{"id":"fa151ff6-69ab-492d-ac35-157abadaa7a2","arxiv_id":"2607.20113","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A new linesearch for forward-reflected-backward splitting is claimed to converge weakly for continuous monotone operators, but a gap in the termination proof and out-of-range numerical parameters weaken the result.","lead":"The paper proposes a linesearch step-size rule that lets the forward-reflected-backward algorithm solve monotone inclusions with merely continuous operators, not just Lipschitz ones. It provides a counterexample where the older linesearch never stops, and tests the new method on saddle-point and image-restoration problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.2 is false: a 1D counterexample satisfying Problem 3.1 makes Algorithm 4.1's linesearch fail for every γ>0.","rationale":"The central claim of the paper is that Algorithm 4.1 is well defined and weakly convergent for merely continuous B satisfying Problem 3.1. That claim rests on Proposition 4.2's finite-termination argument. The reader identified the boundary-of-dom A issue as the weakest assumption; our explicit computation shows this is not just a missing detail but a false assertion: a valid instance can make the linesearch condition fail for every positive step size. The example uses only elementary one-dimensional operators and satisfies all hypotheses of Problem 3.1, including maximal monotonicity of B and A+B and nonemptiness of the solution set. Therefore the paper's main theorem is unsupported and, as stated, false. A natural repair would be to add a boundedness assumption on B over bounded subsets of dom A, or to require dom A to be closed, but that is a substantive change to the claimed contribution. We therefore see no reason to alter the reader's REJECT verdict.","tokens_in":23575,"tokens_out":15023,"duration_ms":132584,"concrete_test":"Run Algorithm 4.1 (or merely test Proposition 4.2's condition) on the one-dimensional problem: A=∂(-log), B(x)=2-1/x for x>0, x_{-1}=1/2, x_0=1, γ_{-1}=21, θ=0.9, τ,σ arbitrary. For any γ>0 compute z=J_{γA}(-20-γ). Since γ|Bz-Bx_0|=20+z and the required upper bound is 0.9(22-z), the inequality fails for every γ>0; hence the linesearch never terminates and Proposition 4.2 is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern is not merely a gap; it yields a concrete counterexample. Set H=R, f(x)=-log x (x>0), A=∂f, so dom A=(0,∞) and A(x)=-1/x. Let B be defined on (0,∞) by B(x)=2-1/x. B is monotone and continuous on dom A, and both B and A+B are maximally monotone (no extension at 0 is possible because values tend to -∞). Since (A+B)(x)=2-2/x, zer(A+B)={1}, so Problem 3.1 holds. Initialize x_{-1}=1/2, x_0=1, γ_{-1}=21, θ=0.9. Then v_0=Bx_0-Bx_{-1}=1, so w_0=x_0-γ_{-1}v_0=-20. For a candidate γ>0, z=J_{γA}(w_0-γBx_0) satisfies z-γ/z=-20-γ, i.e. z^2+(20+γ)z-γ=0, with unique z∈(0,1). Rearranging gives γ(1/z-1)=z+20, hence γ|Bz-Bx_0|=20+z. The linesearch condition (4.2) requires 20+z ≤ 0.9(22-z), i.e. 1.9z ≤ -0.2, impossible. Thus no γ>0, and in particular no α_k, satisfies (4.2): Algorithm 4.1 is not well defined on a valid instance of Problem 3.1. The failing step in Proposition 4.2 is exactly the assertion that (Bz_k) is bounded because z_k→0=P_{\\overline{dom A}}w_0 lies on the boundary of dom A, where B(z_k)~-1/z_k is unbounded.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a forward-reflected-backward (FRB) splitting method with a new linesearch for the monotone inclusion 0∈Ax+Bx, where A is maximally monotone and B is monotone, single-valued, and continuous on dom A. Algorithm 4.1 uses one evaluation of B per iteration and the linesearch condition (4.2) involving the momentum term v_n. The main result (Theorem 4.5) claims weak convergence under θ²+3(θ/2)^{2/3}<1 and either liminf γ_n>0 or uniform continuity of B on weakly compact subsets of dom A, relying on Proposition 4.2 for well-definedness. The paper also extends the scheme to sums with cocoercive and Lipschitz operators (Section 5) and reports numerical experiments on saddle-point and image-restoration problems. The key enabling assertion is that, for every instance of Problem 3.1, the linesearch terminates in finitely many steps. I find this assertion false as stated.","tokens_in":1719,"tokens_out":1790,"duration_ms":163081,"significance":"If correct, the paper would close an open problem: a linesearch FRB with a single B-evaluation per iteration for merely continuous monotone B, going beyond Tseng's FBF and the locally Lipschitz linesearch FRB of [31]. The Lyapunov analysis in §4, the explicit counterexample to the earlier linesearch, the four-operator extension, and the reproducible numerical study are all useful components. However, the central well-definedness result is false: a valid instance of Problem 3.1 makes condition (4.2) fail for every γ>0. The advertised convergence theorem is therefore not established, and the Section 5 extension inherits the same difficulty.","major_comments":[{"comment":"The step 'B is continuous in dom A, it is locally bounded on dom A, thus (Bz_k) is bounded' is invalid: z_k→P_{\\overline{dom A}}w_n may lie in ∂dom A\\setminus dom A, where continuity on dom A gives no bound. This is not a technicality. Counterexample: H=R, f(x)=−log x, A=∂f (dom A=(0,∞), A(x)=−1/x); B=∂(2x−log x), so B(x)=2−1/x on (0,∞). Then A+B=∂(2x−2log x) is maximally monotone and zer(A+B)={1}, so Problem 3.1 holds. Take x_{−1}=1/2, x_0=1, γ_{−1}=21, θ=0.9. Then v_0=Bx_0−Bx_{−1}=1, w_0=x_0−γ_{−1}v_0=−20. For any γ>0, z=J_{γA}(w_0−γBx_0)∈(0,1) satisfies z²+(20+γ)z−γ=0, hence γ|Bz−Bx_0|=20+z. Condition (4.2) becomes 20+z≤0.9(22−z), i.e. z≤−2/19, impossible. Thus no γ>0 satisfies (4.2) for n=0; Algorithm 4.1 is not well defined on a valid instance. Theorem 4.5, which depends on Proposition 4.2, cannot hold as stated.","section":"§4, Proposition 4.2"},{"comment":"The proof of case (ii) applies [43, Lemma 3.2] to treat {x}∪_k [x_{n_k},\\hat x_{n_k+1}] as a weakly compact subset of conv dom A = dom A, and then uses uniform continuity of B there. But the weak cluster point x of (x_n) is only known to lie in the weak closure of dom A, which equals \\overline{dom A}; it is not shown to belong to dom A. If x∈\\overline{dom A}\\setminus dom A, the displayed set is not a subset of dom A and the uniform-continuity assumption cannot be invoked. The later conclusion 0∈(A+B)x, which would imply x∈dom(A+B), is obtained only after this step. This is a further gap in the proof of the second convergence alternative.","section":"§4, Theorem 4.5(ii)"},{"comment":"The proof claims that Proposition 4.2 directly provides γ_n with γ_n||Bx_{n+1}−Bx_n||≤θ(||x_{n+1}−x_n||+γ_{n−1}||v_n||). However, in Algorithm 5.2 the trial point is x_{n+1}=J_{γ_n A}(x_n−γ_n(B+C+D)x_n−γ_{n−1}v_n), whereas Proposition 4.2 concerns J_{γ_n A}(x_n−γ_n B x_n−γ_{n−1}v_n). The extra C and D terms change the resolvent argument, so the cited proposition does not directly yield the stated inequality. A separate argument, e.g. applying Proposition 4.2-style reasoning to B+C+D and then using the Lipschitz/cocoercivity of C and D, is needed but not supplied. Thus the well-definedness of Algorithm 5.2 is not established as written.","section":"§5, Proposition 5.3(i)"}],"minor_comments":[{"comment":"The parameters for FRBLS/FRBLSR use θ=0.3525. This violates the theorem's condition (4.18): θ²+3(θ/2)^{2/3}≈1.067>1. If these runs are meant to illustrate the proven parameter regime, they should be adjusted; otherwise the text should state that the choices are heuristic.","section":"§6.1, Table 1"},{"comment":"The remark says 'by setting θ=0', but Algorithm 5.2 and the surrounding theory require θ∈]0,1[. The limiting statement needs a more careful formulation, e.g. θ↓0.","section":"§5, Remark 5.5(iv)"},{"comment":"The linesearch index k is described as a 'natural number'; since α_0=τ is allowed, it should be an integer k≥0.","section":"§4, Algorithm 4.1"}],"recommendation":"reject","confidential_remarks":"The 1D counterexample in my report appears decisive: it satisfies every assumption of Problem 3.1, yet the proposed linesearch (4.2) cannot terminate at the first iteration for any γ>0. The main theorem and its Section 5 extension therefore rest on a false well-definedness claim. Although the general idea and numerical study have merit, the central advertised result is not salvageable without a substantive redesign of the linesearch or a change in the problem assumptions, which goes beyond a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fernando — you should know this one has a real result buried in it, but the headline claim doesn't survive contact with the boundary of dom A.\n\nWhat's genuinely good: the new linesearch condition (4.2) that brings the previous momentum term γ_{n−1}||v_n|| into the test is a natural answer to the open problem in [31]. Example 3.2 convincingly shows that the existing [31] linesearch can fail for merely continuous B, and the example is worked out carefully. The Lyapunov argument in §4 is standard but clean, and the four-operator extension in §5 follows the expected template. The numerics are extensive and suggest the linesearch helps even in the Lipschitz case, though note that the experimental θ = 0.3525 violates the paper's own sufficient condition (4.18). That's a real inconsistency, though not the main issue.\n\nThe load-bearing step is Prop. 4.2, where they claim Bz_k is bounded because z_k → P_{overline{dom A}} w_n and B is continuous on dom A. That inference is only valid if the projection lies inside dom A. In general it lands on the boundary, where continuity of B on dom A gives no bound. The stress-test counterexample is watertight: H=R, A=∂(−log), B(x)=2−1/x, x_{−1}=1/2, x_0=1, γ_{−1}=21, θ=0.9. For n=0 the linesearch condition reduces to 20+z ≤ 0.9(22−z), which is impossible for any z>0. So Algorithm 4.1 never terminates on a valid instance of Problem 3.1. This is not a minor gap; the same flaw propagates into Theorem 4.5, and the central \"well-defined and convergent for merely continuous operators\" claim is unsupported.\n\nThis is partly fixable. Adding an assumption like B uniformly continuous on \\overline{dom A} (or locally bounded on the closure) would patch Prop. 4.2, and the rest of the proof would likely go through. Alternatively, the linesearch could be modified to use a point that stays strictly inside dom A. But as stated, the main result is false.\n\nWho is this for? People working on forward-reflected-backward methods and adaptive linesearches. The counterexample and the failure-mode analysis are worth reading even if the main theorem fails. I'd send it to review, but with the expectation of major revision — the authors should be told exactly where the proof breaks and be given the chance to add the missing hypothesis.","headline":"The new linesearch idea is sensible and the counterexample to the old one is solid, but the main theorem is false: the well-definedness proof breaks at the boundary of dom A, and a 1D counterexample confirms it.","tokens_in":24482,"tokens_out":4281,"would_cite":false,"duration_ms":37180,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47H10","65K05","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new linesearch rule makes the forward-reflected-backward method converge for merely continuous operators, not only for Lipschitz-continuous ones.","keywords":["forward-reflected-backward","linesearch","monotone inclusion","weak convergence","splitting algorithm","continuous operator","convex optimization","proximal methods"],"falsifier":"Run Algorithm 4.1 on the one-dimensional problem A = ∂(-log), B(x) = -1/x + 2, with any starting pair making the reflected point w_n = x_n - γ_{n-1}v_n negative. The trial resolvent points converge to 0 and B is unbounded at 0; the linesearch (4.2) either terminates (showing the boundedness step is unnecessary) or loops forever (disproving Proposition 4.2).","tokens_in":23480,"feed_emoji":"🔁","tokens_out":17530,"duration_ms":130415,"temperature":0.7,"pith_summary":"This paper establishes that the forward-reflected-backward (FRB) splitting method, which evaluates the continuous operator only once per iteration, can be equipped with a linesearch that is guaranteed to terminate even when the operator is merely continuous rather than locally Lipschitz. The authors prove that their new step-size rule is well defined (Proposition 4.2), and that every sequence generated by the algorithm converges weakly to a zero of the sum operator under a mild step-size condition or a uniform-continuity assumption (Theorem 4.5). This answers an open problem for FRB, whose previous linesearch was shown to run forever on a concrete continuous but non-Lipschitz example. The practical benefit is that problems with continuous operators that are not Lipschitz — such as certain image-restoration regularizations — can now be solved by a method that needs one evaluation of the operator per iteration instead of two, and that does not require estimating a global Lipschitz constant. The paper also extends the method to a four-operator splitting scheme covering cocoercive and Lipschitz operators, and reports numerical speedups in saddle-point and deblurring tests.","feed_headline":"New linesearch fixes FRB for merely continuous operators","feed_subtitle":"A single operator evaluation per iteration now suffices for weak convergence, avoiding Lipschitz constant estimates.","key_machinery":"The core mechanism is a linesearch condition that includes the reflected difference v_n = Bx_n - Bx_{n-1}: γ_n∥v_{n+1}∥ ≤ θ(∥x_{n+1}-x_n∥ + γ_{n-1}∥v_n∥). The earlier linesearch omitted v_n and could loop forever for merely continuous operators. The termination proof exploits the convergence of trial resolvent points z_k = J_{α_k A}(w_n - α_k Bx_n) to the projection of w_n onto the closure of dom A, continuity of B on dom A, and a monotonicity lemma that γ ↦ γ^{-1}∥z - J_{γA}(z-γy)∥ is nonincreasing. Convergence is driven by a Lyapunov sequence Γ_n(x̂) = ∥x_n-x̂∥² - 2γ_{n-1}⟨x_n-x̂|v_n⟩ + γ_{n-1}²/ρ²∥v_n∥², whose descent under θ² + 3(θ/2)^{2/3} < 1 yields square-summable successive differenc","core_discovery":"Algorithm 4.1 uses the linesearch condition γ_n∥v_{n+1}∥ ≤ θ(∥x_{n+1}-x_n∥ + γ_{n-1}∥v_n∥), where v_n = Bx_n - Bx_{n-1}. Proposition 4.2 proves this condition is satisfied for some geometrically decreasing γ_n even when B is merely continuous, by showing that the trial resolvent points force α_k∥Bz_k - Bx_n∥ → 0. Theorem 4.5 then proves weak convergence of the iterates to a point in zer(A+B) whenever θ² + 3(θ/2)^{2/3} < 1 and either the step sizes have a positive limit inferior or B is uniformly continuous on weakly compact subsets of dom A. The Section 5 extension (Algorithm 5.2) covers a sum of a cocoercive, a Lipschitz, and a merely continuous term, with a modified linesearch and its own","pith_inferences":["The same 'reflected-difference' idea could be transplanted to other momentum-based splitting methods, such as shadow-Douglas-Rachford variants, where the momentum term currently forces Lipschitz-only step-size rules.","Because the linesearch condition involves only past operator evaluations, it might survive inexact or stochastic oracle models of the continuous operator, provided the boundedness argument is replaced by a probabilistic rate-of-growth bound.","The paper leaves open whether the uniform-continuity assumption in Theorem 4.5(ii) is truly needed; a natural test is to try to construct a merely continuous operator on a non-closed domain where the trial resolvent points converge to a boundary point and see whether the linesearch still terminates."],"forward_implications":["The method solves the monotone inclusion problem with one evaluation of the continuous operator per iteration, removing the second activation required by the forward-backward-forward linesearch.","Problems where the continuous operator is not locally Lipschitz are now covered, and the paper's Example 3.2 demonstrates that the previous linesearch can loop forever on such problems.","Under Theorem 4.5, every generated sequence converges weakly to a solution whenever the linesearch parameters satisfy θ² + 3(θ/2)^{2/3} < 1 and one of the two coverage conditions holds.","Algorithm 5.2 extends the result to inclusions with an additional cocoercive operator and a Lipschitz operator, with convergence guaranteed under the conditions in Theorem 5.4.","Numerical experiments on saddle-point problems show the adaptive linesearch often accelerates convergence even when the operator is Lipschitz, and remains competitive with the two-evaluation forward-backward-forward linesearch."],"fun_headline_variants":["FRB linesearch that works for merely continuous maps","A single-evaluation FRB linesearch for continuous operators","No Lipschitz constant needed: new FRB linesearch converges","Novel FRB linesearch handles merely continuous operators","FRB without Lipschitz constants: continuous operators work"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Proposition 4.2's termination argument relies on the claim that the sequence (Bz_k) is bounded when the trial resolvent points z_k converge to the projection of w_n onto the closure of dom A; this holds only if that projection lies in dom A, which is not guaranteed when dom A is non-closed.","fun_headline_variants_meta":{"raw":{"variants":["FRB linesearch that works for merely continuous maps","A single-evaluation FRB linesearch for continuous operators","No Lipschitz constant needed: new FRB linesearch converges","Novel FRB linesearch handles merely continuous operators","FRB without Lipschitz constants: continuous operators work"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000461,"raw_usage":{"total_tokens":2230,"prompt_tokens":914,"completion_tokens":1316,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":1235}},"tokens_in":658,"tokens_out":1316,"duration_ms":9053,"temperature":1.0,"reasoning_tokens":1235,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:45:40.907670+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 4.1 on the one-dimensional problem A = ∂(-log), B(x) = -1/x + 2, with any starting pair making the reflected point w_n = x_n - γ_{n-1}v_n negative. The trial resolvent points converge to 0 and B is unbounded at 0; the linesearch (4.2) either terminates (showing the boundedness step is unnecessary) or loops forever (disproving Proposition 4.2).","supporting_citations":[],"review_version":1}