Pith. sign in

REVIEW 2 major objections 4 minor 32 references

Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Compact R-continuity, a condition as weak as closed graph at a point, is shown to guarantee approximate convergence of broad classes of algorithms for inclusions and nonsmooth optimization.

desk verdict Compact R-continuity is a good new idea with a clean closed-graph characterization, but boundedness of iterates is doing real work and Definition 2 has a domain/codomain typo. read the letter →

arxiv 2509.01872 v2 pith:7DWN26LH submitted 2025-09-02 math.OC

Ba Khiet Le , Boris S. Mordukhovich , Michel A. Thera This is my paper · ORCID
classification math.OC MSC 49J5249J5390C26
keywords compactR-continuityset-valuedmappingsinclusionproblemsnonsmoothoptimizationconvergenceofalgorithmsLojasiewiczinequalityproximalpointalgorithmDCprogramming
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 compact R-continuity for set-valued mappings, a strictly weaker relative of R-continuity that asks the mapping to behave well only on compact pieces. Its central result is that if the inverse mapping has closed graph at the reference point and takes a closed value there, then it is compactly R-continuous, and this property is enough to convert 'residuals tend to zero' into 'distance to the solution set tends to zero' for a broad class of iterative algorithms. This covers generic descent methods in nonsmooth optimization without requiring Polyak–Lojasiewicz–Kurdyka conditions, and it covers general inclusions 0 in A(x) through a newly identified R-class of algorithms. In finite dimensions, the compact Holder R-continuity of solution mappings for analytic equations is derived from the Lojasiewicz inequality, and applications to the proximal point algorithm and to difference-of-convex algorithms follow from boundedness plus closedness. The payoff is a unified, easy-to-check convergence principle that explains why bounded iterates plus a closed graph are often enough.

What carries the argument

The central object is compact R-continuity: a set-valued mapping A: X => Y is compactly R-continuous at xbar if for every compact set K, the values A(x) intersect K lie within A(xbar) + rho(||x - xbar||)B for a nondecreasing modulus rho with rho(r) -> 0 as r -> 0. Theorem 2 identifies this property, when A(xbar) is closed, with closedness of the graph of A at xbar. The companion machinery is the R-class algorithm, defined by requiring residuals w_k in A(x_k) with ||w_k|| <= alpha xi(k)^beta for some xi(k) -> 0. Together they carry the argument: the algorithm supplies vanishing residuals, closed graph supplies the control rho, and the inclusion x_k in A^{-1}(w_k) intersect K subset A^{-1}(0)

What would settle it

Try to drop boundedness in Theorem 11(ii). Let A: R => R have A(0) = {0} and A(x) = {1/x} for x != 0. The graph of A^{-1} is closed at 0 and A^{-1} is compactly R-continuous at 0, but the iterates x_k = k with residuals w_k = 1/k in A(x_k) form an R-class sequence with w_k -> 0 while d(x_k,S) = k -> infinity. This shows the boundedness assumption carries the theorem.

Watch

Extended reading notes

Core claim

The paper's central discovery is a convergence principle for inclusions 0 in A(x), with A: R^n => R^n. For any R-class algorithm, meaning iterates x_k for which some residual w_k in A(x_k) tends to zero, if A^{-1} is R-continuous at zero, then the distance d(x_k,S) to the solution set S = A^{-1}(0) tends to zero. Under only compact R-continuity, the same conclusion holds whenever the iterates are bounded. Because compact R-continuity at a point is characterized by closedness of the graph at that point, the condition is often automatic: it holds for analytic equations via the Lojasiewicz inequality, for maximally monotone operators, and for DC programs with closed subgradient graph. The same

Load-bearing premise

The compact-version convergence theorem rests on boundedness of the iterates; if a generated sequence escapes to infinity, closed graph at the reference point gives no control over distance to the solution set.

Editorial extensions

If this is right

  • Generic descent algorithms in nonsmooth optimization, those satisfying sufficient decrease and relative error, converge in distance to the stationary set whenever the inverse subdifferential is R-continuous at zero, with no PLK condition and no continuity-on-subsequence assumption.
  • In finite-dimensional problems, boundedness of the iterates plus closed graph of the inverse subdifferential is enough: gradient descent and the convex proximal point algorithm both fall under this umbrella.
  • For analytic finite-dimensional equations, the Lojasiewicz inequality implies compact Holder R-continuity of the solution mapping, so approximate solutions near zeros are guaranteed.
  • For general inclusions, any R-class algorithm with bounded iterates and a compactly R-continuous inverse converges in distance to the solution set; this yields convergence guarantees for PPA on maximally monotone operators and for a DCA version in DC programming.

Reading between the lines

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

  • Because Theorem 2 makes compact R-continuity equivalent to a closed graph at the reference point, one could check convergence of a new algorithm by verifying closedness of the inverse mapping plus boundedness, even when the algorithm is neither descent nor monotone; this suggests a template for non-descent, inertial, or stochastic inclusion solvers.
  • The paper leaves convergence rates open; a natural next step is to read the modulus rho off the Lojasiewicz exponent in Theorem 7, which would turn qualitative distance convergence into explicit rates for analytic inclusions without PLK machinery.
  • The R-class condition only requires residuals w_k in A(x_k) to vanish, not objective decrease; this could be tested directly on numerical traces of split, primal-dual, or accelerated methods to see whether bounded iterates plus local closedness predicts observed convergence.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces a relaxed notion of R-continuity for set-valued mappings between Banach spaces, called compact R-continuity, defined by requiring the excess estimate A(x)∩K ⊂ A(x̄)+ρ(||x−x̄||)B for each compact set K. It proves a sequential characterization via closedness of the graph at the reference point (Theorem 2), derives a version from the Lojasiewicz inequality for analytic equations (Theorem 7), and applies the concept to convergence of descent algorithms (Theorem 8), to a generic R-class of algorithms for inclusions (Theorem 11), and to PPA and DCA (Theorems 13–15). The central claim is that compact R-continuity, being weaker than R-continuity and often checkable by closed-graph arguments, provides a broad sufficient condition for approximate-solution convergence in optimization and inclusion problems.

Significance. If the framework is fully repaired, it offers a useful alternative to PLK-type conditions: convergence conclusions follow from qualitative properties of the solution mapping rather than from local desingularizing inequalities. Theorem 2 gives a crisp characterization, and the Lojasiewicz connection in Theorem 7 is elegant. The PPA and DCA applications show that the framework can reproduce and extend known convergence facts. Explicit credit is due for the clean sequential characterization and for clearly stating the conditional nature of the convergence theorems. On the other hand, the paper contains no numerical experiments and no machine-checked proofs, and the central convergence results require either full R-continuity or boundedness of the iterates. The boundedness hypothesis is essential and is acknowledged in the text to be hard to verify, which significantly tempers the Introduction's advertised breadth of the results.

major comments (2)
  1. [Section 2 (Definition 2) and Theorem 2] Definition 2 as printed is not meaningful in the stated Banach-space setting. It quantifies over compact sets K⊂X, but A(x)∩K is a subset of Y; when X≠Y the intersection is undefined. The intended quantifier is evidently over compact K⊂Y (the codomain): the reverse implication in Theorem 2 forms K={y_k}⊂Y, and Theorem 7 takes K⊂U⊂R^n, which is the codomain of f^{-1}. Since compact R-continuity is the paper's central object and Theorem 2 is its characterization, this must be corrected in Definition 2 and in the forward part of Theorem 2, which currently says 'Given a compact set K⊂X'.
  2. [Section 5, Theorem 11(ii) (also Theorem 8(ii), Corollary 9, Theorem 15)] The boundedness of (x_k) is an essential hypothesis, not a technical convenience. For A:R→R with A(x)=x/(1+x^2), A is continuous, A^{-1}(0)={0}, and A^{-1} has closed graph at 0, so by Theorem 2 A^{-1} is compactly R-continuous at 0. The sequence x_k=k is R-class: w_k=k/(1+k^2)∈A(x_k) and ||w_k||≤1/k→0, yet d(x_k,S)=|k|→∞. Thus compact R-continuity alone cannot deliver the advertised convergence; the boundedness assumption carries the conclusion. The paper's own acknowledgment before Corollary 10 that boundedness 'is not easy to verify' should be reflected in the Introduction's claims and in the statements of Corollary 9 and related results, where boundedness is imposed but no verifiable sufficient condition is given.
minor comments (4)
  1. [Theorems 4 and 5] In the statements and proofs, y should range over R^m (the codomain of f), not R^n. The current text 'Take y∈R^n' is a typo that can confuse readers.
  2. [Section 5, R-class definition] The sentence 'We say that an algorithm belongs to the R-class if the there exist a function ξ' contains a grammatical typo. Also, because ξ may be chosen after the sequence, condition (16) is essentially equivalent to ||w_k||→0; stating this equivalence would clarify the definition.
  3. [Proof of Theorem 13] The quantity shown decreasing is ||x_k−x̄||, not ||x_{k+1}−x̄||. Please correct the phrase 'the decreasing property and convergence of the sequence (||x_{k+1}−x̄||)'.
  4. [Sections 2–3 versus 6] The conclusion lists extension to infinite-dimensional spaces as future research, but Sections 2–3 claim results for general Banach spaces. After correcting Definition 2, it would help to state explicitly which characterization and Lojasiewicz-related results are valid in general Banach spaces and which are finite-dimensional.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main characterization is proven from definitions and the convergence theorems are conditional applications of R-continuity, not fitted or self-referential predictions.

full rationale

The central result (Theorem 2) is proved directly from Definition 2: compact R-continuity plus closed value is shown equivalent to closed graph using only compactness and the definition of the modulus function. Theorems 8 and 11 are conditional: they assume R-continuity or compact R-continuity of the inverse mapping plus a residual condition (wk in A(xk), ||wk|| -> 0), and then conclude d(xk,S) -> 0 by substituting wk into the modulus inclusion. That is an application of the definition, not a circular reduction; the conclusion is not hidden in the assumptions. Theorem 7 uses the external Lojasiewicz inequality [21] to obtain compact R-Hölder continuity, which is independent support. The paper does cite the first author's own [18] for the background notion of R-continuity and for a DCA convergence lemma (Theorem 15), but these are not used to force the paper's central claim and are not uniqueness or ansatz imports; they are ordinary citations to prior work. The formal mismatch in Definition 2 (quantifying over compact K ⊂ X while A(x) ⊂ Y) is a correctness concern, not a circularity, and does not affect this verdict.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper fits no parameters and introduces no new physical entities. It relies on standard variational analysis results and on R-continuity facts from the authors' prior papers, which are cited but not re-derived.

assumptions (4)
  • standard math Lojasiewicz inequality (Theorem 6): for analytic f and compact K, there exist theta,c > 0 with [d(x,S)]^theta <= c|f(x)| on K.
    Imported as a classical result to prove Theorem 7; the paper cites [21,22,10,11] and does not prove it.
  • domain assumption Closed graphs for limiting subdifferential maps in broad function classes (Remark 4(ii)): convex, prox-regular, subdifferentially continuous functions, and sums/differences with C^1 functions.
    Needed to obtain compact R-continuity of (∂f)^(-1) in Theorem 8(ii), Corollary 10, and Theorem 15.
  • standard math Classical PPA convergence and monotone operator properties from Rockafellar [29] (decreasing ||x_k - xbar||, closed graph of maximally monotone operators).
    Used in Theorems 13 and 14 to establish boundedness and compact R-continuity.
  • domain assumption Convergence fact for DCA iterates imported from [18, Theorem 4.5] (self-cited), used to get ||x_{k+1}-x_k|| -> 0 and the residual inclusion in Theorem 15.
    External result by the same research group; the paper does not reproduce the proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms." pith.science (2026). https://pith.science/paper/7DWN26LH

@misc{pith2026250901872,
  author       = {Pith},
  title        = {Pith review of: Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DWN26LH}},
  note         = {Machine review of arXiv:2509.01872}
}
read the original abstract

This paper investigates the notion of compact R-continuity and its specifications for set-valued mappings between Banach spaces. We reveal several important properties of compact R-continuity in general settings and show that in finite dimensions, this notion is supported by the classical Lojasiewicz inequality for analytic functions. An application of compact R-continuity and the obtained results is given to convergence analysis for a broad class of descent algorithms in nonsmooth optimization. We also show that this notion is instrumental for the design and justification of a novel R-class of algorithms to solve inclusion problems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [18]

    B. K. Le , R-Continuity with applications to convergence analysis of Tikhonov regularization and DC Programming, J. Convex Anal. 31, 243–254 (2024)

    work page 2024

  2. [1]

    Adly, Understanding qualitative robust stability of composite variational systems, Set-Valued Var

    S. Adly, Understanding qualitative robust stability of composite variational systems, Set-Valued Var. Anal. 33, 16 (2025)

    work page 2025

  3. [2]

    Attouch, J

    H. Attouch, J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features , Math. Program. 116, 5–16 (2009)

    work page 2009

  4. [3]

    Attouch, J

    H. Attouch, J. Bolte, 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. 16

    work page 2013

  5. [4]

    F. J. Arag´on-Artacho, P. T. Vuong, The boosted difference of convex functions algorithm for nonsmooth functions , SIAM J. Optim. 30, 980–1006 (2020)

    work page 2020

  6. [5]

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

    work page 2017

  7. [6]

    G. C. Bento, B. S. Mordukhovich, T. S. Mota, Yu. Nesterov , Convergence of descent optimization algorithms under Polyak- Lojasiewicz-Kurdyka conditions , J. Optim. Theory. Appl. 207:41, DOI: 10.1007/s10957-o25-02816-z (2025)

    work page doi:10.1007/s10957-o25-02816-z 2025

  8. [7]

    Characterizations of Lojasiewicz inequalities and applications

    J. Bolte, A. Daniilidis, O. Ley, L. Mazet , Characterizations of Lojasiewicz inequalities and applications, arXiv.0802.0826 (2008)

    work page Pith review arXiv 2008

Show all 32 references
  1. [8]

    Br´ezis, Op´ erateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Math

    H. Br´ezis, Op´ erateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Math. Studies 5, North-Holland American Elsevier, 1973

    1973

  2. [9]

    Br ´ezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations , Springer New York (2010)

    H. Br ´ezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations , Springer New York (2010)

    2010

  3. [10]

    S. T. Dinh, T. S. Pham , Lojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices , Inter. J. Math. 27, 1650012 (2016)

    2016

  4. [11]

    S. T. Dinh, T. S. Pham, Lojasiewicz inequalities with explicit exponents for smallest singular value functions , J. Complex. 41, 58–71 (2017)

    2017

  5. [12]

    A. L. Dontchev, R. T. Rockafellar , Implicit Functions and Solution Mappings. A View from Variational Analysis , 2nd edition, Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)

    2014

  6. [13]

    A. J. Hoffman, On approximate solutions of systems of linear inequalities , J. Research Nat. Bureau Standards 49, 263–265 (1952)

    1952

  7. [14]

    A. D. Ioffe , Metric regularity–a survey. Part I. Theory , J. Aust. Math. Soc., 101: 188–243, (2016)

    2016

  8. [15]

    Karimi, J

    H. Karimi, J. Nutini, M. Schmidt , Linear convergence of gradient and proximal-gradient methods under the Polyak- Lojasiewicz condition. In: Machine Learning and Knowledge Discovery in Databases, Part 1, pp. 795–811. Cham, Springer International Publishing (2016)

    2016

  9. [16]

    A. Y. Kruger, Nonlinear metric subregularity, J. Optim. Theory Appl. 171, 820–855, (2016)

    2016

  10. [17]

    Kurdyka, On gradients of functions definable in o-minimal structures , Ann

    K. Kurdyka, On gradients of functions definable in o-minimal structures , Ann. Inst. Fourier 48, 769–783 (1998)

    1998

  11. [19]

    B. K. Le, M. Th ´era, Explicit convergence rate of the proximal point algorithm under R- continuity, Evolution Equations Control Theory, DOI: 10.3934/eect.2025016 (2025). 17

    2025 doi

  12. [20]

    T. H. A. Le, V. N. Huynh, D. T. Pham , Convergence analysis of difference-of-convex algorithm with subanalytic data , J. Optim. Theory Appl. 179, 103–126 (2018)

    2018

  13. [21]

    Lojasiewicz,Division d’une distribution par une fonction analytique de variables r´ eelles , C

    S. Lojasiewicz,Division d’une distribution par une fonction analytique de variables r´ eelles , C. R. Acad. Sci. Paris 246, 683–686 (1958)

    1958

  14. [22]

    Lojasiewicz, Sur la g´ eom´ etrie semi- et sous-analytique, Annales de l’institut Fourier, tome 43, no 5, p

    S. Lojasiewicz, Sur la g´ eom´ etrie semi- et sous-analytique, Annales de l’institut Fourier, tome 43, no 5, p. 1575–1595 (1993)

    1993

  15. [23]

    M. A. L ´opez, Measuring the stability. A paradigmatic problem in optimization , Set-Valued Var. Anal 33, 18 (2025)

    2025

  16. [24]

    B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I: Basic Theory, II: Applications . Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vols. 330 and 331, Springer, Berlin (2006)

    2006

  17. [25]

    B. S. Mordukhovich, Second-Order Variational Analysis in Optimization, Variational Sta- bility, and Control: Theory, Algorithms, Applications , Springer Series in Operations Research and Financial Engineering, Springer, Cham, Switzerland (2014)

    2014

  18. [26]

    Nesterov, Lectures on Convex Optimization , 2nd edition, volume 137 of Springer Opti- mization and Its Applications, Springer, Cham, Switzerland (2018)

    Y. Nesterov, Lectures on Convex Optimization , 2nd edition, volume 137 of Springer Opti- mization and Its Applications, Springer, Cham, Switzerland (2018)

    2018

  19. [27]

    H. V. Ngai, M. Th ´era, Error bounds in metric spaces and application to the perturbation stability of metric regularity , SIAM J. Optim. 19, 1–20 (2008)

    2008

  20. [28]

    B. T. Polyak, Gradient methods for the minimization of functionals, USSR Comput. Math. Math. Phys. 3, 864–878 (1963)

    1963

  21. [29]

    R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization 14, 877–898 (1976)

    1976

  22. [30]

    R. T. Rockafellar and R. J-B. Wets , Variational Analysis. Grundlehren der Mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer, Berlin (1998)

    1998

  23. [31]

    Tanwani, B

    A. Tanwani, B. Brogliato, C. Prieur, Well-posedness and output regulation for implicit time-varying evolution variational inequalities , SIAM J. Control Optim. 56, 751–781 (2018)

    2018

  24. [32]

    Thibault, Unilateral Variational Analysis in Banach Spaces , published in two volumes

    L. Thibault, Unilateral Variational Analysis in Banach Spaces , published in two volumes. World Scientific, Singapore (2023). 18

    2023

Pith tools

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