Homogeneous Self-Dual Embedding via Perspective Functions

Goran Banjac

REVIEW 2 major objections 4 minor 29 references

Homogeneous Self-Dual Embedding via Perspective Functions

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A generalized self-dual embedding reduces any sum-of-convex problem to one homogeneous inequality.

desk verdict 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. read the letter →

arxiv 2607.22278 v1 pith:IWJBO34W submitted 2026-07-24 math.OC

Goran Banjac This is my paper · ORCID

classification math.OC MSC 90C2590C4665K05

keywords homogeneousself-dualembeddingperspectivefunctionsoperatorsplittingconvexoptimizationinfeasibilitycertificatesFencheldualityquadraticconeprogrammingDouglas-Rachfordalgorithm

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Two 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.

Overall, 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[§4.5.1, condition (18)] 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.
[§4.5.1] 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.

minor comments (4)

[§4.5] 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.
[§4.4, Algorithm 1] 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.
[§6.2] 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.
[§5.3, Remark 5.1] 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'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: HSDE properties and DRA simplification are derived from first principles; the SCS recovery is a downstream check, not an input.

full rationale

The paper's central derivation is self-contained rather than circular. The HSDE model in Section 3 is constructed directly from definitions of F(x,ν)=f(x)+f*(ν), G(x,ν)=g(x)+g*(-ν), and perspective functions, and the optimality/infeasibility interpretations are proved from weak duality and recession calculus (§3.2–3.3). The self-duality claim is proved via Lemma A.1 and Proposition 2.1, not assumed. The Douglas-Rachford simplification in §4.1 follows from the standard Moreau decomposition and the structure of the cones F and G; the reduction to scalar equations in §4.2–4.3 is a derived consequence. The recovery of SCS in §5.3 is presented as an outcome ('we have arrived to the same algorithm taking a very different path'), not as an input to the derivation. The cited projection/proximity results are external published results with independent proofs. There is one minor self-citation, [Ban21, App. A], used for an auxiliary identity in evaluating a projection variant; it is not load-bearing. The paper's own limitation — condition (18) is only verified for conic g (Remark 5.1) and shifted structured g (Remark 6.2), so the nonzero-convergence guarantee is not established for general g — is a genuine generality gap, but it is not a circularity: the missing assumption is not hidden inside any fitted parameter or renamed prediction, and the derivation does not assume the conclusion it claims to prove.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central construction is self-contained; the main extra assumptions are strong duality and the non-degeneracy condition (18). No free parameters are fitted to data; algorithm initialization (p^1, r^1) is a hyperparameter. No invented entities.

assumptions (5)

domain assumption Strong duality holds for the primal-dual pair (P)-(D)
Stated at the start of §3. Needed for the τ>0 case to encode a primal-dual solution with zero duality gap.
standard math f and g are in Γ0(H) (proper, lower semicontinuous, convex)
Standard class of functions; perspective and recession function theory relies on this.
domain assumption Condition (18): eG(x,ν,τ) ≥ 0 for all (x,ν,τ)
Used in §4.5.1 to prove that DRA iterates do not converge to zero. Verified only for conic g (Remark 5.1) and for the structured g when 0 ∈ [l,u] (Remark 6.2); not established for general g.
domain assumption If 0 ∉ [l,u], problem (22) can be reformulated by shifting so that 0 ∈ [l−e, u−e] for some e ∈ [l,u]
Remark 6.2 relies on this to ensure eG ≥ 0. This changes the data (b, l, u) but not the feasible set.
standard math Projection formulas of Proposition 2.2 and Proposition 2.3 are valid
Used in §4.2–§4.3 to compute P_F and P_G; these are cited results from [BAVV24b].

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Pith. "Pith review of Homogeneous Self-Dual Embedding via Perspective Functions." pith.science (2026). https://pith.science/paper/IWJBO34W

@misc{pith2026260722278,
  author       = {Pith},
  title        = {Pith review of: Homogeneous Self-Dual Embedding via Perspective Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWJBO34W}},
  note         = {Machine review of arXiv:2607.22278}
}

read the original abstract

We present a generalization of the well-known homogeneous self-dual embedding model, which is widely used in conic optimization. The new embedding applies to a problem of minimizing the sum of two proper lower-semicontinuous convex functions and can be represented as a single inequality that uses perspectives of these functions and of their conjugates. A solution to the proposed embedding encodes a primal-dual solution to the original problem when available, or an infeasibility certificate otherwise. We then use the Douglas-Rachford algorithm to find a solution to the embedding and discuss its efficient implementation by exploiting the problem structure. The resulting algorithm recovers an existing method for solving quadratic cone programs as a special case. We demonstrate the generality and effectiveness of the algorithm on a class of convex optimization problems with non-smooth objective function and non-conic constraints.

Figures

Figures reproduced from arXiv: 2607.22278 by the authors.

**Figure 2.** Convergence of primal infeasibility residuals for problem ( [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 3.** Convergence of dual infeasibility residuals for problem ( [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Convergence of primal and dual infeasibility residuals for problem ( [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Applegate, M

D. Applegate, M. D\' az, O. Hinder, H. Lu, M. Lubin, B. O'Donoghue, and W. Schudy. PDLP : a practical first-order method for large-scale linear programming. Mathematical Programming Computation , 2024

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Applegate, M

D. Applegate, M. D\' az, H. Lu, and M. Lubin. Infeasibility detection with primal-dual hybrid gradient for large-scale linear programming. SIAM Journal on Optimization , 34(1):459--484, 2024

2024

[3] [3]

M. S. Andersen, J. Dahl, and L. Vandenberghe. CVXOPT : a P ython package for convex optimization , 2026

2026

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E. D. Andersen and Y. Ye. On a homogeneous algorithm for the monotone complementarity problem. Mathematical Programming , 84(2):375--399, 1999

1999

[5] [5]

L. M. Brice \ n o-Arias, P. L. Combettes, and F. J. Silva. Proximity operators of perspective functions with nonlinear scaling. SIAM Journal on Optimization , 34(4):3212--3234, 2024

2024

[6] [6]

G. Banjac. On the minimal displacement vector of the D ouglas- R achford operator. Operations Research Letters , 49(2):197--200, 2021

2021

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L. M. Brice \ n o-Arias and C. Vivar-Vargas. Enhanced computation of the proximity operator for perspective functions. Journal of Optimization Theory and Applications , 200(3):1078--1099, 2024

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L. M. Brice \ n o-Arias and C. Vivar-Vargas. Projection onto cones generated by epigraphs of perspective functions. arXiv:2411.08000 , 2024

arXiv 2024