{"id":"b8cfd015-7eaa-4c8d-9fe8-cd94da8222d5","arxiv_id":"2508.20872","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A Tikhonov/proximal double-loop method for hierarchical VIs is proposed, but its main proof requires a coefficient bound the assumptions do not provide.","lead":"This paper proposes a double-loop algorithm for hierarchical variational inequalities, where the allowed set is itself the solution set of a lower-level problem. The convergence proof has an unsupported coefficient bound, and the advertised strong convergence is stronger than what is proven.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8's inner-loop convergence proof applies Lemma 5 to a Lyapunov recurrence with coefficient Q_k not guaranteed to lie in (0,1); strong convergence guarantee is unproven.","rationale":"The reader's weakest assumption identifies the most load-bearing concern: the proof of Theorem 8 applies Lemma 5 to a recurrence whose coefficient Q_k is not guaranteed to be less than 1 under the stated assumptions. This is not a minor technicality—it invalidates the inner-loop strong convergence claim, which is the core algorithmic guarantee. The outer-loop result Theorem 9 depends on the inner loop being able to reach a stopping time K(ε) with error bound (10); without the inner-loop convergence, the outer loop's approximation property is unsupported. I considered other potential issues, such as the abstract overclaiming strong convergence to the hierarchical solution when Theorem 9 only gives weak limit points, and a missing factor of 1/γ in the error bound (10). These are secondary; the Q_k gap is a clear mathematical error in the proof. The paper's algorithmic idea is plausible and the applications are relevant, but as written the central convergence theorem is not established. The reader's REJECT verdict is appropriate, so no adjustment is needed.","tokens_in":14305,"tokens_out":8426,"duration_ms":75839,"concrete_test":"Independently verify whether there exists a parameter set satisfying Assumptions 3–4 and the step-size rule with q_k^2 ≥ 1/2. For example, take α=0.1, L=1.2, γ=α/L^2, θ_k≡0.5, τ_k≡0. Then q^2 = 1 - α^2/L^2 ≈ 0.993, Q = 1 - 0.5 + 2*0.5*0.993 = 1.493 > 1. This shows the assertion 'Q_k ∈ (0,1)' is false. Then check whether the proof of Theorem 8 contains any alternative argument controlling V_k for Q_k ≥ 1 (e.g., boundedness of V_k, summability of positive parts) that would make Lemma 5 unnecessary; if no such argument exists, the inner-loop strong convergence result is unproven as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 8 (Section IV-A), the authors derive the Lyapunov recurrence V_{k+1} ≤ Q_k V_k + E(δ_k) with Q_k = 1 - θ_k + 2θ_k q_k^2, and assert 'Q_k ∈ (0,1)' before applying Lemma 5. However, Q_k < 1 iff q_k^2 < 1/2, i.e., γ_k(2α - γ_k L^2_{α,β_t}) > 1/2. This condition is not stated in Assumptions 3–4; the only parameter restriction given is equation (11), which does not control Q_k's magnitude. The numerical example (Section V) uses α=0.1, L≈0.1+β_t+α, γ=α/L^2, giving q^2 ≈ 1 - α^2/L^2 ≈ 0.993, so Q_k > 1 for any θ_k ≥ θ > 0. Thus Lemma 5 cannot be invoked. The conclusion that ∑∥v_k-ū∥^2 < ∞ and strong convergence of (v_k, z_k) to ū is not established. This inner-loop convergence is essential: the stopping time K(ε) and the outer-loop error bound (10) rely on the iterates reaching a neighborhood of ū; without it, the hierarchical convergence theorem (Theorem 9) lacks a foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a double-loop algorithm (IKM) for hierarchical variational inequalities in real Hilbert spaces: the lower-level problem is a monotone inclusion zer(A+F), and the upper-level problem is VI(G,S0) with S0 = zer(A+F). The inner loop is a relaxed-inertial inexact forward-backward method aimed at the unique solution of the strongly monotone auxiliary inclusion zer(A + F + βG + α(I - w_t)), with a stopping rule based on the discrepancy ∥v_{k+1} - z_k∥. The outer loop updates the anchor w_t and the Tikhonov parameter β_t. The paper claims strong convergence of the inner iterates and weak convergence of the outer sequence to the solution set S1 = zer(G + N_{S0}), together with numerical experiments on a zero-sum game. The main technical innovation is a Lyapunov analysis of the inertial-relaxed inner scheme without assuming summability of the discrete velocity.","tokens_in":14740,"tokens_out":8923,"duration_ms":80840,"significance":"If the proofs were correct, the paper would extend proximal-penalization methods for hierarchical VIs to monotone Lipschitz operators without cocoercivity, in infinite-dimensional Hilbert spaces, with inexact inner solves. The Lyapunov treatment of the inertial-relaxed scheme and the explicit stopping criterion are potentially useful. However, the central inner-loop convergence proof contains a gap: it invokes Lemma 5 with a coefficient Q_k that is not shown to lie in (0,1) and, in the authors' own numerical setting, actually satisfies Q_k > 1. Consequently, the inner-loop strong convergence, the finiteness of the stopping time, and the error bound used in the outer loop are not established. The outer-loop proof also has unjustified limit-passing steps. The abstract overstates the result by claiming strong convergence to the nested solution when only weak convergence of the outer sequence is proved.","major_comments":[{"comment":"The proof asserts 'Q_k ∈ (0,1)' without support. With Q_k = 1 - θ_k + 2θ_k q_k^2, Q_k < 1 requires q_k^2 < 1/2, i.e., γ_k(2α - γ_k L^2_{α,β_t}) > 1/2. This condition is not part of Assumptions 3–4; equation (11) does not control Q_k's magnitude. In the numerical experiment (Section V), α=0.1, L ≈ 0.2 + β_t, γ = α/L^2, giving q^2 = 1 - α^2/L^2 ≈ 0.993 at t=0 and still > 0.5 later, so Q_k > 1 for any θ_k > 0. Lemma 5 cannot be applied, and the conclusions ∑∥v_k - ū∥^2 < ∞ and strong convergence of (z_k,v_k) are unproven. This also undermines the finiteness of K(ε), on which the outer-loop update relies.","section":"Section IV-A, Theorem 8"},{"comment":"The paper's abstract claims 'strong convergence guarantees towards a solution of the nested VI problem', but Theorem 9 only shows that weak limit points of the outer sequence lie in S1. This is a qualitative mismatch. More importantly, the proof of Theorem 9 depends on condition (6) and the error bound (10), which rest on the unproven inner-loop convergence. Without a finite stopping time, the outer loop is not well-defined and the subsequent limit-passing arguments lack foundation.","section":"Section IV-B, Theorem 9"},{"comment":"The passage 'By the definition of h_m, which we know to converges, we deduce that Π_{S1}(w_m) − w_m converges weakly and in norm Therefore it converges strongly' is not justified. Convergence of h_m gives only convergence of the norm of the difference; it does not identify the norm limit with the norm of the weak limit, so strong convergence does not follow. This step is used to obtain 0 = ⟨G(ẅ), ŵ − ẅ⟩, which is essential for concluding ẅ ∈ S1 via Lemma 2. The subsequent invocation of Lemma 2 is also unclear, since Lemma 2 requires an element u* ∈ zer(G+N_{S0}), which is exactly what is being proved.","section":"Section IV-B, Case 1 proof"},{"comment":"The experiment uses α=0.1 and γ=α/L^2 with L≈0.2+β_t, which yields q_k^2 > 1/2 and hence Q_k > 1. Thus the tested parameter regime is outside the (missing) sufficient condition for Theorem 8. The experiment therefore does not illustrate the convergence regime claimed by the theory, and the paper should either add the missing condition and choose parameters satisfying it, or revise the theory to cover the tested regime.","section":"Section V, Numerical experiments"}],"minor_comments":[{"comment":"In the display after the monotonicity inequality, the intermediate term γ⟨g, Φ(v,w) - Φ(ū,w)⟩ is upper-bounded by γL_{α,β}∥v-ū∥, omitting the factor ∥g∥. The final bound is correct, but the line is a typo that should be fixed.","section":"Lemma 7 proof"},{"comment":"Typo: 'various application' should be 'various applications'. Also the phrase 'strong convergence guarantees towards a solution of the nested VI problem' overstates Theorem 9, which establishes weak convergence of the outer sequence.","section":"Abstract"},{"comment":"The derivation before the stopping time uses a bound ε/[θ_k(1 - q_k)] but the stopping time K(ε) is defined by ∥v_{k+1} - z_k∥ ≤ ε. Please clarify the relation between the two; as written, the bound with 1/(1-q_k) is not reflected in the stopping rule or in the error estimate (10).","section":"Section III-B"},{"comment":"The sentence 'Across inner loops, we consider the acceleration parameter τ_k ≡ τ to be constant and to be the largest value satisfying equation (11)' is ambiguous because equation (11) involves Q_k, which depends on β_t and therefore varies with the outer iteration t. Please specify whether τ is chosen uniformly over all t or re-tuned per inner loop.","section":"Section V"},{"comment":"Algorithm 1 sets v_0 = v_1 = w_1 and returns v_{K+1}; the update w_{t+1} = v_{K+1} is consistent with the text, but the pseudocode should specify how K_t is computed in practice and how the parameter sequences θ_k, τ_k are reset or continued across outer loops.","section":"Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a working note that extends a known template ([1]) to monotone Lipschitz operators with an inertial-relaxed inexact inner solver. The main gap (Q_k < 1) is not merely a technicality: in the numerical setup Q_k > 1, so the current proof cannot cover the algorithm as tested. The proofs of Theorem 8 and Theorem 9 need substantial revision, and the abstract's claim of 'strong convergence' must be aligned with the weak convergence result. I would not reject outright, because the framework is plausible and the Lyapunov technique may be repairable, but the present version is not correct as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things: the Hilbert-space, inexact, inertial inner loop with non-summable momentum is a real extension over Facchinei et al., and the application to monotone Lipschitz lower-level operators is wider than the cocoercive cases in BiG-SAM. But the main convergence theorem doesn't hold as written. Theorem 8 applies Lemma 5 to V_{k+1} ≤ Q_k V_k + E(δ_k) with Q_k = 1-θ_k+2θ_k q_k^2, asserting Q_k ∈ (0,1). That inequality is equivalent to q_k^2 < 1/2, i.e. γ_k(2α-γ_k L^2) > 1/2. Neither Assumption 3, 4 nor equation (11) enforce it. In their own α=0.1 experiment, L=0.1+β_t+α and γ=α/L^2 give q^2≈0.993, so Q_k>1 for any positive θ. Lemma 5 cannot be invoked. This isn't a symbolic gap: the stopping-time bound (10) and the outer-loop error control both assume inner iterates reach a neighborhood of the auxiliary solution, so Theorem 9 rests on an unproven foundation.\n\nWhat the paper does well: the Lyapunov construction is clever and avoids the usual summability assumption on momentum; the inexact oracle model is honest; the formulation in Hilbert space is useful. Lemma 7 has a dropped norm factor in an intermediate line (γ∥g∥ becomes γL∥v−ū∥), but the final bound is still correct. The numerical example is preliminary—a single low-dimensional game—and, notably, runs in a regime where Q_k>1, so it doesn't exercise the theorem.\n\nThe abstract also overstates the outer-loop result: Theorem 9 produces bounded sequences with weak limit points in S1, not strong convergence of the outer iterates.\n\nBottom line: this is a plausible research direction, and the paper is worth a serious referee because the fix might be a parameter condition (e.g. q^2<1/2, or a different Lyapunov inequality that tolerates Q_k≥1). But in current form the central inner-loop convergence guarantee is unproven, and the numerical evidence doesn't cover the theorem's assumptions. I'd send it to review, but I'd expect major revision; if the authors can't close the Q_k gap, the paper shouldn't be accepted.","headline":"The inertial inexact inner-loop idea is a genuine extension, but Theorem 8's Lyapunov argument needs Q_k<1, which the assumptions don't ensure; the main convergence guarantee is unproven as written.","tokens_in":15116,"tokens_out":3135,"would_cite":false,"duration_ms":28449,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J40","47J20","90C33","65K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A double-loop Tikhonov-penalty algorithm is claimed to converge, weakly in the outer loop and strongly in the inner loop, to a solution of a hierarchical variational inequality.","keywords":["hierarchical variational inequality","Tikhonov regularization","monotone inclusion","forward-backward splitting","inertial methods","bilevel optimization","equilibrium selection","proximal penalty"],"falsifier":"Run the paper's zero-sum game with its reported setting α=0.1, L≈1.2, γ=α/L² and any constant θ_k=θ∈(0,1); compute q²=1−α²/L²≈0.993 and Q=1−θ+2θq²>1. If the inner loop is run without the stopping criterion, this puts the iterates in the regime where the proof's Lyapunov inequality does not apply. A concrete divergence, or a proof of convergence in this regime, would settle whether the unconditional strong-convergence claim holds.","tokens_in":14217,"feed_emoji":"🎯","tokens_out":9670,"duration_ms":94656,"temperature":0.7,"pith_summary":"This paper tackles hierarchical variational inequalities: find a point satisfying an upper-level inequality VI(G, S0) over the solution set S0 of a lower-level monotone inclusion 0 ∈ A(v) + F(v). The authors propose a double-loop algorithm in real Hilbert space. The inner loop approximates, with inexact inertial forward-backward steps, the unique solution of a strongly monotone auxiliary problem built by adding a Tikhonov term βG and a proximal anchor term α(v−w). The outer loop updates the anchor point and shrinks the Tikhonov parameter. They prove strong convergence of the inner iterates and weak convergence of the outer iterates to a solution of the nested problem, and they show the framework covers bilevel convex optimization and equilibrium selection in Nash games, with a numerical illustration on a two-player zero-sum game.","feed_headline":"Double-loop solver reaches nested equilibrium solutions","feed_subtitle":"Inner loops converge strongly; outer loops land in the upper-level solution set despite inexact resolvent steps.","key_machinery":"The central object is the parametric auxiliary operator Φ_{α,β}(v,w) = F(v) + βG(v) + α(v−w), which is α-strongly monotone and Lipschitz, so the auxiliary inclusion zer(A + Φ_{α,β}(·,w)) has a unique solution ū_{α,β}(w). The inner loop tracks this solution with an inexact, inertial, relaxed forward-backward iteration v_{k+1} = (1−θ_k)z_k + θ_k(J_{γA}(z_k − γΦ_{α,β}(z_k,w)) + δ_k), z_k = v_k + τ_k(v_k−v_{k−1}). The proof builds a Lyapunov function V_k from squared distances and successive-difference penalties, converting the recurrence V_{k+1} ≤ Q_k V_k + E(δ_k) into strong convergence. The outer loop updates w_{t+1} to the stopping-time iterate and sends β_t to zero slowly (β_t not summable,","core_discovery":"For a maximally monotone operator A with bounded domain and monotone Lipschitz maps F and G, the paper's double-loop method IKM produces a bounded outer sequence (w_t) whose weak limit points all lie in S1 = zer(G + N_{S0}), the solution set of the upper-level variational inequality over the lower-level solution set. Inside each loop, when the forward-backback operator is a contraction and resolvent errors are summable, the inertial relaxed iterates converge strongly to the unique solution of the auxiliary inclusion zer(A + F + β_t G + α(·−w_t)). The claimed novelty is that F need not be cocoercive, the inner computations may be inexact, the discrete velocities need not be assumed summable,","pith_inferences":["The successful numerics in a regime where the proof's contraction-strength condition fails suggest the algorithm may converge beyond the provable range; that is an empirical hint rather than a claim of the paper.","If the Q_k < 1 condition is genuinely necessary, the strong-convergence theorem covers only strongly contractive inner maps; extending it to weakly contractive maps would require a different Lyapunov construction or an averaged stopping rule.","The same outer-loop anchor update could plausibly be paired with other inner splittings for zer(A + F), such as Douglas–Rachford or ADMM, yielding hierarchical algorithms for problems where forward-backward contractions are unavailable."],"forward_implications":["Hierarchical problems with merely monotone Lipschitz lower-level operators become algorithmically tractable, including structured bilevel convex problems and saddle-point reformulations where cocoercivity-based methods fail.","The inner loop tolerates inexact resolvent evaluations, so proximal operators computed by auxiliary internal algorithms can be used without breaking the convergence guarantees.","Outer iterates asymptotically approach the upper-level solution set: any weak cluster point of the bounded outer sequence lies in S1, so the algorithm selects a solution of the nested variational inequality.","The Tikhonov parameter β_t may decay nonsummably as long as the inner accuracy e_t goes to zero faster than β_t, permitting a persistent regularizing effect that drives the hierarchical selection."],"supporting_citations":[{"why":"Supplies the Tikhonov-plus-proximal auxiliary operator Φ_{α,β} and the VI-constrained problem template that the double loop adapts.","marker":"[1]"},{"why":"Introduces the inertial forward-backward iteration for monotone inclusions whose momentum-and-relaxation structure the inner loop uses.","marker":"[15]"},{"why":"Inspires the Lyapunov analysis that lets the paper drop the usual summability assumption on discrete velocities.","marker":"[16]"},{"why":"Grounds the inexact forward-backward error model in which each resolvent step is computed with a perturbation δ_k.","marker":"[17]"},{"why":"Supplies inertial forward-backward rate results that the inertial inner scheme extends to the hierarchical setting.","marker":"[18]"},{"why":"Contains the technical lemma (Lemma 5 here) that converts the inner-loop Lyapunov recurrence into strong convergence.","marker":"[23]"}],"fun_headline_variants":["Nested VI solver drops cocoercivity, keeps strong convergence","Inexact inner iterates still reach nested equilibrium set","Hierarchical VI solved without summable velocity assumption","Double-loop method for nested inequalities tolerates inner errors","Strong convergence to upper-level solution with non-cocoercive maps"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The inner-loop convergence proof depends on the forward-backward step reducing the squared error by at least half at every iteration; the assumptions stated in the paper do not guarantee this, and in parameter regimes where it fails the key recurrence becomes non-contracting, so the strong-convergence conclusion is not established there.","fun_headline_variants_meta":{"raw":{"variants":["Nested VI solver drops cocoercivity, keeps strong convergence","Inexact inner iterates still reach nested equilibrium set","Hierarchical VI solved without summable velocity assumption","Double-loop method for nested inequalities tolerates inner errors","Strong convergence to upper-level solution with non-cocoercive maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000874,"raw_usage":{"total_tokens":3556,"prompt_tokens":621,"completion_tokens":2935,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":2853}},"tokens_in":365,"tokens_out":2935,"duration_ms":18672,"temperature":1.0,"reasoning_tokens":2853,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:45:11.669400+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's zero-sum game with its reported setting α=0.1, L≈1.2, γ=α/L² and any constant θ_k=θ∈(0,1); compute q²=1−α²/L²≈0.993 and Q=1−θ+2θq²>1. If the inner loop is run without the stopping criterion, this puts the iterates in the regime where the proof's Lyapunov inequality does not apply. A concrete divergence, or a proof of convergence in this regime, would settle whether the unconditional strong-convergence claim holds.","supporting_citations":[{"cited_title":"Vi- constrained hemivariational inequalities: distributed algorithms and power control in ad-hoc networks,","cited_arxiv_id":null,"evidence_quote":"Supplies the Tikhonov-plus-proximal auxiliary operator Φ_{α,β} and the VI-constrained problem template that the double loop adapts."},{"cited_title":"Accelerated and inexact forward-backward algorithms,","cited_arxiv_id":null,"evidence_quote":"Grounds the inexact forward-backward error model in which each resolvent step is computed with a perturbation δ_k."},{"cited_title":"Convergence rate of inertial forward–backward algorithm beyond nesterov’s rule,","cited_arxiv_id":null,"evidence_quote":"Supplies inertial forward-backward rate results that the inertial inner scheme extends to the hierarchical setting."},{"cited_title":"An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping,","cited_arxiv_id":null,"evidence_quote":"Contains the technical lemma (Lemma 5 here) that converts the inner-loop Lyapunov recurrence into strong convergence."}],"review_version":1}