{"id":"83dec2c3-1d6d-4378-9417-f1d151f89f92","arxiv_id":"1908.07398","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Three outer approximation variants using Landweber half-spaces converge in norm to the unique solution of a variational inequality over a split convex feasibility solution set.","lead":"This paper designs iterative algorithms that solve variational inequalities over the solution set of a split convex feasibility problem, replacing difficult projections with easy projections onto half-spaces. It proves norm convergence of three variants under regularity assumptions, extending outer approximation methods to a new constraint class.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The closed-range assumption behind inequality (3.31) is the decisive scope limit; it is explicit in Theorem 3.1, so the stated result stands.","rationale":"The paper's central claim is Theorem 3.1: under explicit assumptions including closed range of A and bounded regularity, the outer-approximation iterates converge in norm to the unique solution of VI(F,S). I checked the proof structurally. Step 3 correctly reduces the three operator constructions to the regularity properties (3.8)–(3.9), and the lower bounds in cases (i)–(iii) are consistent with Theorems 2.5, 2.6 and 2.12. Step 4 is the unique point where the split structure is handled: it needs the quantitative comparison d(x, A^{-1}(Q)) <= (1/|A|) d(Ax, R(A) ∩ Q), which is valid only when R(A) is closed. The reader's weakest-assumption identification is therefore correct and is the same one I would flag. The concern is not a defect in the theorem because the assumption is stated explicitly; rather, it is a scope restriction that matters for infinite-dimensional applications where A may have non-closed range. The proof relies on previously published results, but those do not contain the target theorem, so there is no circularity. No numerical implementation is provided, but this affects confidence in practical behavior, not the correctness of the conditional convergence statement. For these reasons, the reader's ACCEPT verdict should stand unchanged.","tokens_in":15339,"tokens_out":25507,"duration_ms":252339,"concrete_test":"Check whether the bridge used in Step 4 can hold without closed range by taking H1 = H2 = ℓ2, A = diag(1/(i+1)), Q = {0}, C = H1, and u_k = e_k. Then S = N(A) = {0}, d(Au_k, R(A) ∩ Q) = ||Au_k|| = 1/(k+1) -> 0, but d(u_k, A^{-1}(Q)) = d(e_k, {0}) = 1 for all k. This shows that the implication d(Au_k, R(A) ∩ Q) -> 0 ⇒ d(u_k, A^{-1}(Q)) -> 0 fails for bounded sequences without closed range, so the estimate (3.31) cannot be replaced by a purely topological argument and the closed-range assumption is genuinely load-bearing in the proof as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Step 4 of Theorem 3.1 is the only place where the convergence to S is completed for the split part: it applies Theorem 2.12(2.18) to assert d(u_k, A^{-1}(Q)) <= (1/|A|) d(Au_k, R(A) ∩ Q). Because R(A) closed is equivalent to |A| > 0 (Theorem 2.1), this estimate is exactly where the closed-range hypothesis is load-bearing. If R(A) is not closed, |A| = 0 and the inequality has no finite right-hand side, so the proof cannot conclude d(u_k, A^{-1}(Q)) -> 0 from d(Au_k, R(A) ∩ Q) -> 0. This is not an internal inconsistency: Theorem 3.1 explicitly assumes R(A) is closed. It is, however, the main restriction in the infinite-dimensional setting advertised by the paper, since many natural bounded linear operators (e.g., compact diagonal operators on ℓ2) have non-closed range. The rest of the proof, including the three cases (i)–(iii) and the bounded-regularity step, is structurally sound under the stated assumptions, and the dependence on prior results [13,15,24] is not circular.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies variational inequalities governed by a strongly monotone and Lipschitz continuous operator F over a closed convex set S that is the solution set of a multiple-set split convex feasibility problem, S = C ∩ A^{-1}(Q). It proposes three variants of an outer approximation method — product, simultaneous, and alternating — in which the difficult projection onto S is replaced by a projection onto a half-space built from a Landweber-type operator acting on the split part. Under explicit hypotheses (strong quasi-nonexpansiveness of the constituent operators, s-intermittent control sequences, bounded regularity of the relevant families, closed range of A, and step sizes λ_k satisfying λ_k → 0 and ∑λ_k = ∞), Theorem 3.1 establishes norm convergence of the iterates to the unique solution of the variational inequality. The proof verifies the regularity condition (1.7) of the general convergence result [24, Theorem 3.1] by showing, in Step 3, that small residuals of T_k imply vanishing distances to the individual sets C_i and Q_j, and then, in Step 4, uses bounded regularity and the closed-range estimate (3.31) to conclude d(u_k, S) → 0.","tokens_in":15510,"tokens_out":18985,"duration_ms":179553,"significance":"If the result holds, the paper provides a useful unified framework for solving variational inequalities over split feasibility sets, covering CQ, simultaneous CQ, and alternating variants, with multiple sets and general quasi-nonexpansive operators. The proof is careful and detailed, and the dependence on the authors' earlier results [15, 24] is transparent citation rather than circular reasoning. The assumptions are stated explicitly, and the three cases are genuinely different algorithmic constructions. The main scope limitation is the closed-range assumption on A, which is load-bearing in Step 4 via inequality (3.31): when R(A) is not closed, |A| = 0 and the estimate d(u_k, A^{-1}(Q)) ≤ (1/|A|) d(Au_k, R(A) ∩ Q) collapses. Because the assumption is explicitly part of Theorem 3.1, this is a limitation rather than an error; nevertheless, it should be made more visible to readers interested in infinite-dimensional applications.","major_comments":[],"minor_comments":[{"comment":"The first displayed operator should be U_k := P_C, not P_Q; as written, the remark says a projection onto Q acts on H_1, which is undefined.","section":"Remark 3.2"},{"comment":"The theorem statement should state explicitly that A is nonzero; Definition 2.9 assumes this, and the factor |A| in inequality (3.31) requires it.","section":"Theorem 3.1"},{"comment":"The notation i_k := argmax_{i∈I} d(u_{n_k}, C_i) is ambiguous when the maximum is attained at several indices; the authors should say that i_k is chosen arbitrarily among the maximizers.","section":"Proof of Theorem 3.1, Step 3"},{"comment":"The positive-part notation (·)_+ is used without definition; a brief definition would improve readability.","section":"Equations (2.20) and (2.23)"},{"comment":"The abstract and introduction advertise the general Hilbert-space setting without qualification; since the closed-range assumption on A in Theorem 3.1 is a substantial restriction in infinite dimensions, it would be helpful to flag this limitation in the introduction.","section":"Introduction and abstract"}],"recommendation":"accept","confidential_remarks":"The paper is a competent extension of the authors' earlier outer approximation framework. The novelty is incremental, but the three algorithmic variants and the explicit treatment of the split feasibility structure are useful and the proof appears sound. I would ask the authors to make the closed-range limitation more visible. No concerns about novelty disclosure or circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent synthesis of three known building blocks — outer-approximation half-spaces, the Landweber transform, and SQNE machinery — into a single theorem (3.1) covering product, simultaneous, and alternating variants. The proof is carefully structured and, on my reading, correct. The closed-range assumption on A is the true scope limit, but it is explicit in the statement, so the theorem as stated stands.\n\nThe new content is the three half-space constructions and the unified convergence guarantee. The operator theory itself is mostly borrowed from [24], [15], and [8]; that is fine, because the contribution is knowing how to put those pieces together. Lemma 2.13 and 2.14 give explicit projection formulas for the Landweber half-spaces, which is genuinely useful for implementation. Step 3 of the proof handles the product, simultaneous, and alternating cases separately and coherently; Step 4 then makes the right use of Theorem 2.12 to pass from distance to R(A) ∩ Q to distance to A^{-1}(Q).\n\nThe soft spots are real but not disabling. The closed-range condition is load-bearing exactly at inequality (3.31): if R(A) is not closed then |A| = 0 and the estimate has no finite right-hand side. The stress-test note is correct about that. Since the theorem states the assumption openly, this is a scope restriction rather than an error, but the paper could be more candid that the advertised infinite-dimensional setting is narrower than it sounds — compact diagonal operators on ℓ2 fail it. Bounded regularity is also assumed on both families; Remark 3.4 notes the finite-dimensional case covers this, but no rates or comparisons are given. There are no numerical experiments, so we learn nothing about whether the three variants behave differently in practice. That is a limitation, not a correctness issue.\n\nThe reliance on earlier self-authored work is heavy but not circular: [13], [15], and [24] are parameter-free theorems that do not contain Theorem 3.1. I agree with the reader's low circularity score.\n\nThis paper is for specialists in projection and fixed-point algorithms, particularly those working on split feasibility problems with variational inequalities. They will find a clean framework and a reusable template. I would want a referee who knows the regularity machinery to double-check conditions (3.8)–(3.9) and the s-intermittent controls, but I found no algebraic or logical gap. Recommendation: send it out; acceptance is defensible, with minor revision possibly asking for a comment on the closed-range condition and, if available, a small numerical illustration.","headline":"A competent synthesis of outer-approximation, Landweber, and SQNE tools into one convergence theorem; the closed-range assumption is the real scope limit, but the proof is sound and the paper deserves referee time.","tokens_in":16089,"tokens_out":2699,"would_cite":true,"duration_ms":26244,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H09","47H10","47J20","47J25","65K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a variational inequality over the split feasibility solution set $S=C\\cap A^{-1}(Q)$ can be solved by outer approximation steps onto half-spaces, with norm convergence under a closed-range condition on $A$.","keywords":["variational inequality","split convex feasibility problem","CQ-method","outer approximation method","Landweber transform","strongly quasi-nonexpansive operators","half-space projection","norm convergence"],"falsifier":"A concrete check of the boundary: take $H_1=\\ell^2$ and $A$ defined by $(Ax)_n=x_n/n$, which is injective with dense, non-closed range and $|A|=0$; set $C=H_1$, $Q=\\{0\\}$, and $F=\\mathrm{Id}$, so $S=\\{0\\}$ is the unique solution. The theorem's closed-range assumption fails exactly at estimate (3.31), which would divide by $|A|$; showing whether the iteration still converges for this $A$, or where the distance estimate breaks, would decide whether the closed-range hypothesis is intrinsic to the claim or only an artefact of the proof.","tokens_in":15097,"feed_emoji":"🎯","tokens_out":14358,"duration_ms":125969,"temperature":0.7,"pith_summary":"This paper proves that a variational inequality governed by a strongly monotone, Lipschitz continuous operator $F$ can be solved over the solution set $S=C\\cap A^{-1}(Q)$ of a split convex feasibility problem by an outer approximation method in which each metric projection onto $S$ is replaced by a metric projection onto a half-space $H_k$ that contains $S$ and is easy to evaluate. The half-spaces are constructed from the split structure in three variants---product, simultaneous, and alternating combinations of operators acting on $C$ and $Q$---using the Landweber transform to pull the constraint $Az\\in Q$ back to $H_1$. The main theorem shows that if the step sizes $\\lambda_k$ satisfy $\\lambda_k\\to 0$ and $\\sum\\lambda_k=\\infty$, if the linear operator $A$ has closed range, and if the relevant families of sets are boundedly regular, then the iterates converge in norm to the unique solution of $\\mathrm{VI}(F,S)$. The significance is that projections onto $S$, or onto $A^{-1}(Q)$, are generally expensive, while the half-space projections used here reduce to closed-form formulas involving $A$, $A^*$, and the current iterate.","feed_headline":"Half-space steps solve split-feasibility variational inequalities","feed_subtitle":"Each iterate projects onto a cheap half-space and still converges in norm to the unique solution.","key_machinery":"The central machinery is the half-space projection step (1.3)--(1.5), where $H_k=\\{z\\in H_1: \\langle u_k-T_k(u_k), z-T_k(u_k)\\rangle\\le 0\\}$ and $T_k$ is a cutter with $S\\subseteq \\mathrm{Fix}\\,T_k$. For the split constraint, the extrapolated Landweber transform $L_\\sigma\\{V\\}(x)=x+\\sigma(x)/\\|A\\|^2\\, A^*(V(Ax)-Ax)$ is the transfer device: under $R(A)\\cap \\mathrm{Fix}\\,V\\neq\\emptyset$ it is $\\rho$-strongly quasi-nonexpansive and has fixed point set $A^{-1}(\\mathrm{Fix}\\,V)$, so applying it to operators on $H_2$ produces operators on $H_1$ that encode the condition $Az\\in Q$. Lemma 2.13 then rewrites $H_k$ as $\\{z: \\langle Au_k-V_k(Au_k), Az-V_k(Au_k)\\rangle\\le 0\\}=A^{-1}(H_2(Au_k,V_k(Au_k)))$, giving a closed-form projection (2.20). The three variants differ only in how $U_k$ and $L_\\sigma\\{V_k\\}$ are combined---as a product, as a convex combination, or in alternation---and the proof's inequalities (3.17) and (3.23) feed the resulting residuals into the regularity conditions (3.8)--(3.9) and $s$-intermittent control sequences, yielding $d(u_k,S)\\to 0$.","core_discovery":"On the paper's own terms, the discovery is that the split structure of $S$ can be turned into the driving device of the algorithm rather than an obstacle. Given two sequences of strongly quasi-nonexpansive operators $U_k$ on $H_1$ with $C\\subseteq \\mathrm{Fix}\\,U_k$ and $V_k$ on $H_2$ with $Q\\subseteq \\mathrm{Fix}\\,V_k$, Theorem 3.1 defines algorithmic operators $T_k$ in three ways and proves that the outer approximation recurrence $u_{k+1}=R_k(u_k-\\lambda_k F(u_k))$, $R_k=\\mathrm{Id}+\\alpha_k(P_{H_k}-\\mathrm{Id})$, with $H_k=\\{z\\in H_1: \\langle u_k-T_k(u_k), z-T_k(u_k)\\rangle\\le 0\\}$, satisfies $d(u_k,S)\\to 0$; with $\\sum\\lambda_k=\\infty$ the sequence converges in norm to the unique solution of $\\mathrm{VI}(F,S)$. The proof is carried by the Landweber transform identity $\\mathrm{Fix}\\,L_\\sigma\\{V_k\\}=A^{-1}(\\mathrm{Fix}\\,V_k)$ and by the closed-range inequality $d(u,A^{-1}(Q))\\le (1/|A|)\\,d(Au,R(A)\\cap Q)$, which together transfer progress on the $Q$-side to progress on the $S$-side.","pith_inferences":["Inference: the closed-range condition on $A$ is likely the real boundary of the theory: for a compact injective $A$ with dense non-closed range, $|A|=0$ and inequality (3.31) cannot hold, so a different distance-transfer argument would be needed to cover such operators.","Inference: Lemma 2.14 suggests a nonsmooth variant of the method in which the split constraint is handled by subgradients of $q\\circ A$ rather than projections onto $Q$; the paper provides the projection formula but does not develop this as a separate algorithm.","Inference: the paper does not compare the product, simultaneous, and alternating variants quantitatively; a finite-dimensional test on a multiple-set split convex feasibility problem with known solution would be a natural way to see whether the differences in composition and relaxation parameters affect practical speed."],"forward_implications":["In finite-dimensional spaces, the closed-range and bounded-regularity assumptions hold automatically, so Theorem 3.1 guarantees norm convergence for every bounded linear $A$ and for all three variants.","When the split part is absent, Theorem 3.7 applies the same half-space construction to ordinary convex feasibility and yields the same norm convergence under the regularity condition (3.38).","Each iteration uses only the closed-form projection onto the half-space $H_k$, so the method avoids any projection onto $A^{-1}(Q)$ or onto $S$; when only subgradients are available, Lemma 2.14 provides an explicit formula of the same type.","The step-size requirement is only $\\lambda_k\\to 0$ with $\\sum\\lambda_k=\\infty$, so the algorithm does not need to know the strong monotonicity or Lipschitz constants of $F$ in order to choose its steps.","For the alternating variant with maximal extrapolation, the half-space is the preimage under $A$ of a half-space in $H_2$, making the method a direct split analogue of the CQ method described in Remark 3.2."],"supporting_citations":[{"why":"Supplies the outer approximation framework and the convergence criterion (Theorem 1.1) that the paper adapts to the split feasibility structure.","marker":"[24]"},{"why":"Provides the Landweber transform estimates, including the fixed-point-set identity, inequality (2.17), and the closed-range distance bounds (2.18).","marker":"[15]"},{"why":"Establishes the quasi-nonexpansiveness and strong-convergence properties of Landweber-type operators used in the proof of Theorem 2.12.","marker":"[13]"},{"why":"Supplies the definitions and composition/convex-combination properties of strongly quasi-nonexpansive operators and cutters used to build the operators T_k.","marker":"[8]"},{"why":"Provides the notion of bounded regularity and the conditions under which it holds, used in the convergence hypotheses.","marker":"[4]"},{"why":"Gives the convex-analytic facts for subgradient and proximal projections, including the adjoint identity behind Lemma 2.14 and the examples.","marker":"[5]"}],"fun_headline_variants":["Half-space projections tame split-feasibility inequalities","Outer approximation half-space method for split VIs","Landweber transform enables half-space VI solver","Three half-space variants for split convex feasibility","Split structure fuels half-space projection algorithm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the linear map $A$ has closed range; without it $|A|$ can be $0$ and the proof's bound $d(u,A^{-1}(Q))\\le (1/|A|)\\,d(Au,R(A)\\cap Q)$ collapses, so the argument cannot transfer convergence from the $Q$-side back to the $S$-side.","fun_headline_variants_meta":{"raw":{"variants":["Half-space projections tame split-feasibility inequalities","Outer approximation half-space method for split VIs","Landweber transform enables half-space VI solver","Three half-space variants for split convex feasibility","Split structure fuels half-space projection algorithm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1614,"prompt_tokens":1014,"completion_tokens":600,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":531}},"tokens_in":630,"tokens_out":600,"duration_ms":5725,"temperature":1.0,"reasoning_tokens":531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:19:44.971247+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check of the boundary: take $H_1=\\ell^2$ and $A$ defined by $(Ax)_n=x_n/n$, which is injective with dense, non-closed range and $|A|=0$; set $C=H_1$, $Q=\\{0\\}$, and $F=\\mathrm{Id}$, so $S=\\{0\\}$ is the unique solution. The theorem's closed-range assumption fails exactly at estimate (3.31), which would divide by $|A|$; showing whether the iteration still converges for this $A$, or where the distance estimate breaks, would decide whether the closed-range hypothesis is intrinsic to the claim or only an artefact of the proof.","supporting_citations":[{"cited_title":"Optimization 66(3), 417–437 (2017)","cited_arxiv_id":null,"evidence_quote":"Supplies the outer approximation framework and the convergence criterion (Theorem 1.1) that the paper adapts to the split feasibility structure."},{"cited_title":"Optimization 65(7), 1463–1476 (2016)","cited_arxiv_id":null,"evidence_quote":"Establishes the quasi-nonexpansiveness and strong-convergence properties of Landweber-type operators used in the proof of Theorem 2.12."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definitions and composition/convex-combination properties of strongly quasi-nonexpansive operators and cutters used to build the operators T_k."},{"cited_title":"SIAM Rev","cited_arxiv_id":null,"evidence_quote":"Provides the notion of bounded regularity and the conditions under which it holds, used in the convergence hypotheses."},{"cited_title":"CMS Books in Mathematics","cited_arxiv_id":null,"evidence_quote":"Gives the convex-analytic facts for subgradient and proximal projections, including the adjoint identity behind Lemma 2.14 and the examples."}],"review_version":1}