pith. sign in

arxiv: 2605.15838 · v1 · pith:GEC63PDAnew · submitted 2026-05-15 · 🧮 math.OC

Finding directional stationary points of DC programs

Hoai An Le Thi , Van Ngai Huynh , Tao Pham Dinh This is my paper

Pith reviewed 2026-05-20 17:21 UTC · model grok-4.3

classification 🧮 math.OC
keywords DC programmingdirectional stationary pointsnon-smooth optimizationDC algorithmLojasiewicz inequalityconvergence analysisstochastic optimization
0
0 comments X

The pith

A unified algorithm computes directional stationary points for non-smooth DC programs by generalizing the classical DC algorithm.

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

The paper develops a method to find directional stationary points in non-convex problems expressed as differences of convex functions, allowing both functions to be non-smooth. This matters because directional stationarity is stronger than standard critical points and does not depend on the chosen decomposition. The framework handles the second convex function as the pointwise maximum of a continuous family of smooth convex functions or any continuous convex function on the whole space. It proves full sequence convergence under the Lojasiewicz inequality when the second function is the maximum of finitely many C^{1,1} convex functions. A randomized variant is shown to converge almost surely.

Core claim

We propose a new and unified approach for identifying directional stationary points of non-smooth DC programs, where both DC components may be non-smooth. Our framework generalizes both the classical DCA and the above recent approaches. It applies when the second DC component is the pointwise maximum of a continuous family of smooth convex functions, and more generally, to any continuous convex function on the whole space. We also establish strong convergence results. Specifically, when the second DC component is the pointwise maximum of finitely many C^{1,1}-smooth convex functions, we prove under the Lojasiewicz inequality that the entire sequence generated by our algorithm converges. This

What carries the argument

The unified algorithmic framework extending DCA to compute directional stationary points using subdifferentials for general non-smooth DC decompositions.

If this is right

  • The method applies to a broader class of non-smooth DC programs than prior approaches restricted to finite maxima of smooth functions.
  • Full sequence convergence holds under the Lojasiewicz inequality without requiring global C^{1,1} smoothness of the second component.
  • The framework unifies the classical DCA with recent directional-stationarity methods for specific non-smooth cases.
  • A stochastic randomized variant of the algorithm converges almost surely to directional stationary points.

Where Pith is reading between the lines

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

  • The unification may allow existing DCA implementations to be adapted for directional stationarity with minimal changes in practice.
  • The convergence analysis could extend to other non-smooth optimization settings where Lojasiewicz-type conditions appear naturally.

Load-bearing premise

The second DC component must be expressible as the pointwise maximum of a continuous family of smooth convex functions or any continuous convex function on the whole space.

What would settle it

Running the algorithm on a DC program with a qualifying second component and finding that a limit point is not directionally stationary, or that the sequence fails to converge when the Lojasiewicz inequality holds for finitely many C^{1,1} functions, would disprove the claims.

read the original abstract

We address the problem of computing stationary points for non-smooth, non-convex optimization problems. While this topic is well studied in the smooth setting, fewer algorithmic and theoretical results exist for the non-smooth case. Within Difference-of-Convex functions (DC) programming, the well-known DC Algorithm (DCA) is a standard method for computing critical points, whose definition depends on the chosen DC decomposition. More recently, some works have focused on computing directional stationary points - a stronger notion that does not depend on any particular DC decomposition - for specific non-smooth DC programs, where the second DC component is the pointwise maximum of finitely many smooth convex functions. In this contribution, we propose a new and unified approach for identifying directional stationary points of non-smooth DC programs, where both DC components may be non-smooth. Our framework generalizes both the classical DCA and the above recent approaches. It applies when the second DC component is the pointwise maximum of a continuous family of smooth convex functions, and more generally, to any continuous convex function on the whole space. We also establish strong convergence results. Specifically, when the second DC component is the pointwise maximum of finitely many $C^{1,1}$ - smooth convex functions, we prove under the \L ojasiewicz inequality that the entire sequence generated by our algorithm converges. This extends previous results requiring global $C^{1,1}$ - smoothness. Finally, we introduce a randomized (stochastic) variant of our method and prove its almost sure convergence, thereby extending our deterministic results to the randomized setting.

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

1 major / 2 minor

Summary. The paper proposes a unified algorithmic framework for computing directional stationary points of non-smooth DC programs in which both components may be non-smooth. The second DC component is assumed to be either the pointwise supremum of a continuous family of smooth convex functions or an arbitrary continuous convex function on the whole space. The method generalizes the classical DCA and recent directional-stationarity algorithms limited to finite maxima of smooth functions. Convergence to a directional stationary point is claimed in the general case; under the additional Lojasiewicz inequality and when the supremum is finite with each function C^{1,1}, the entire sequence is shown to converge. A randomized stochastic variant is introduced with an almost-sure convergence guarantee.

Significance. If the derivations hold, the work meaningfully extends the scope of DC programming by accommodating broader non-smooth structures while recovering classical and recent results as special cases. The unification, the full-sequence convergence result under Lojasiewicz (extending prior global C^{1,1} assumptions), and the stochastic extension constitute clear contributions to the literature on non-smooth non-convex optimization.

major comments (1)
  1. [§3] §3 (or the main convergence theorem): the statement that the algorithm converges to a directional stationary point in the general continuous-convex case appears to rely on a limiting argument that must carefully handle the subdifferential of the supremum; the manuscript should explicitly verify that the directional derivative condition is preserved in the limit without additional regularity.
minor comments (2)
  1. [Introduction and §2] The notation for the continuous family of convex functions (e.g., the index set and the pointwise max operator) should be introduced once and used consistently; currently it varies slightly between the abstract, introduction, and algorithmic section.
  2. [§2.2] A short remark clarifying how the new framework reduces exactly to the classical DCA when the second component is smooth would strengthen the generalization claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comment on our manuscript. We address the major comment below and will revise the paper accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§3] §3 (or the main convergence theorem): the statement that the algorithm converges to a directional stationary point in the general continuous-convex case appears to rely on a limiting argument that must carefully handle the subdifferential of the supremum; the manuscript should explicitly verify that the directional derivative condition is preserved in the limit without additional regularity.

    Authors: We appreciate the referee pointing out this subtlety in the limiting argument for the general case where the second DC component is an arbitrary continuous convex function. The proof relies on the fact that, for a continuous convex function, the subdifferential mapping is outer semicontinuous and the directional derivative is upper semicontinuous. Consequently, if a sequence of iterates satisfies an approximate directional stationarity condition, the limit point satisfies the exact condition. To make this step fully explicit and transparent without invoking additional regularity, we will insert a short clarifying lemma (or expanded remark) in §3 that recalls the relevant properties of convex functions and shows directly that the directional derivative condition passes to the limit. This addition strengthens the exposition while leaving the theorem statement and proof strategy unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new unified algorithmic framework for directional stationary points in non-smooth DC programs, explicitly generalizing classical DCA and prior directional-stationary methods under stated structural assumptions on the second DC component (pointwise max of continuous family of smooth convex functions or arbitrary continuous convex function). Convergence claims, including full sequence convergence under Lojasiewicz for the finite C^{1,1} case, are derived from these assumptions and standard analysis techniques rather than reducing to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain remains self-contained with independent content beyond prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard DC-programming assumptions plus the Lojasiewicz inequality for sequence convergence and continuity/smoothness conditions on the second DC component.

axioms (2)
  • domain assumption The second DC component is the pointwise maximum of a continuous family of smooth convex functions or any continuous convex function
    Stated in the abstract as the setting where the new approach applies.
  • domain assumption Lojasiewicz inequality holds
    Invoked to prove entire sequence convergence when the second component is finite max of C^{1,1} functions.

pith-pipeline@v0.9.0 · 5819 in / 1287 out tokens · 81940 ms · 2026-05-20T17:21:57.106282+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Optim., 30(1), 56-79, (2020)

    Beck A., Hallak N., On the Convergence to Stationary Points of Deterministic and Random- ized Feasible Descent Directions Methods, Siam J. Optim., 30(1), 56-79, (2020)

    work page 2020

  2. [2]

    Amer Math

    Cucker F., Smale S., On the mathematical foundations of learning, Bull. Amer Math. Soc., 39(1), 1-49, (2001)

    work page 2001

  3. [3]

    Optim., 16, 531-547 (2005)

    Absil P.-A., Mahony M., Andrews B., Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16, 531-547 (2005)

    work page 2005

  4. [4]

    F., Descent methods for composite nondifferentiable optimization problems, Math

    Burke, J. F., Descent methods for composite nondifferentiable optimization problems, Math. Program., Serie A, 33, 260-279, (1985)

    work page 1985

  5. [5]

    Ioffe A., Tikhominov V., Theory of extremal problems, North Holland, (1979)

    work page 1979

  6. [6]

    H., Optimization and Nonsmooth Analysis, Wiley, New York, (1983)

    Clarke F. H., Optimization and Nonsmooth Analysis, Wiley, New York, (1983)

    work page 1983

  7. [7]

    Attouch, J

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

    work page 2009

  8. [8]

    Attouch, J

    H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kyrdyka- Lojasiewicz inequality, Mathematics of Operations research, 35, 438-457 (2010)

    work page 2010

  9. [9]

    , 17 (4), 1205-1223 (2007)

    Bolte J., Daniliidis A., Lewis A., Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamic systems, SIAM Optim. , 17 (4), 1205-1223 (2007)

    work page 2007

  10. [10]

    Bolte J., Daniliidis A., Ley O., Mazet L., Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity, Trans. Amer. Math. Soc., 362, 3319-3363 (2010)

    work page 2010

  11. [11]

    Bolte, A

    J. Bolte, A. Daniilidis, A. Lewis and M. Shiota, Clarke subgradients of stratifiable functions, SIAM Optim., 18, 556-572 (2007)

    work page 2007

  12. [12]

    Bierstone, P

    E. Bierstone, P. Milman, Semianalytic and subanalytic sets, IHES Publ. Math. , 67, 5-42 (1988)

    work page 1988

  13. [13]

    , Lemar´ echal C., Convex Analysis and Minimization Algorithms , Parts I&II, Springer Verlag, (1991)

    Hiriart-Urruty J.B. , Lemar´ echal C., Convex Analysis and Minimization Algorithms , Parts I&II, Springer Verlag, (1991)

    work page 1991

  14. [14]

    Le Thi H. A. and Pham Dinh T., The DC (difference of convex functions) Programming and DCA revisited with DC models of real world nonconvex optimization problems. Annals of Operations Research, 133, pp. 23-48, (2005)

    work page 2005

  15. [15]

    In the special issue on DC programming: theory, algorithms and applications, on the occasion of 30th birthday of DC programming and DCA, Math

    Le Thi, H.A., Pham Dinh, T., DC programming and DCA: thirty years of developments. In the special issue on DC programming: theory, algorithms and applications, on the occasion of 30th birthday of DC programming and DCA, Math. Program. 169, 5–68 (2018). https://doi.org/10.1007/s10107-018-1235-y

    work page doi:10.1007/s10107-018-1235-y 2018

  16. [16]

    Open issues and recent advances in DC programming and DCA

    Le Thi, H.A., Pham Dinh, T. Open issues and recent advances in DC programming and DCA. J Glob Optim (2023). https://doi.org/10.1007/s10898-023-01272-1

    work page doi:10.1007/s10898-023-01272-1 2023

  17. [17]

    In: Le Thi, H.A., Pham Dinh, T., Le, H.M

    Le Thi, H.A., Pham Dinh, T., DC Programming and DCA: 40 Years of Breakthroughs Driving Intelligence and Technology. In: Le Thi, H.A., Pham Dinh, T., Le, H.M. (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. MCO 2025. Lecture Notes in Networks and Systems, vol 1688. Springer, Cham. https://doi.org/10.1007/978-3-...

    work page doi:10.1007/978-3-032-08381-4 2025

  18. [18]

    A., Ngai H

    Le Thi H. A., Ngai H. V., and Pham Dinh T., Convergence Analysis of DC Algorithm for DC programming with subanalytic data, J. Optim. Theory and Appl., 109, 103-126 (2018)

    work page 2018

  19. [19]

    A., Ngai H

    Le Thi H. A., Ngai H. V., and Pham Dinh T., DC programming and DCA for general DC programs. Advances in Intelligent Systems and Computing, ISBN 978-3-319-06568-7, 15-35, Springer, (2014)

    work page 2014

  20. [20]

    L., Minimum-support solutions of polyhedral concave programs, Optimiza- tion, 45, 149-162, (1999)

    Mangasarian O. L., Minimum-support solutions of polyhedral concave programs, Optimiza- tion, 45, 149-162, (1999)

    work page 1999

  21. [21]

    Lojasiewicz, Sur le probl` eme de la division, Studia Mathematica, 18, 87-136(1959)

    S. Lojasiewicz, Sur le probl` eme de la division, Studia Mathematica, 18, 87-136(1959)

    work page 1959

  22. [22]

    Lojasiewicz, Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels,in les Equa- 24 H

    S. Lojasiewicz, Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels,in les Equa- 24 H. A. LE THI, V. N. HUYNH, T. PHAM DINH tions aux d´ eriv´ ees Partielles, Editions du Centre National de la Recherche Scientifique, Paris, 87-89 (1963)

    work page 1963

  23. [23]

    Lojasiewicz, Sur la g´ eom´ etrie semi- et sous-analytique, Anal de l’institute Fourier, 43, 1575-1595 (1993)

    S. Lojasiewicz, Sur la g´ eom´ etrie semi- et sous-analytique, Anal de l’institute Fourier, 43, 1575-1595 (1993)

    work page 1993

  24. [24]

    S., Variational analysis and generalized differentiation

    Mordukhovich B. S., Variational analysis and generalized differentiation. I. Basic theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe- matical Sciences], 330. Springer-Verlag, Berlin, 2006

    work page 2006

  25. [25]

    Mordukhovich B.S, Variational analysis and generalized differentiation. II. Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe- matical Sciences], 331. Springer-Verlag, Berlin, 2006

    work page 2006

  26. [26]

    Nesterov Y., Gradient Methods for Minimizing Composite Ojective Function, CORE Dis- cussion Paper 2007/76, Universit´ e Catholique de Louvain, Belgium, (2007)

    work page 2007

  27. [27]

    V., Th´ era M, Error bounds for systems of lower semicontinuous functions in Asplund spaces, Math

    Ngai H. V., Th´ era M, Error bounds for systems of lower semicontinuous functions in Asplund spaces, Math. Program. , 116, 397-427 (2009)

    work page 2009

  28. [28]

    Optim, 28(2), 1640- 1669 (2018)

    Pang J.-S., Tao M., Decomposition methods for computing directional stationary solutions of a class of nonsmooth nonconvex optimization problems, SIAM J. Optim, 28(2), 1640- 1669 (2018)

    work page 2018

  29. [29]

    Pang J.-S., Razaviyan M., and Alvarado A., Computing B−Stationary Points of Nonsmooth DC Program, Math. Oper. Res., 42(1), 95-118 (2016)

    work page 2016

  30. [30]

    Optim., 28(4), 3344-3374, (2018)

    Cui Y., Pang J.-S., and Sen B., Composite difference-max programs for modern statistical estimation problems, SIAM J. Optim., 28(4), 3344-3374, (2018)

    work page 2018

  31. [31]

    Lu Z., Zhou Z., Sun Z., Enhanced proximal DC algorithms with extrapolation for a class of strutured nonsmooth DC minimization, Mathematical Programming, 176(1-2), 369-401, (2019)

    work page 2019

  32. [32]

    Optim., 29(4), 2725-2752, (2019)

    Lu Z., Zhou Z., Nonmonotone enhanced procimal DC algorithms for a class of strutured nonsmooth DC programming, SIAM J. Optim., 29(4), 2725-2752, (2019)

    work page 2019

  33. [33]

    Programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22, 289-355(1997)

    Pham Dinh Tao, Le Thi Hoai An, Convex analysis approach to D.C. Programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22, 289-355(1997)

    work page 1997

  34. [34]

    Optim., 8, 2, 476-505(1998)

    Pham Dinh T., Le Thi H.A., A DC Optimization algorithm for solving the trust region subproblem, SIAM J. Optim., 8, 2, 476-505(1998)

    work page 1998

  35. [35]

    Pham Dinh, T., Le Thi, H.A. (2014). Recent Advances in DC Programming and DCA. In: Nguyen, NT., Le-Thi, H.A. (eds) Transactions on Computational Intelligence XIII. Lecture Notes in Computer Science, vol 8342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54455-21

    work page doi:10.1007/978-3-642-54455-21 2014

  36. [36]

    Methods of subgradients

    Pham Dinh, T., Souad, E.B.: Algorithms for solving a class of nonconvex optimization prob- lems. Methods of subgradients. In: Hiriart-Urruty, J.B. (ed.) Fermat Days 85: Math- ematics for Optimization, North-Holland Mathematics Studies, vol. 129, pp. 249–271. North-Holland, Amsterdam (1986)

    work page 1986

  37. [37]

    T., Convex Analysis, Princeton University Press, 1970

    Rockafellar R. T., Convex Analysis, Princeton University Press, 1970

    work page 1970

  38. [38]

    T., Penalty methods and augmented Lagrangians in nonlinear programming, Lecture Notes in Comput

    Rockafellar R. T., Penalty methods and augmented Lagrangians in nonlinear programming, Lecture Notes in Comput. Sci., 3, 418-425, Springer-Verlag, Berlin, (1973)

    work page 1973

  39. [39]

    T., Wets R

    Rockafellar R. T., Wets R. J-B., Variational Analysis, Springer, New York (1998)

    work page 1998

  40. [40]

    Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Math., 150

    M. Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Math., 150. Birkhauser Boston, Inc., Boston, MA (1997)

    work page 1997