{"id":"6bba8955-f7ee-43aa-a8b5-66510e649b95","arxiv_id":"2607.22278","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A single perspective-function inequality encodes primal-dual solutions and infeasibility certificates for minimizing a sum of two convex functions, and Douglas-Rachford on that inequality recovers and extends SCS.","lead":"This paper generalizes the homogeneous self-dual embedding framework — a core tool in conic optimization solvers — to any problem that minimizes the sum of two convex functions, encoding optimality and infeasibility in a single inequality built from perspective functions. If the construction holds, it subsumes the existing SCS solver and opens a route to first-order solvers for non-smooth, non-conic problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence to a nonzero HSDE point is only proved under condition (18), which is not established for general g∈Γ0(H) and can fail, leaving the central non-degeneracy guarantee unsupported.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: Section 4.5.1 depends on condition (18) to guarantee that the Douglas-Rachford iterates stay bounded away from the zero solution. The reader's verification of this gap is correct: for arbitrary f,g∈Γ0(H), condition (18) is not derived, is not an elementary property of perspectives, and is false for simple functions such as g(x)=x. Since zero is always a solution of the embedding, without a lower bound the central claim that Algorithm 1 yields a solution or certificate is not proven for the full problem class. The paper is otherwise strong: the embedding construction in Section 3 is internally consistent, the self-duality argument is clean, and the reduction to SCS in Section 5.3 gives independent support. The residual issues are a proof gap around non-degeneracy, an abstract that overstates the infeasibility dichotomy by not foregrounding the non-conclusive case of Section 3.4, and the absence of code and a larger numerical study. These are addressable and do not break the core construction, so the conditional verdict stands unchanged.","tokens_in":18534,"tokens_out":26093,"duration_ms":223307,"concrete_test":"Run Algorithm 1 on H=R with f(x)=0, g(x)=x, a pair in Γ0(R) violating (18) and whose HSDE solutions all have τ*=0 and κ*≤0, so no solution has r*>0. Track (p^k,r^k) from (p1,r1)=(0,1). If the iterates converge to (0,0), the non-degeneracy claim fails for a general Γ0 problem. If they converge to a nonzero point with r*=0 (as a hand calculation for this case suggests), then (18) is sufficient but not necessary and the remaining issue is a proof gap, not a demonstrated failure. Repeat for g(x)=-x and g(x)=e^x to probe whether any violating g drives the iterates to zero.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.5.1 tries to rule out convergence to the trivial zero solution, which always satisfies (HSDE). The argument invokes Lemma 4.1 with initialization (p1,r1)=(0,1), requiring a fixed point w* with r*=τ*+κ*>0. To get κ*≥0, the paper imposes condition (18): 0≤eG(x,ν,τ) for all (x,ν,τ). This is not a consequence of f,g∈Γ0(H). It is verified only for conic g in Remark 5.1 and for the shifted box/absolute-value g in Remark 6.2. Natural non-conic examples violate it: take g(x)=x, then eG can be negative and certificate solutions can have κ*≤0, hence r*≤0, so Lemma 4.1 gives no lower bound and the DRA iterates could converge to zero, producing neither a primal-dual solution nor an infeasibility certificate. The paper also does not prove existence of a solution with r*>0 even when (18) holds. Thus the advertised guarantee for arbitrary f,g∈Γ0(H) is not established; at best it is proved under additional assumptions on g.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a homogeneous self-dual embedding (HSDE) for the unconstrained convex problem of minimizing f+g, where f,g∈Γ0(H). The embedding is formulated as a single inequality involving perspective functions of f,g and their conjugates, and is shown to be a convex cone. The authors prove that solutions with τ>0 encode zero-duality-gap primal-dual solutions, while strict solutions with τ=0 encode strong infeasibility certificates, and they establish self-duality via a polar-cone formula. They then apply Douglas-Rachford to the feasibility form, simplify the iteration using Moreau decomposition, and show that the resulting algorithm recovers the SCS algorithm for quadratic cone programs. For a structured class with nonsmooth objective and box constraints, they derive explicit scalar subproblems and demonstrate optimality/infeasibility certificates on small numerical examples.","tokens_in":18819,"tokens_out":17241,"duration_ms":164004,"significance":"If fully established, this is a valuable framework: it extends HSDE-style modeling well beyond conic constraints, the polar-cone derivation is clean, and the Moreau-based simplification gives a practical iteration whose special case reproduces SCS. The paper is also commendably explicit about infeasibility certificates and provides small reproducible experiments. The main weakness is that the central non-degeneracy guarantee for the algorithm is conditional on an assumption that is neither proved for general f,g∈Γ0(H) nor shown to be satisfied by all problem classes claimed in the abstract; this limits the advertised generality.","major_comments":[{"comment":"The argument excluding convergence to the trivial zero solution requires (18): 0≤eG(x,ν,τ) for all (x,ν,τ). This is not a consequence of f,g∈Γ0(H) and is not true in general. For example, take H=R, g(x)=x. Then G(x,ν)=x if ν=-1 and +∞ otherwise, so eG(-1,0,0)=-1, violating (18). For f=0, the embedding has nonzero infeasibility certificates, but every fixed point of the DRA has τ=0 and κ≤0, hence r=τ+κ≤0; Lemma 4.1 then gives no lower bound and the algorithm may converge to zero. Thus the abstract's claim that the method applies to arbitrary proper lsc convex functions and finds a useful solution is not supported by the stated convergence result.","section":"§4.5.1, condition (18)"},{"comment":"Even when (18) holds, the paper does not prove existence of a fixed point w* with r*=τ*+κ*>0. The text says 'if there exists a solution to the embedding where either τ* or κ* is nonzero', but no construction is given for the infeasible case, and no argument shows that a primal-dual solution or an infeasibility certificate for the original problem yields such a fixed point. If the only nonzero certificates in a problem have κ=0 and τ=0, then r*=0 and the lower bound collapses. This is not merely a technicality: the numerical infeasibility examples in §7 satisfy (18) after shifting, but the theory still does not establish that the iterates stay away from zero in those cases. This needs to be either proved or made an explicit hypothesis of the convergence theorem.","section":"§4.5.1"}],"minor_comments":[{"comment":"The convergence statement invokes Lions–Mercier without specifying the mode of convergence. In infinite-dimensional Hilbert space DRA converges weakly, not strongly; in finite dimensions it is strong. Please state this and explain how the returned certificate is extracted from the weakly convergent sequence.","section":"§4.5"},{"comment":"Step 4 gives two possible choices for x̂k that are not always equal. The text after the algorithm explains that the resulting iterates agree unless τk>0, but this caveat should appear before the pseudocode so that the reader is not misled.","section":"§4.4, Algorithm 1"},{"comment":"The formula for φ(τ̃) and the active-set update are stated without derivation. Since the algorithm's efficiency depends on solving this scalar equation, a brief derivation from (26) would substantially improve readability.","section":"§6.2"},{"comment":"The phrase 'either τ*≥0 or κ*≥0 is nonzero' is confusing. Since τ is always nonnegative, this should read 'τ*>0 or κ*>0'.","section":"§5.3, Remark 5.1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid and lands on a load-bearing point. The embedding construction and the DRA simplification are sound, but the advertised guarantee for arbitrary f,g∈Γ0(H) is not established; the paper currently proves the non-degeneracy result only under condition (18) and does not prove the existence of a positive-r fixed point even then. I would not reject, because the core ideas are useful and the gap is localizable: the authors can either prove the missing existence/nonzero claims under explicit hypotheses or restrict the scope of the convergence and algorithmic claims accordingly. The numerical section does not by itself cure the gap, since all tested instances satisfy (18) after shifting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main idea here is real. Embedding the full primal–dual pair for min f+g through perspectives of f, f*, g, and g* into a single homogeneous inequality is not something I have seen before, and it is a natural extension of the HSDE framework beyond conic problems. The self-duality argument in §3.5 is clean, and the recovery of SCS as a special case of the Douglas–Rachford iteration is a nice validation rather than a restatement. The simplification of the DRA step via Moreau decomposition and the reduction of each projection to a scalar equation is elegant and should be useful.\n\nThe soft spots are real but not fatal. The convergence proof in §4.5.1 relies on condition (18), 0 ≤ eG(x,ν,τ) for all (x,ν,τ), to guarantee κ* ≥ 0 and hence r* = τ* + κ* > 0 for some solution. That condition is not a consequence of f,g ∈ Γ0(H); it fails for simple non-conic g, as the stress-test note shows with g(x)=x. The paper verifies it only for conic g and for the shifted box/absolute-value structure in §6.2. Moreover, even when (18) holds, the paper does not actually prove that a nonzero solution with r* > 0 exists — it just assumes there is one. So the advertised guarantee for arbitrary f,g is not established. This is a gap, but it is a gap in the non-degeneracy argument, not in the embedding itself.\n\nTwo smaller issues. The abstract overstates the infeasibility dichotomy: §3.4 itself concedes a non-conclusive case where τ=0 and equality holds. And the numerical section is a single parametric example with no shipped code, so the implementation claims are not independently checkable from the paper alone. Neither of these undermines the central construction.\n\nOverall, this is a serious paper for people working on operator splitting solvers and HSDE theory. It deserves a proper referee. The core embedding is sound, the derivation of SCS is a genuine byproduct, and the open questions around non-degeneracy and existence of nonzero solutions are well-posed and addressable. I would conditionally accept after requiring the authors to either prove a nonzero-solution existence result under (18) or clearly restrict the convergence claims, and to fix the abstract so it does not promise certificates in the non-conclusive case.","headline":"The perspective-function HSDE embedding is genuinely new and the core theory in §3 is sound, but the advertised guarantee for arbitrary f,g is not established: the non-degeneracy proof depends on condition (18), which is only verified for special g, and the paper never proves a nonzero solution with r*>0 exists even when (18) holds.","tokens_in":19288,"tokens_out":2106,"would_cite":true,"duration_ms":22651,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C46","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A generalized self-dual embedding reduces any sum-of-convex problem to one homogeneous inequality.","keywords":["homogeneous self-dual embedding","perspective functions","operator splitting","convex optimization","infeasibility certificates","Fenchel duality","quadratic cone programming","Douglas-Rachford algorithm"],"falsifier":"Find a proper convex lower-semicontinuous function g for which eG(x, ν, 0) < 0 for some (x, ν) and a problem instance whose only nonzero embedding solution has τ* + κ* ≤ 0; then run the proposed algorithm from the zero point with r = 1 and observe whether the normalized iterates collapse to zero instead of producing a positive-τ solution or a separating hyperplane. If they do, the non-degeneracy condition is essential and the algorithm as stated can fail to return a certificate.","tokens_in":18419,"feed_emoji":"🧮","tokens_out":8604,"duration_ms":81694,"temperature":0.7,"pith_summary":"The paper claims that the classical homogeneous self-dual embedding, long used for conic optimization, can be rebuilt for any problem of minimizing the sum of two proper lower-semicontinuous convex functions. The new embedding uses perspective functions of the two functions and their conjugates, and compresses optimality and infeasibility into a single inequality. A solution with τ > 0 gives a primal-dual point with zero duality gap; a solution with τ = 0 and strict inequality gives a separating-hyperplane certificate that the feasible sets are disjoint. The paper shows that an operator-splitting iteration converges to such a point, reduces each projection to scalar root-finding plus proximity evaluations, and demonstrates the machinery on a non-smooth, box-constrained problem. This broadens the reach of self-dual embeddings beyond quadratic cone programs and non-conic constraints.","feed_headline":"Perspective functions lift self-dual embedding past conic limits","feed_subtitle":"Operator splitting on the new embedding solves sum-of-convex problems and certifies infeasibility.","key_machinery":"The perspective function e f(x, τ) = τ f(x/τ) for τ > 0, extended at τ = 0 by the recession function, turns the epigraph of f into a cone. Proposition 2.1 identifies the polar of this cone as the epigraph of the perspective of f*, which is what makes the embedding self-dual. The solving algorithm applies an operator-splitting iteration to the intersection of the two cones F and G; using Moreau decomposition, each projection reduces to solving a scalar equation and evaluating the proximity operator of f or g. This is the mechanism that converts a general convex-sum problem into a tractable feasibility problem.","core_discovery":"The central discovery is the embedding (HSDE): find (x, ν, τ, κ) such that eF(x, ν, τ) ≤ −κ and eG(x, ν, τ) ≤ κ, where F(x, ν) = f(x) + f*(ν), G(x, ν) = g(x) + g*(−ν), and e denotes the perspective function. This single feasibility problem is self-dual: its polar cones reproduce the same embedding under a permutation of arguments. If τ > 0, then x/τ solves the primal problem and ν/τ solves the dual problem with equal optimal values. If τ = 0 and the inequality is strict, the data yield a hyperplane strongly separating dom f from dom g, or dom f* from −dom g*, certifying primal or dual infeasibility. The paper applies an operator-splitting iteration to the intersection of the two perspective","pith_inferences":["For a generic convex function g, the non-negativity condition on eG may fail, so the bounded-away-from-zero guarantee is not unconditional; the paper's own remark suggests recentering constraints so that 0 lies in the relevant set, which is likely the practical recipe users should follow.","The same perspective-cone construction could extend to problems with more than two additive terms by intersecting additional cones; the scalar-equation reduction would likely need a Moreau argument for each new cone.","The algorithm's practical cost is dominated by the proximity operator of f and projection onto the recession cone of dom g, so implementations should focus on making those two oracles fast.","The scalar root-finding form suggests that the method is especially well suited to structured objectives where the active-set geometry is simple, as in the box/L1 example, and less so when the recession sets are complicated."],"forward_implications":["If the paper is right, one homogeneous feasibility system decides, for any sum-of-convex problem with strong duality, whether a primal-dual solution exists or an infeasibility certificate can be produced.","A strict solution with τ = 0 gives a concrete separating hyperplane, so infeasibility is certified by the algorithm rather than merely detected through diverging iterates.","The method applies to objective terms that are not conic, including non-smooth functions like absolute values, as long as the relevant projections and proximity operators are computable.","For quadratic cone programs, the resulting algorithm reduces to an existing operator-splitting method, meaning the generalized embedding is a strict extension of the conic case.","For box-constrained L1-regularized quadratic problems, the paper supplies explicit residuals for primal infeasibility, dual infeasibility, and simultaneous infeasibility, and shows these residuals converge in numerical tests."],"fun_headline_variants":["Perspective functions generalize self-dual embedding","Self-dual embedding via perspectives beyond conic","Embedding solves convex sums with infeasibility proofs","New self-dual embedding: beyond conic optimization","Perspective embedding handles sum of convex problems"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof that the iterates stay away from the trivial zero solution assumes the perspective quantity eG is never negative for any (x, ν, τ); the paper verifies this only for cone indicators and for the box-constrained model after shifting the box to contain zero.","fun_headline_variants_meta":{"raw":{"variants":["Perspective functions generalize self-dual embedding","Self-dual embedding via perspectives beyond conic","Embedding solves convex sums with infeasibility proofs","New self-dual embedding: beyond conic optimization","Perspective embedding handles sum of convex problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000947,"raw_usage":{"total_tokens":3862,"prompt_tokens":707,"completion_tokens":3155,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":3098}},"tokens_in":451,"tokens_out":3155,"duration_ms":19193,"temperature":1.0,"reasoning_tokens":3098,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T05:15:21.078294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a proper convex lower-semicontinuous function g for which eG(x, ν, 0) < 0 for some (x, ν) and a problem instance whose only nonzero embedding solution has τ* + κ* ≤ 0; then run the proposed algorithm from the zero point with r = 1 and observe whether the normalized iterates collapse to zero instead of producing a positive-τ solution or a separating hyperplane. If they do, the non-degeneracy condition is essential and the algorithm as stated can fail to return a certificate.","supporting_citations":[],"review_version":1}