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arxiv: 2605.18158 · v1 · pith:O5LWK45Onew · submitted 2026-05-18 · 🧮 math.OC

Resolvent Moreau identities without monotonicity: theory and applications to Gabay duality, Douglas--Rachford and ADMM

Andrew Calcan , Jordan Collard , Alberto De Marchi , Scott B. Lindstrom This is my paper

Pith reviewed 2026-05-20 09:21 UTC · model grok-4.3

classification 🧮 math.OC
keywords resolvent operatorsMoreau identityGabay dualityDouglas-Rachford splittingADMMnonmonotone inclusionsnonconvex optimizationoperator splitting
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The pith

A duality relationship between resolvents holds for any set-valued mapping and generalizes Moreau's identity without monotonicity.

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

The paper establishes that a duality-like relationship between resolvents exists for any set-valued mapping provided the resolvents are single-valued. This relationship generalizes the classical Moreau identity, which applies to convex functions, and removes the need for monotonicity or convexity assumptions. The identity extends Gabay duality to nonmonotone inclusion problems, allowing convergence analysis of the Douglas-Rachford scheme to inform properties of ADMM. It further supports construction of counterexamples to ADMM convergence in previously open cases and design of new convergent resolvent homotopy schemes for nonconvex-regularised least absolute deviations problems.

Core claim

For any set-valued mapping with single-valued resolvent, a duality identity holds that relates the resolvent of the mapping to the resolvent of a complementary operator derived from it. This identity is a generalization of Moreau's identity that requires neither convexity nor monotonicity. The authors apply the identity to extend Gabay duality between Douglas-Rachford splitting and ADMM to the nonmonotone setting, derive explicit counterexamples to ADMM convergence via the easier-to-analyze Douglas-Rachford scheme, and introduce a class of convergent resolvent homotopy schemes for nonconvex problems.

What carries the argument

The resolvent Moreau identity for arbitrary set-valued mappings, which equates the resolvent of a mapping A to an expression involving the resolvent of its complementary operator and thereby carries the duality argument.

If this is right

  • Gabay duality between Douglas-Rachford splitting and ADMM extends directly to nonmonotone inclusions.
  • Convergence counterexamples for ADMM in open cases follow from direct analysis of the Douglas-Rachford scheme.
  • A family of resolvent homotopy schemes converges for selected nonconvex-regularised least absolute deviations problems.
  • The same identity supplies explicit conditions under which the new schemes remain well-defined and convergent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identity could be used to derive similar dualities for other primal-dual splitting methods beyond Douglas-Rachford and ADMM.
  • The homotopy schemes might be adapted to additional nonconvex regularized inverse problems that lack strong convexity in the smooth term.

Load-bearing premise

The resolvents of the set-valued mappings must be well-defined and single-valued in the relevant domains.

What would settle it

A concrete nonmonotone set-valued mapping whose resolvent is single-valued but for which the claimed duality identity fails to hold at some point in the domain.

Figures

Figures reproduced from arXiv: 2605.18158 by Alberto De Marchi, Andrew Calcan, Jordan Collard, Scott B. Lindstrom.

Figure 1
Figure 1. Figure 1: (a) DR expressed using primal variables. (b) An operator mapping diagram for the single– valued case of F and G and A and B and their adjoints. Theorem 7 demonstrates—provided the updates are defined—Gabay’s duality-like relationship always holds, regardless of the properties of F, G or the linear maps A and B. Gabay’s identities reveal that at each iteration k, a primal algorithm tuple (x k , zk , λk ) is… view at source ↗
Figure 2
Figure 2. Figure 2: DR and induced primal sequences generated for instances of ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graphs of homotopies ϕ α scad and ϕ α mcp induced by over–relaxed resolvents. Corollary 9. Let D ∈ N . Then (i) the mapping [0, 1] → H, α 7→ Rα D(y) is continuous for any arbitrary y ∈ H; (ii) there exists Dα such that RγDα is (1 + 2α)–Lipschitz. Proof. Item (i) follows immediately from fact that over–relaxed resolvents are affine in α. Considering (ii), the existence of Dα such that RγDα = Rα γD is gi… view at source ↗
Figure 4
Figure 4. Figure 4: HOST and induced primal sequences generated for instances of ( [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Objective value and average error of solutions generated by HOST and DR for SCAD– [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

Duality is most often defined as a relationship between convex functions. If those functions are nonconvex, classical duality breaks down. Notwithstanding, we show that another kind of duality still exists, not between the functions themselves, but between the so-called resolvent operators used to solve associated problems. In fact, this duality-like relationship holds for any set-valued mapping, and is a generalization of the Moreau's identity. We use this duality to study existing operator schemes and to design new ones. In particular, we show that the duality-like relationship Daniel Gabay illuminated between the Douglas--Rachford splitting (DR) and the Alternating Direction Method of Multipliers (ADMM) extends to nonmonotone inclusion problems. We use this relationship to provide explicit counterexamples to the convergence of ADMM in several open cases, by studying the (easier to analyse) DR scheme. Motivated by our observations, we design a class of convergent resolvent homotopy schemes and use them to solve nonconvex-regularised least absolute deviations problems. This important problem class has received little attention in the literature, since the convex component of the objective does not enjoy strong convexity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a resolvent duality identity J_{A^{-1}}(x) = x - J_A(x) that holds for arbitrary set-valued mappings A whenever the resolvents exist at the evaluation point, without requiring monotonicity or convexity. This identity is used to generalize Gabay duality between Douglas-Rachford splitting and ADMM to nonmonotone inclusions, to construct explicit counterexamples showing nonconvergence of ADMM in previously open cases, and to design a family of convergent resolvent homotopy schemes that are applied to nonconvex-regularized least absolute deviations problems.

Significance. If the algebraic identity and its applications are correctly derived, the work is significant because it removes the monotonicity barrier that has limited operator-splitting methods to convex settings, supplies concrete counterexamples that clarify the boundary of convergence for ADMM, and introduces practical convergent schemes for an important nonconvex problem class that lacks strong convexity. The direct, parameter-free character of the core identity is a strength.

major comments (2)
  1. [§3.1, Theorem 3.2] §3.1, Theorem 3.2: the claim that the duality identity extends Gabay duality to nonmonotone inclusions requires explicit verification that the fixed-point sets of the DR and ADMM operators remain in bijection under the resolvent map when monotonicity is dropped; the current argument only shows the algebraic relation but does not address whether the iteration sequences are correspondingly related.
  2. [§4.2, Example 4.5] §4.2, Example 4.5: the counterexample for ADMM nonconvergence is constructed via the DR scheme; it is necessary to confirm that the chosen nonmonotone operator A satisfies the domain conditions under which both resolvents are single-valued everywhere, otherwise the reduction from ADMM to DR may not be valid for the reported divergence.
minor comments (2)
  1. [§2] The notation distinguishing single-valued resolvents from set-valued ones is introduced only in §2; repeating the single-valuedness assumption in the statements of the main applications (Theorems 3.2 and 5.1) would improve readability.
  2. [Figure 2] Figure 2 caption does not specify the step-size schedule used for the homotopy scheme; adding this detail would allow direct reproduction of the plotted trajectories.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive and we address them point by point below, making the necessary clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.1, Theorem 3.2]: the claim that the duality identity extends Gabay duality to nonmonotone inclusions requires explicit verification that the fixed-point sets of the DR and ADMM operators remain in bijection under the resolvent map when monotonicity is dropped; the current argument only shows the algebraic relation but does not address whether the iteration sequences are correspondingly related.

    Authors: The referee correctly notes that the algebraic identity alone does not automatically guarantee the full correspondence of iteration sequences. However, because the DR and ADMM operators are defined pointwise via the resolvents, and the identity J_{A^{-1}}(x) = x - J_A(x) holds at every evaluation point where the resolvents exist, the fixed-point sets are in bijection by direct substitution: if z^* satisfies z^* = T_DR(z^*), then u^* = J_A(z^*) satisfies the fixed-point equation of the ADMM operator. The same pointwise relation maps each iterate of one scheme to the corresponding iterate of the other, so the sequences remain related throughout. We will insert a short clarifying paragraph after Theorem 3.2 to spell out this step-by-step correspondence. revision: yes

  2. Referee: [§4.2, Example 4.5]: the counterexample for ADMM nonconvergence is constructed via the DR scheme; it is necessary to confirm that the chosen nonmonotone operator A satisfies the domain conditions under which both resolvents are single-valued everywhere, otherwise the reduction from ADMM to DR may not be valid for the reported divergence.

    Authors: We agree that explicit domain verification is required for the reduction to be rigorous. In Example 4.5 the operator A is constructed so that both J_A and J_{A^{-1}} are single-valued and defined on all of R^2; this is verified by direct computation showing that for every x there exists a unique y satisfying x ∈ y + A(y). The same holds for A^{-1}. We will add a sentence in the example stating these domain conditions explicitly so that the validity of the ADMM–DR equivalence is immediate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; identities follow directly from resolvent definitions

full rationale

The central claim is the resolvent duality identity J_{A^{-1}}(x) = x - J_A(x), which holds for any set-valued operator A whenever the resolvents exist at the relevant points. This follows immediately by algebraic rearrangement from the definition of the resolvent: if y = J_A(x) then x - y ∈ A(y), which is exactly the condition that x - y = J_{A^{-1}}(x). No monotonicity, convexity, or additional structure is required, and the paper presents this as a direct operator identity rather than a fitted or self-referential construction. Subsequent applications to Gabay duality, DR/ADMM extensions, counterexamples, and new schemes are built on this algebraic relation plus standard operator-splitting arguments; no load-bearing step reduces to a self-citation, ansatz, or renaming of a known empirical pattern. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard properties of resolvent operators and the assumption that the duality identity can be stated without monotonicity; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)
  • domain assumption Resolvents of set-valued mappings are well-defined operators to which the identity applies.
    Required for the duality relationship to be stated for arbitrary mappings.

pith-pipeline@v0.9.0 · 5752 in / 1232 out tokens · 36548 ms · 2026-05-20T09:21:22.292480+00:00 · methodology

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Reference graph

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