{"id":"e08f8c43-72ed-43d4-ae68-6b0b18131b92","arxiv_id":"1908.05912","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A one-evaluation-per-iteration reflected forward-backward method is shown to converge weakly for monotone inclusions with Lipschitzian operators, with a new three-operator semi-reflected variant and improved stepsizes for cocoercive operators.","lead":"This paper proves that a reflected forward-backward splitting algorithm, which evaluates the Lipschitzian operator only once per iteration, converges weakly for general monotone inclusions, and extends it to a new three-operator splitting method. The results provide a simpler and potentially cheaper alternative to classic splitting methods for a broad class of nonsmooth convex optimization problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the convergence proofs in Theorems 2.1 and 3.3 are internally consistent, and the cocoercivity assumptions are explicit and used correctly.","rationale":"The reader's weakest assumption identifies cocoercivity as the fragile premise. I agree that cocoercivity is restrictive and that the three-operator theorem would fail if C were merely monotone Lipschitzian, but this is not a hidden flaw: it is stated in the abstract, in Problem 3.1, and in Theorem 3.3. The proof uses the assumption in exactly the way advertised, so it does not undermine the central claim. The reader's conditional verdict is nevertheless reasonable because the paper has small but real presentation gaps: the unstated maximal monotonicity sum theorem in the limiting arguments, the T_n indexing typo, and the absence of numerical validation. Since these issues are local and repairable, I would keep the CONDITIONAL verdict rather than escalate or downgrade it.","tokens_in":13087,"tokens_out":45138,"duration_ms":345119,"concrete_test":"As a verification step, symbolically re-derive Eq. (3.26) from (3.8), (3.23), and (3.24), keeping the cancellations in (3.16) through (3.19) explicit; if the coefficient of ‖x_n−x_{n+1}‖² does not equal γ(1+ξ)/(2β)−2ζ, there is a hidden sign error in the SRFB descent. This is the same hand-check I performed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After working through the proofs, I do not find a load-bearing concern. The cocoercivity of B in Theorem 2.1(i) and of C in Theorem 3.3 is an explicit hypothesis, not a hidden one; it is used precisely to produce the dissipative term −βγ‖Cx_n−Cx‖² in (3.15) and to relax the stepsize in (2.14). The algebra in the key inequalities (2.14) and (3.26) checks out: the apparent mismatch in (2.14) disappears when the terms are regrouped, and the SRFB descent inequality (3.26) follows from (3.8), (3.23), and (3.24) without an unaccounted sign. The only genuine weaknesses are minor: the proof invokes maximal monotonicity of A+B and A+B+C in the limit steps without stating the standard sum theorem, the index in the definition of T_n in Theorem 2.1(i) is off by one, and no numerical experiments or code are supplied. None of these unsettle the central convergence claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies reflected forward-backward splitting (RFBS) for monotone inclusions on a Hilbert space. For the inclusion 0 in Ax + Bx, with A maximally monotone and B monotone and mu-Lipschitzian, it proves weak convergence of the one-evaluation-per-iteration iteration (1.6) for gamma in (0, (sqrt(2)-1)/mu), and improves the stepsize range to gamma in (0, beta/2] when B is beta-cocoercive. It then proposes a semi-reflected forward-backward (SRFB) method for the three-operator inclusion 0 in Ax + Bx + Cx, where C is cocoercive, and proves weak convergence under explicit stepsize bounds. A primal-dual application to composite monotone inclusions is given in Section 4. The main results are Theorem 2.1, Theorem 3.3, and Corollary 4.2.","tokens_in":13225,"tokens_out":15127,"duration_ms":115855,"significance":"If correct, the paper contributes a simple, cheap-to-implement splitting method that extends Malitsky's projected reflected gradient method from normal-cone operators to general maximal monotone operators. The improved cocoercive stepsize and the three-operator SRFB combination are useful additions to the operator-splitting toolkit. The proofs are self-contained and the Lyapunov algebra is coherent; I checked the key inequalities (2.14), (2.19), and (3.26) and found them consistent. The main limitation, namely the explicit reliance on global cocoercivity in Theorem 2.1(i) and Theorem 3.3, is a real restriction on applicability but it is not hidden and is used correctly to provide the dissipative terms. The paper is an incremental but solid theoretical contribution; it does not require numerical experiments for its claims to be credible.","major_comments":[],"minor_comments":[{"comment":"The definition of T_n is misindexed. As written, T_n = ||x_{n+1}-x||^2 - 2 <p_{n+1}+gamma Bx | x_{n+1}-x_n>, but the inequality (2.10) requires T_n = ||x_n-x||^2 - 2 <p_n+gamma Bx | x_n-x_{n-1}>. Please adjust the index so that the subsequent inequality reads consistently.","section":"Section 2, after Eq. (2.9)"},{"comment":"The theorem allows gamma = beta(1-epsilon)/2, but the proof says 'Since gamma < (1-epsilon)beta/2, we obtain ...'. The endpoint is harmless because the coefficient 2beta - 4gamma/(1-epsilon) vanishes there, but the proof should use 'less than or equal to' or explain that the strict inequality is not essential.","section":"Theorem 2.1(i), proof after Eq. (2.14)"},{"comment":"The proofs invoke maximal monotonicity of A+B (in Theorem 2.1(ii)) and of A+B+C (in Theorem 3.3) without stating why these sums are maximally monotone. This is a standard consequence of the sum theorem for a maximally monotone operator and a Lipschitz monotone (hence maximally monotone, full-domain) operator, but it should be stated explicitly or cited, since it is load-bearing for the cluster-point argument.","section":"Theorem 2.1(ii) and Theorem 3.3, limit passages"},{"comment":"In the sentence 'Since B is maximally monotone, its graph is closed ..., we obtain Bx = Bx and thus By_{k_n} -> Bx', the equality 'Bx = Bx' appears to be a typo for 'B xbar = Bx' where xbar is the weak cluster point. As written it is tautological.","section":"Theorem 2.1(i), cluster-point step"},{"comment":"The inequality (3.24) uses the constant (1+sqrt(2)) but the Young-expansion details that lead to the three quadratic terms with coefficients gamma mu(1+sqrt(2)), gamma mu, and gamma mu sqrt(2) are not shown. Please spell out the bound on ||y_n - y_{n-1}|| in terms of the increments y_n - x_n and y_{n-1} - x_{n-1}, so that the reader can verify the coefficients.","section":"Theorem 3.3, inequality (3.24)"},{"comment":"There are several typographical and presentation issues: 'Lipschizian' in the abstract, 'Optial's result' in the proof of Theorem 3.3, 'sing' for 'using' in the proof of Corollary 4.2, and the reference [16] is listed as 'Y. Malitsky, Y. and M. K.Tam'. In Corollary 4.2, the formula for mu in (4.4) is stated without derivation; a brief justification or explicit citation to [8] for this Lipschitz bound would help.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, proof-based contribution. I found no load-bearing errors; the issues are local (a misindexed definition, a few omitted justifications for standard maximal-monotonicity facts, and typos). The absence of numerical experiments is not a blocker for a theory paper of this type. The novelty relative to [14,15,16] is modest but genuine, especially the semi-reflected three-operator method. I would be comfortable with acceptance after the minor revisions are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it takes the projected reflected gradient method from variational inequalities and proves weak convergence for the general monotone inclusion 0 ∈ Ax + Bx, with A maximally monotone and B monotone and Lipschitzian. The two-operator result is a legitimate extension of Malitsky's work, and when B is linear it reduces exactly to the forward-reflected-backward method of Malitsky–Tam. The improved stepsize under cocoercivity is a small but useful addition. Section 3's SRFB method for three operators is the most novel piece; it combines the reflected idea with forward-backward in a way I have not seen before, and the convergence conditions are explicit and checkable. The proofs are self-contained, the Lyapunov arguments are standard, and I found no circularity. The citations to [14,15,16] are appropriate and not doing load-bearing work beyond context and special cases.\n\nThe soft spots are real but mostly minor. The cocoercivity assumption in Theorem 2.1(i) and Theorem 3.3 is explicit and used correctly to produce the dissipative term, but it does restrict the practical range: many monotone Lipschitz operators of interest are not globally cocoercive, and for the three-operator method you need C to be cocoercive on the whole space. That is a genuine limitation of scope, not a hidden flaw, and it should be stated more prominently. The proof also invokes maximal monotonicity of A+B and A+B+C in the limit steps without stating the standard sum theorem; that is a small gap that a referee should ask the authors to fix. The definition of T_n after (2.9) has a misindexed subscript; again minor. No numerical experiments or code are provided, which is fine for a theory paper but leaves the practical behavior of SRFB untested.\n\nI agree with the stress-test note more than with the reader's conditional verdict: the central claims hold up, and the concerns are presentation-level rather than load-bearing. This is a useful paper for people working on operator splitting and nonsmooth optimization. It deserves a serious referee and, after minor revision, publication. I would take it to reading group and would probably cite it for the three-operator method.","headline":"A sound, honest generalization of Malitsky's reflected gradient method to monotone inclusions; the three-operator variant is genuinely new but leans on global cocoercivity, which is a limitation to state, not a flaw.","tokens_in":13818,"tokens_out":1529,"would_cite":true,"duration_ms":16343,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","49M29","49M27","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $0\\in Ax+Bx$ with $A$ maximally monotone and $B$ monotone and $\\mu$-Lipschitzian, the reflected forward-backward iteration converges weakly to a zero for every step size below $(\\sqrt{2}-1)/\\mu$.","keywords":["monotone inclusions","operator splitting","reflected forward-backward splitting","cocoercive operators","weak convergence","Lipschitzian monotone operators","primal-dual algorithms","composite monotone inclusions"],"falsifier":"Set $H=\\mathbb{R}^2$, take $A$ to be the normal cone of the positive orthant, and take $B$ to be the 90-degree rotation matrix; run the two-operator iteration with $\\gamma$ just below $(\\sqrt{2}-1)/\\mu$ from several starting points—if any such run fails to converge weakly, the theorem's central bound is unsound. For the three-operator claim, replace $C$ by the same rotation matrix (monotone Lipschitzian but not cocoercive) and run the semi-reflected scheme under (3.3); divergence would show the cocoercivity assumption is doing essential work.","tokens_in":12815,"feed_emoji":"🔁","tokens_out":12910,"duration_ms":103614,"temperature":0.7,"pith_summary":"The paper claims that a 'reflected' version of forward-backward splitting—where the Lipschitzian operator is evaluated at the reflected point $y_n=2x_n-x_{n-1}$ instead of at $x_n$—converges weakly to a zero of $A+B$, for any maximally monotone operator $A$ and any monotone $\\mu$-Lipschitzian operator $B$, as long as the step size is below $(\\sqrt{2}-1)/\\mu$. This matters because it gives a provably convergent splitting that needs only one evaluation of the Lipschitzian operator per iteration, unlike older forward-backward-forward schemes that required two. The same idea is combined with ordinary forward-backward splitting to handle sums of three operators $A+B+C$, and is applied to composite primal-dual inclusions. A careful reader would care because these methods are the workhorses of large-scale convex optimization, where operator evaluations dominate cost.","feed_headline":"Reflected splitting reaches zeros with a single evaluation per step","feed_subtitle":"Weak convergence holds for step sizes below (√2−1)/µ, rising to β/2 when the operator is cocoercive.","key_machinery":"The load-bearing object is the reflected extrapolation $y_n=2x_n-x_{n-1}$, which lets the method update with one evaluation of $B$ per step instead of two. The proof tracks a Lyapunov function that is small at a solution: bounds on $\\|x_n-x\\|^2$, $\\|x_{n-1}-x_n\\|^2$, $\\|p_n+\\gamma Bx\\|^2$, and related reflected-difference terms. A telescoping inequality forces $\\sum_n\\|x_n-y_n\\|^2<\\infty$ and $\\sum_n\\|x_{n+1}-y_n\\|^2<\\infty$, hence $x_{n+1}-x_n\\to0$. In the three-operator case, cocoercivity of $C$ contributes the dissipative term $-\\beta\\gamma\\|Cx_n-Cx\\|^2$, and the Lyapunov function $\\alpha_{n+1}$ is shown nonincreasing and coercive, so the iterates stay bounded. The argument finishes with the standard lemma that bounded iterates with converging norms and only zeros of $A+B$ (or $A+B+C$) as weak cluster points must converge weakly.","core_discovery":"For the inclusion $0\\in Ax+Bx$ with $A$ maximally monotone and $B$ monotone and $\\mu$-Lipschitzian, the reflected forward-backward splitting $x_{n+1}=(I+\\gamma A)^{-1}(x_n-\\gamma B(2x_n-x_{n-1}))$ produces a sequence that converges weakly to some solution for every $\\gamma\\in(0,(\\sqrt{2}-1)/\\mu)$. When $B$ is $\\beta$-cocoercive the admissible range enlarges to $\\gamma\\in(0,\\beta/2)$, matching the natural forward-backward range. For the three-operator inclusion $0\\in Ax+Bx+Cx$ with $C$ $\\beta$-cocoercive, the semi-reflected iteration $x_{n+1}=J_{\\gamma A}(x_n-\\gamma B(2x_n-x_{n-1})-\\gamma Cx_n)$ also converges weakly under explicit step-size conditions. The paper further embeds the two-operator method in a product space to solve composite inclusions involving parallel sums, where the iteration remains a single evaluation of each Lipschitzian component per step.","pith_inferences":["Inference: the two-operator step-size bound $\\gamma<(\\sqrt{2}-1)/\\mu$ may be sharp for general monotone Lipschitzian $B$; constructing a divergence example at the critical value would test that.","Inference: the cocoercivity of $C$ in Theorem 3.3 is used only through a quadratic dissipation term; one could try to replace it by a weaker 'dissipation at the solution' condition, but the paper does not.","Inference: because each iteration uses one evaluation of each operator, variance-reduced or stochastic variants of $B$ might inherit the same Lyapunov structure; the paper does not address randomness.","Inference: the reflected construction could likely be combined with variable-metric resolvents to give a preconditioned method, but no such extension is claimed here."],"forward_implications":["Any $\\beta$-cocoercive $B$ can be handled with step sizes up to $\\beta/2$, the same range as ordinary forward-backward splitting, while still using only one evaluation of $B$ per iteration.","For convex minimization $\\min f+h$ with $h$ convex and $\\mu$-Lipschitz-smooth, the reflected proximal-gradient method converges for $\\gamma<1/(2\\mu)$.","The three-operator method solves $0\\in Ax+Bx+Cx$ with $A$ maximally monotone, $B$ monotone $\\mu$-Lipschitzian, and $C$ $\\beta$-cocoercive, under the explicit bounds in (3.3).","In the product-space formulation of Section 4, the same iteration yields weak convergence of primal and dual sequences for composite inclusions involving parallel sums, with one evaluation of each Lipschitzian operator per step."],"supporting_citations":[{"why":"Supplies the working definitions and background results—monotone operators, resolvents, cocoercivity, and the gradient cocoercivity used in Example 2.3.","marker":"[1]"},{"why":"Proposes a forward-backward-half-forward method for three-operator inclusions; the paper's Section 3 method is situated against it.","marker":"[6]"},{"why":"Establishes the product-space primal-dual framework that Section 4 uses to reduce composite inclusions with parallel sums to a two-operator inclusion.","marker":"[8]"},{"why":"Introduces the projected reflected gradient method for variational inequalities, the special case that this paper generalizes to arbitrary maximally monotone A.","marker":"[14]"},{"why":"Introduces proximal extrapolated gradient methods and line-search versions; motivates the reflected evaluation at $2x_n-x_{n-1}$.","marker":"[15]"},{"why":"Proposes a forward-reflected-backward splitting with the same one-evaluation structure; Theorem 2.1(ii) re-proves and extends its convergence range.","marker":"[16]"},{"why":"Provides the standard lemma that turns bounded iterates, norm convergence, and cluster-point zeros into weak convergence; used twice in Section 2.","marker":"[17]"},{"why":"The classical forward-backward-forward splitting method, the two-evaluation benchmark the reflected method improves to one evaluation.","marker":"[24]"}],"fun_headline_variants":["Reflected splitting widens stepsize range for cocoercive inclusions","One evaluation per step: reflected forward-backward for monotone inclusions","Reflected splitting: weak convergence for three monotone operators","Improved step size for reflected splitting with cocoercive operators","Single evaluation per step: reflected splitting for composite inclusions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof for the three-operator method and the enlarged-stepsize case requires the operator to be cocoercive on the entire space, a property strictly stronger than both monotonicity and Lipschitz continuity; if that assumption fails, the step-size ranges and convergence guarantees in those results have no basis.","fun_headline_variants_meta":{"raw":{"variants":["Reflected splitting widens stepsize range for cocoercive inclusions","One evaluation per step: reflected forward-backward for monotone inclusions","Reflected splitting: weak convergence for three monotone operators","Improved step size for reflected splitting with cocoercive operators","Single evaluation per step: reflected splitting for composite inclusions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001168,"raw_usage":{"total_tokens":4843,"prompt_tokens":969,"completion_tokens":3874,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":3788}},"tokens_in":585,"tokens_out":3874,"duration_ms":27755,"temperature":1.0,"reasoning_tokens":3788,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:03:32.605154+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $H=\\mathbb{R}^2$, take $A$ to be the normal cone of the positive orthant, and take $B$ to be the 90-degree rotation matrix; run the two-operator iteration with $\\gamma$ just below $(\\sqrt{2}-1)/\\mu$ from several starting points—if any such run fails to converge weakly, the theorem's central bound is unsound. For the three-operator claim, replace $C$ by the same rotation matrix (monotone Lipschitzian but not cocoercive) and run the semi-reflected scheme under (3.3); divergence would show the cocoercivity assumption is doing essential work.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the working definitions and background results—monotone operators, resolvents, cocoercivity, and the gradient cocoercivity used in Example 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proposes a forward-backward-half-forward method for three-operator inclusions; the paper's Section 3 method is situated against it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the product-space primal-dual framework that Section 4 uses to reduce composite inclusions with parallel sums to a two-operator inclusion."},{"cited_title":"Malitsky , Projected reflected gradient methods for monotone variatio nal inequalities , SIAM J","cited_arxiv_id":null,"evidence_quote":"Introduces the projected reflected gradient method for variational inequalities, the special case that this paper generalizes to arbitrary maximally monotone A."},{"cited_title":"Meth- ods Softw., 33 (2018), pp","cited_arxiv_id":null,"evidence_quote":"Introduces proximal extrapolated gradient methods and line-search versions; motivates the reflected evaluation at $2x_n-x_{n-1}$."},{"cited_title":"Malitsky, Y","cited_arxiv_id":null,"evidence_quote":"Proposes a forward-reflected-backward splitting with the same one-evaluation structure; Theorem 2.1(ii) re-proves and extends its convergence range."},{"cited_title":"Opial , Weak convergence of the sequence of successive approximati ons for nonexpansive mappings, Bull","cited_arxiv_id":null,"evidence_quote":"Provides the standard lemma that turns bounded iterates, norm convergence, and cluster-point zeros into weak convergence; used twice in Section 2."},{"cited_title":"Tseng , A modified forward-backward splitting method for maximal mo notone mappings , SIAM J","cited_arxiv_id":null,"evidence_quote":"The classical forward-backward-forward splitting method, the two-evaluation benchmark the reflected method improves to one evaluation."}],"review_version":1}