Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version

Ba Khiet Le; Boris S. Mordukhovich; Michel A. Thera

REVIEW 2 major objections 4 minor 25 references

Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves linear convergence of the generalized proximal point algorithm for nonmonotone inclusions under adaptive strong monotonicity, and shows its inexact version remains stable with uniformly bounded errors.

desk verdict Correct, useful, and reviewable — just don't let the abstract's 'linear convergence' oversell the iterate result without R-Lipschitz F^{-1}. read the letter →

arxiv 2608.01584 v1 pith:H2CNSXKI submitted 2026-08-03 math.OC

Ba Khiet Le , Boris S. Mordukhovich , Michel A. Thera This is my paper · ORCID

classification math.OC MSC 65K0565K1049J5249J53

keywords adaptivestrongmonotonicitygeneralizedproximalpointalgorithminexactnonmonotoneinclusionswarpedresolventR-continuitylinearconvergencenondiminishingerrors

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 introduces a condition called adaptive strong monotonicity for pairs of operators (F,v): rather than requiring F to be monotone in the original space, it only requires F's variation to track the variation of a kernel map v. Under this condition, the generalized proximal point algorithm produces residuals in v-coordinates that shrink at a fixed linear rate. If the inverse of F is also regular at zero, that residual shrinkage lifts to the actual iterates, whose distance to the solution set vanishes linearly. The inexact version tolerates computational errors that never shrink, provided the error bound is paired with a regularization parameter, and the eventual error goes to zero as the parameter goes to zero. A reader should care because monotonicity is traditionally the price of such guarantees; this extends them to problems where F itself is not monotone.

What carries the argument

The warped resolvent J^v_{γF}=(γF+v)⁻¹∘v is the iteration map; it makes the algorithm take steps in v-coordinates rather than original coordinates, so a nonmonotone F can become monotone in pair with v. Adaptive strong monotonicity supplies the algebraic inequality that turns each step into a contraction: combining the inclusion v(x_{k+1})−v(x_k)∈−γF(x_{k+1}) with 0∈−γF(x*) and the inequality gives a polarization estimate whose squared v-residual rate is 1/(1+2γε). R-continuity of F⁻¹ at 0 is the bridge that converts a small v-residual into a small distance from x_{k+1} to S. In the regularized problem, adding εv to F makes the pair automatically ε-adaptively strongly monotone, which is why

What would settle it

Take H=R², v(x1,x2)=x1, and F(x1,x2)=(x1, x2 − 1/x1) for x1≠0, with F(0,x2)=(0,x2). The pair (F,v) is 1-adaptively strongly monotone and ran v ⊆ ran(γF+v), so the warped resolvent is well-defined. The GPPA gives x1_{k+1}=x1_k/(1+γ) and forces x2_{k+1}=1/x1_{k+1}, so the v-residual shrinks linearly while the iterates escape to infinity. The solution set is S={(0,0)}, so d(x_k,S) does not go to 0, isolating the missing R-continuity of F⁻¹ at 0 as the condition that turns v-space contraction into convergence to the solution set.

Watch

Extended reading notes

Core claim

The central claim is that proximal-point-type methods can solve nonmonotone inclusions at a guaranteed linear rate, provided the operator F is paired with a kernel map v so that (F,v) is adaptively strongly monotone: ⟨F(x)−F(y), v(x)−v(y)⟩ ≥ ε‖v(x)−v(y)‖². Under this inequality and the range condition ran v ⊆ ran(γF+v), the GPPA step contracts the v-residual by the factor 1/√(1+2γε) (Theorem 2(a)). If, in addition, F⁻¹ is R-Lipschitz at 0, the same contraction transfers to d(x_k,S), so the iterates themselves converge linearly to the solution set (Theorem 2(b)). For the inexact version with errors bounded by δ=ε² on the regularized problem 0∈F(x)+εv(x), the paper proves limsup d(x_k,S)→0 as

Load-bearing premise

The load-bearing premise is that the inverse operator F⁻¹ is R-Lipschitz at 0 in Theorem 2(b) and merely R-continuous at 0 in Theorem 4—a regularity condition asserting that points with small residual remain proportionally close to the solution set; this must be verified problem by problem and is not implied by adaptive strong monotonicity, and without it the linear v-space contraction says nothing about where the actual iterates go.

Editorial extensions

If this is right

Any inclusion satisfying adaptive strong monotonicity and the range condition has approximate solutions computed at a guaranteed linear rate, with no monotonicity or bijectivity of v required.
When F⁻¹ is R-Lipschitz at 0, the distance from the iterates to the solution set converges to zero linearly; this is the practical sense in which the algorithm solves the original problem.
The inexact version permits errors that never tend to zero, as long as the error tolerance δ=ε² is tied to the regularization parameter; the asymptotic distance to S can be made arbitrarily small by shrinking ε.
The contraction estimate holds uniformly for every solution x* in S, so the linear rate is not tied to a particular limiting solution.
In finite-dimensional settings with bounded iterates, the R-Lipschitz requirement can be relaxed to compact R-Lipschitz continuity, widening the class of admissible operators.

Reading between the lines

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

An implication the paper leaves implicit: the same εv regularization plus R-continuity step should let bounded-error results carry over to forward-backward and other splitting algorithms for nonmonotone operators, not just the proximal iteration.
The contraction factor 1/√(1+2γε) exposes a trade-off the paper does not quantify: larger step size γ speeds up v-space contraction, but may make the required range condition harder to verify for a given v.
When v is not injective, the linear residual theorem alone certifies only approximate solutions in v-coordinates; applications should report d(x_k,S) only after checking R-continuity of F⁻¹, since otherwise iterates may wander within a fiber of v.

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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

Bottom line: this is a solid paper with a genuine new definition and correct proofs, but the advertised 'linear convergence' is only for the v-residual unless you add R-Lipschitz continuity of F^{-1}.

The new notion (adaptive strong monotonicity) is a real weakening of the old strong monotonicity of pairs, and Theorem 2(a) gives a clean contraction argument in the v-metric. The inexact analysis in Theorem 4 also checks out: the regularization trick and the geometric-sum argument are sound, and the extension of bounded-error stability from the monotone case to this nonmonotone setting is a legitimate step forward. I verified the algebra; the only real presentation gap is that u_k is defined using y_0, which never appears in the iteration — set y_0=0 or start the index at k=1.

The soft spot is framing. The abstract says 'linear convergence under adaptive strong monotonicity' without qualification. What is actually proven without extra hypotheses is linear decay of ||v(x_k) - v(x^*)||. If F^{-1} is R-Lipschitz at 0, then d(x_k,S) also decays linearly, but that condition is not implied by adaptive strong monotonicity and is not verified for any concrete problem class in the paper. The stress-test example is correct and worth taking seriously: take v(x_1,x_2)=(x_1,0), F(x_1,x_2)=(x_1, x_2 + 1/log x_1) on a small neighborhood of 0. The pair is adaptively strongly monotone with epsilon=1, F is not monotone, and GPPA with gamma=1 produces v-residuals of size 2^{-k} while d(x_k,S) decays only like 1/(k log 2). So the iterate distance can be sublinear. The authors are not hiding this — Theorem 2(b) states the R-Lipschitz assumption clearly — but the abstract and intro blur the distinction between residual convergence and iterate convergence.

I would send this to a serious referee. The core contributions are real, the proofs are short enough to check, and the paper honestly states its assumptions. Before acceptance, the authors should qualify the abstract, add a remark that residual linear convergence is the baseline and R-Lipschitz is an extra condition needed for iterates, and fix the minor indexing issue with y_0. I'd be happy to see it in print after those adjustments.

Referee Report

2 major / 4 minor

Summary. The paper studies the Generalized Proximal Point Algorithm (GPPA) of Le, Dao, and Théra for nonmonotone inclusions 0∈F(x) in Hilbert spaces. It introduces a new notion, adaptive strong monotonicity of a pair (F,v), i.e. ⟨F(x)−F(y), v(x)−v(y)⟩ ≥ ε‖v(x)−v(y)‖², and proves in Theorem 2(a) that the v-residual sequence converges linearly with rate factor (1+2γε)^{-1}. In Theorem 2(b), if F^{-1} is R-Lipschitz continuous at 0, then the distance to the solution set S converges linearly. The paper then studies the Inexact GPPA with uniformly bounded computational errors. Applying the GPPA to the Tikhonov-regularized inclusion 0∈F(x)+εv(x) and choosing δ=ε², Theorem 4 establishes a limsup bound on d(x_k,S) in terms of the R-continuity modulus of F^{-1}, and consequently lim_{ε→0} limsup_{k→∞} d(x_k,S)=0.

Significance. If the advertised claims are read with the correct qualifiers, this is a useful and technically sound contribution. The new adaptive strong monotonicity condition is strictly weaker than pair strong monotonicity and sufficient for linear convergence of the v-residual. The stability result for the inexact algorithm with constant error bounds is a meaningful extension of the summable-error literature to nonmonotone problems. The proofs are self-contained, direct, and do not rely on fitted parameters or empirically tuned constants. The main caveat, which I develop below, is that the paper's title and abstract phrase 'linear convergence under adaptive strong monotonicity' in a way that can be misread as linear convergence of the iterates; in fact, iterate convergence requires an additional and separate regularity condition on F^{-1}.

major comments (2)

[Abstract and §3 (Theorem 2)] The advertised claim 'linear convergence under adaptive strong monotonicity' is, strictly speaking, only linear convergence of the v-residuals. Theorem 2(a) proves Q-linear decay of ‖v(x_k)−v(x*)‖; Theorem 2(b) proves linear decay of d(x_k,S) only under the additional hypothesis that F^{-1} is R-Lipschitz at 0. This hypothesis is not implied by adaptive strong monotonicity, and the gap is not merely cosmetic. For example, take H=R², v(x₁,x₂)=(x₁,0), F(x₁,x₂)=(x₁,x₂−φ(x₁)) with φ(0)=0 and φ(r)=−1/log r for 0<r<e^{-1}. Then (F,v) is 1-adaptively strongly monotone, F is not monotone, F^{-1} is not R-Lipschitz at 0, and the GPPA with γ=1 from x₀=(1,0) gives x_{k+1}=(2^{-(k+1)}, φ(2^{-(k+1)})). The residual ‖v(x_{k+1})−v(x_k)‖=2^{-(k+1)} decays linearly, while d(x_k,S) ~ 1/(k log 2) is sublinear. Thus the abstract and Section 1 should explicitly distinguish 'linear convergence of residuals' f
[§4, Theorem 4] The statement of Theorem 4 does not explicitly include the condition ran v ⊆ ran(γ(F+εv)+v) that is needed for the warped resolvent J^v_{γ(F+εv)} to be well-defined and for the iterates to exist. Since the algorithm is written with J^v, this is presumably intended as an implicit standing assumption, but it should be stated. Moreover, no example or verifiable sufficient condition is provided for the R-continuity of F^{-1} at 0 in the nonmonotone setting; the result is conditional on a regularity condition that is known to be delicate. Adding a concrete class of operators satisfying the hypotheses would significantly strengthen the applicability claims.

minor comments (4)

[§4, proof of Theorem 4] The definition of u_k omits y_0. Since the error sequence is defined for k≥1, the natural correction is u_0:=x_0 and u_{k+1}:=x_{k+1}−y_{k+1} for k≥0. As written, the equation u_k := x_k − y_k is undefined at k=0.
[§4, Theorem 4] The phrase 'nondiminishing errors' should be qualified. For each fixed ε>0, the bound δ=ε² is constant in k, so the errors do not vanish along the iteration; however, in the limiting statement ε→0, the allowed error also tends to zero. This is standard in Tikhonov-type results but should be stated explicitly to avoid confusion with a fixed, truly nondiminishing error independent of the regularization parameter.
[Definition 3] For set-valued F, the notation ⟨F(x)−F(y), v(x)−v(y)⟩ is ambiguous. It should be quantified over all selections, e.g. 'for all (x,x*),(y,y*)∈graF, ⟨x*−y*, v(x)−v(y)⟩≥ε‖v(x)−v(y)‖².' This will also make the proof of Theorem 2(a) formally airtight.
[§4, proof of Theorem 4] There is a minor typographical issue in the display containing ρ( ‖v(u_{k+1})−v(x_k)‖/γ + ε‖v(u_{k+1}‖ ), where the second norm has an unmatched parenthesis. The notation for the regularized operator also alternates between ilde F and eF; please make it uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: derivations follow from stated assumptions, not from their conclusions.

full rationale

The paper introduces adaptive strong monotonicity (Definition 3) and proves Theorem 2(a) directly from the GPPA inclusion and the definition: using 0 in F(x*), adaptive strong monotonicity gives the inner-product inequality, and polarization yields the contraction in v-space. This is a self-contained derivation; no quantity in the inequality is fitted and no prediction is defined in terms of the result. Theorem 2(b) adds the separate hypothesis that F^{-1} is R-Lipschitz at 0 (Definition 5) and converts the linearly decaying v-residual into distance decay by the elementary inclusion x_{k+1} in S + (L/gamma)||v(x_{k+1})-v(x_k)||B. The R-Lipschitz condition is not implied by adaptive strong monotonicity and is not derived from the conclusion; it is an extra, explicitly stated regularity assumption. Theorem 4 similarly uses the fact that (F+epsilon v, v) is epsilon-adaptively strongly monotone, verified in one line, followed by a contraction estimate in v-space plus R-continuity of F^{-1}; the regularization argument is explicit and does not assume the limsup bound it aims to prove. The paper's self-citations to the GPPA framework and to R-continuity refer to published, stated results; the relevant definitions are restated in the paper, and the new theorems do not reduce to those citations. The residual-versus-iterate distinction in the word linear convergence is a presentation point, not circularity. No fitted parameters, self-referential definitions, or imported uniqueness/ansatz claims occur. Hence score 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proofs depend on the imported framework (pair monotonicity, warped resolvents, R-continuity) and the newly stated adaptive strong monotonicity condition. No parameters are fitted to data; the constants epsilon, gamma, L are given problem data or design choices, and delta = epsilon^2 is chosen by the authors to make the final bound vanish.

assumptions (7)

standard math H is a real Hilbert space and v, F are defined on all of H with dom v = H
Section 2 Preliminaries; needed for inner product and norm arguments.
domain assumption (F,v) is monotone, i.e., <F(x)-F(y),v(x)-v(y)> >= 0
Definition 1; fundamental for GPPA framework from [13].
domain assumption The pair (F,v) is epsilon-adaptively strongly monotone for some epsilon > 0
Definition 3; this is the new assumption that drives Theorem 2.
domain assumption ran v subset of ran(gamma*F + v) for the chosen gamma > 0
Needed for warped resolvent J^v_{gamma*F} to be defined; stated in Theorem 1 and Theorem 2.
domain assumption F^{-1} is R-continuous (R-Lipschitz) at 0
Definition 5; used in Theorem 2(b) and Theorem 4 to convert v-space residuals into distances to S.
domain assumption zer(F+epsilon*v) is nonempty for sufficiently small epsilon > 0
Assumed in Theorem 4; guarantees the regularized problem has a solution.
domain assumption v is L-Lipschitz continuous
Used in Theorem 4 to bound ||v(x_k) - v(u_k)|| <= L*delta.

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Pith. "Pith review of Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version." pith.science (2026). https://pith.science/paper/H2CNSXKI

@misc{pith2026260801584,
  author       = {Pith},
  title        = {Pith review of: Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2CNSXKI}},
  note         = {Machine review of arXiv:2608.01584}
}

read the original abstract

In this paper, we introduce the notion of adaptive strong monotonicity and establish a linear convergence of the recently proposed Generalized Proximal Point Algorithm (GPPA) for computing approximate solutions to nonmonotone inclusion problems in Hilbert spaces. We also investigate the Inexact Generalized Proximal Point Algorithm (IGPPA) in the presence of nondiminishing errors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

S. Adly, M. G. Cojocaru and B. K. Le , State-dependent sweeping processes: asymptotic behavior and algorithmic approaches. J. Optim. Theory Appl. , 202 (2024), 932--948

work page 2024

[2] [2]

F. J. Arag\'on-Artacho and P. T. Vuong , The boosted difference of convex functions algorithm for nonsmooth functions, SIAM J. Optim. 30 (2020), 980--1006

work page 2020

[3] [3]

Attouch, J

H. Attouch, J. Bolt\'e and B. F. Svaiter , Convergence of descent methods for semialgebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss--Seidel methods, Math. Program. 137 (2013), 91--129

work page 2013

[4] [4]

H. H. Bauschke and P. L. Combettes , Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2nd edition, Springer, Cham, Switzerland, 2017

work page 2017

[5] [5]

H. H. Bauschke, J. M. Borwein and P. L. Combettes , Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003), 596--636

work page 2003

[6] [6]

M. N. B\`ui and P. L. Combettes , Warped proximal iterations for monotone inclusions, J. Math. Anal. Appl. 491 (2020), 124315

work page 2020

[7] [7]

Br\'ezis , Op\' e rateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert , North-Holland American Elsevier, New York, 1973

H. Br\'ezis , Op\' e rateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert , North-Holland American Elsevier, New York, 1973

work page 1973

[8] [8]

E. L. Buscaglia, P. A. Lotito, L. A. Parente and M. V. Solodov , A general framework for symmetric and asymmetric variable metric proximal point methods, with relations between relative errors and summable errors, SIAM J. Optim. 35 (2025), 2518--2543

work page 2025