total observed work scales worse than a conventional inexact ALM with an accelerated-gradient inner loop, then the single-loop efficiency advantage is limited to structured $\\Omega$, exactly as the weakest assumption suggests.","tokens_in":2121,"feed_emoji":"🧮","tokens_out":2265,"duration_ms":73570,"temperature":0.7,"pith_summary":"This paper proposes a bundle-based augmented Lagrangian framework (BALA) for constrained convex problems of the form $\\min\\{\\langle c, x\\rangle : Ax = b, x \\in \\Omega\\}$. The claim is that replacing the exact inner solve of classical ALM by an exact solve over a simple inner approximation $\\Omega_k \\subset \\Omega$, built from a bundle of past primal and dual iterates, yields a single-loop algorithm whose dual iterates converge at rate $O(\\epsilon^{-3})$ and whose primal residuals converge at rate $O(\\epsilon^{-6})$, with $\\rho = \\epsilon$ improving these to $O(\\epsilon^{-2})$ and $O(\\epsilon^{-4})$. Under quadratic growth of the dual function and quadratic closeness of the approximations, every iteration is a descent step and the method converges linearly, including for semidefinite programs with rank-one optimal solutions. If correct, this gives a projection-free, subroutine-free ALM with provable rates and a per-iteration cost that can be as low as a one-dimensional line search. The proof works by showing BALA is exactly the proximal bundle method applied to the dual problem, in analogy with the classical equivalence between ALM and the proximal point method.","feed_headline":"Single-loop ALM: sublinear to ε, linear for SDPs","feed_subtitle":"A bundle of past iterates makes each step a closed-form update; rank-one SDPs converge linearly.","key_machinery":"The load-bearing object is the inner approximation set $\\Omega_k$, chosen as a line segment (or a small convex hull) spanned by two points: $v_k$, the point where the Lagrangian realizes the current dual function value, and $w_k$, the previous approximate primal solution. This set satisfies Assumption 3.1 (proper, inner, containing $v_{k+1}$, and containing $w_{k+1}$ on null steps), and the induced dual function $g_k$ is a valid bundle model. Solving the quadratic subproblem over such a line segment reduces to a scalar saturation formula, so each iteration is a closed-form computation; the descent test (15) decides whether to accept or reject the candidate, exactly as in a proximal bundle method.","core_discovery":"The central discovery is that a bundle of past iterates can define a sequence of inner approximations $\\Omega_k \\subset \\Omega$ whose induced approximate dual functions $g_k(y) = -\\min_{x \\in \\Omega_k} L(x,y)$ satisfy exactly the three requirements of a proximal bundle model: they stay below the true dual function, they carry the subgradient of $g$ at the candidate point, and they aggregate the previous model's subgradient. Consequently the BALA update $z_{k+1} = y_k + \\rho(b - A w_{k+1})$ is precisely a proximal step on $g_k$, and the descent test $g(y_k) - g(z_{k+1}) \\ge \\beta\\bigl(g(y_k) - g_k(z_{k+1})\\bigr)$ is the bundle method's test. This equivalence lets the authors transfer the convergence machinery of proximal bundle methods to ALM: null steps are bounded by $O(\\epsilon^{-2})$, giving the $O(\\epsilon^{-3})$ dual rate, and the inexact-ALM interpretation gives the primal average-iterate rate $O(\\epsilon^{-3})$. Under the additional regularity conditions (18) and (19), a proximal-point contraction in the distance to the dual optimal set shows that every iteration is a descent step and the method converges linearly, a regime that includes SDPs with rank-one solutions.","pith_inferences":["The efficiency claim is strongest for the three symmetric cones with trace bounds; for a general compact convex $\\Omega$, the exact dual-oracle step in Assumption 3.1 is itself an optimization problem, so the practical single-loop advantage is an oracle property rather than a property of the bundle framework alone.","The equivalence with the proximal bundle method means that any improvement to bundle methods (better models, adaptive step sizes, second-order information) automatically yields an improved BALA; conversely, the average-iterate result proven for BALA transfers to PBM, which the paper notes is new.","The linear convergence for rank-one SDPs hints at a practical two-phase protocol: a coarse solver reaches a moderately accurate solution, then BALA refines it to high accuracy cheaply — the paper's experiments reach a $10^{-11}$ dual gap, beyond typical interior-point precision, though this use is only implicit.","The line-segment inner approximation gives BALA a Frank-Wolfe-like flavor with a safeguard test; connecting the two families, e.g. through momentum or weak-proximal oracles, is a natural next step that the paper only sketches."],"forward_implications":["BALA reaches a dual gap $g(y_k) - g^\\star \\le \\epsilon$ in $O(\\epsilon^{-3})$ iterations, and primal feasibility and cost gap $\\le \\epsilon$ in $O(\\epsilon^{-6})$ iterations; setting $\\rho = \\epsilon$ improves both to $O(\\epsilon^{-2})$ and $O(\\epsilon^{-4})$.","Averaging the primal iterates over descent steps yields $O(\\epsilon^{-3})$ for both primal residuals, giving the first average-iterate convergence result for the proximal bundle method.","Under quadratic growth (18) and quadratic closeness (19), every iteration is a descent step and the dual distance to optimality contracts by a constant factor; the same linear rate transfers to primal feasibility and cost gap.","When applied to SDPs with rank-one optimal solutions, BALA enjoys these linear rates while each subproblem over the line-segment approximation is solved in closed form, matching the structure of spectral bundle methods.","BALA matches the convergence behavior of classical inexact ALM, so subsequent iALM developments can be adapted to a single-loop scheme without an inner solver."],"supporting_citations":[{"why":"Supplies the ALM-PPM equivalence and the summability criterion (Theorem 1) that underlies the asymptotic convergence and telescoping argument for BALA.","marker":"[15]"},{"why":"Supplies the proximal bundle method convergence rates that yield the $O(\\epsilon^{-3})$ dual bound through the BALA-PBM equivalence.","marker":"[38]"},{"why":"Supplies the average-iterate inexact-ALM bound used for Theorem 5's $O(\\epsilon^{-3})$ primal average rate.","marker":"[21]"},{"why":"Establishes quadratic growth and quadratic closeness for spectral bundle methods on SDPs, which are the regularity conditions behind Theorem 6's linear convergence.","marker":"[5]"},{"why":"Establishes quadratic growth for conic programs in inexact ALM, supporting the linear-convergence regime.","marker":"[22]"},{"why":"Provides the spectral bundle method construction for SDPs and the rank-one capture used in the inner approximation.","marker":"[4]"},{"why":"Is the classical spectral bundle method that BALA reproduces for SDPs, giving the subproblem structure and update rules.","marker":"[49]"}],"fun_headline_variants":["Bundle-based ALM: single-loop, linear for SDPs","Bundle trick makes ALM single-loop, linear on SDPs","ALM with bundles: one loop, linear SDP rates","Bundle-based ALM: one loop, sublinear to ε, linear on SDPs","Single-loop ALM, bundle-powered: linear SDP convergence"],"cache_read_input_tokens":34560,"weakest_assumption_plain":"The convergence rates assume that at every iteration the algorithm can evaluate the true dual function exactly and can find the exact point in $\\Omega$ where the Lagrangian attains the dual value; for a general compact convex set this is itself a convex optimization problem, so the advertised cheap per-iteration cost holds only for structured sets such as the nonnegative orthant, second-order cone, and positive semidefinite cone with a trace bound.","fun_headline_variants_meta":{"raw":{"variants":["Bundle-based ALM: single-loop, linear for SDPs","Bundle trick makes ALM single-loop, linear on SDPs","ALM with bundles: one loop, linear SDP rates","Bundle-based ALM: one loop, sublinear to ε, linear on SDPs","Single-loop ALM, bundle-powered: linear SDP convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002059,"raw_usage":{"total_tokens":8037,"prompt_tokens":991,"completion_tokens":7046,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":6952}},"tokens_in":607,"tokens_out":7046,"duration_ms":47988,"temperature":1.0,"reasoning_tokens":6952,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:32:11.872160+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an SDP with a rank-one optimal solution that satisfies quadratic growth (18) and quadratic closeness (19) with known constants $\\alpha, \\gamma$, and run BALA with $\\beta = 1/2$ and $\\rho \\ge 1/\\gamma$ from a point satisfying the hypotheses. If any null step occurs after the iteration $T$ where (19) is guaranteed, then the claim that every iteration is a descent step is false. Alternatively, implement BALA with line-segment approximations on a general compact polytope $\\Omega$ in dimension $n \\ge 100$, counting the cost of the exact dual evaluations; if the total observed work scales worse than a conventional inexact ALM with an accelerated-gradient inner loop, then the single-loop efficiency advantage is limited to structured $\\Omega$, exactly as the weakest assumption suggests.","supporting_citations":[{"cited_title":"Augmented Lagrangians and applications of the proximal point algorithm in convex programming","cited_arxiv_id":null,"evidence_quote":"Supplies the ALM-PPM equivalence and the summability criterion (Theorem 1) that underlies the asymptotic convergence and telescoping argument for BALA."},{"cited_title":"Optimal convergence ratesfor the proximal bundle method","cited_arxiv_id":null,"evidence_quote":"Supplies the proximal bundle method convergence rates that yield the $O(\\epsilon^{-3})$ dual bound through the BALA-PBM equivalence."},{"cited_title":"Iteration complexity of inexact augmented Lagr angian methods for constrained convex programming","cited_arxiv_id":null,"evidence_quote":"Supplies the average-iterate inexact-ALM bound used for Theorem 5's $O(\\epsilon^{-3})$ primal average rate."},{"cited_title":"Revisiting spectral bundle metho ds: Primal-dual (sub) linear convergence rates","cited_arxiv_id":null,"evidence_quote":"Establishes quadratic growth and quadratic closeness for spectral bundle methods on SDPs, which are the regularity conditions behind Theorem 6's linear convergence."},{"cited_title":"Inexact augmented Lagrangian methods for conic opti- mization: Quadratic growth and linear convergence","cited_arxiv_id":null,"evidence_quote":"Establishes quadratic growth for conic programs in inexact ALM, supporting the linear-convergence regime."},{"cited_title":"A spectral bundle method for semidefinite programming","cited_arxiv_id":null,"evidence_quote":"Is the classical spectral bundle method that BALA reproduces for SDPs, giving the subproblem structure and update rules."}],"review_version":1}