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Difference of high-order Moreau envelopes yields an inexact first-order oracle for DC optimization.

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2026-07-01 01:43 UTC pith:2HMUAO6D

load-bearing objection HOME-DC extends high-order Moreau smoothing to DC problems with an inexact oracle and convergence analysis that holds up internally.

arxiv 2606.30991 v1 pith:2HMUAO6D submitted 2026-06-30 math.OC

Difference-of-Convex Optimization via Inexact Smoothing Descent Methods: Difference of High-Order Moreau Envelopes

Alireza Kabgani , Moslem Zamani , Masoud Ahookhosh This is my paper
classification math.OC
keywords difference-of-convex optimizationMoreau envelopesmoothing descentinexact proximal methodsnonconvex optimizationfirst-order oracleconvergence analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces the difference of high-order Moreau envelopes, termed HOME-DC, and derives its basic properties along with first-order information. Proximal-point approximations produce a controllable inexact gradient oracle whose error bounds are established explicitly. These oracles support a family of inexact descent algorithms for general DC problems, with a convergence theory that tolerates approximate subproblem solutions. The construction widens the reach of envelope smoothing to structured nonconvex objectives that arise in clustering and related tasks.

Core claim

We introduce the difference of high-order Moreau envelopes (HOME-DC) and establish its fundamental and differential properties. Approximating the underlying proximal points, we generate an inexact first-order oracle for HOME-DC and characterize its accuracy guarantees. Building upon this oracle, we propose a class of inexact descent methods for minimizing DC functions and provide a convergence analysis.

What carries the argument

The difference of high-order Moreau envelopes (HOME-DC), which supplies a smooth surrogate to the DC objective together with an inexact gradient oracle obtained from proximal approximations.

Load-bearing premise

Proximal points of the convex summands can be approximated accurately enough that the resulting oracle error remains controllable and compatible with the convergence proof of the outer descent iteration.

What would settle it

A DC objective for which no sequence of proximal approximations yields oracle errors small enough to satisfy the descent-method convergence conditions, or a sparse-clustering instance on which the proposed algorithms fail to converge under the stated accuracy tolerances.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • HOME-DC inherits well-defined smoothness and subdifferential properties from its high-order envelope components.
  • Accuracy guarantees on the inexact oracle translate directly into convergence rates for the family of descent methods.
  • The framework accommodates inexact proximal solves, broadening applicability to DC problems where exact proximal operators are unavailable.
  • Numerical behavior on sparse clustering instances aligns with the theoretical convergence guarantees.

Where Pith is reading between the lines

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

  • The same oracle construction could be combined with acceleration schemes to improve practical speed on large-scale DC instances.
  • Error-control techniques developed here may transfer to other envelope-based smoothings that rely on proximal subproblems.
  • The approach suggests a route for embedding HOME-DC surrogates inside branch-and-bound or cutting-plane frameworks for global DC optimization.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 0 minor

Summary. The paper introduces the difference of high-order Moreau envelopes (HOME-DC) for difference-of-convex (DC) optimization. It establishes fundamental and differential properties of HOME-DC, generates an inexact first-order oracle by approximating proximal points of the convex components while characterizing accuracy guarantees, proposes a class of inexact descent methods for DC minimization, and provides a convergence analysis. The framework is tested via preliminary numerical experiments on a sparse clustering problem.

Significance. If the results hold, the work extends envelope-based smoothing techniques to a wider range of structured nonconvex DC problems by explicitly accommodating inexact proximal subproblem solutions. Strengths include the explicit constructions of HOME-DC and its differential properties, the derivation of controlled error bounds for the inexact oracle, and the embedding of those bounds into a standard inexact-oracle convergence framework for the outer descent methods. These elements supply a coherent, falsifiable theoretical foundation together with initial empirical support.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation to accept the manuscript. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces HOME-DC as a difference of high-order Moreau envelopes, establishes its properties from standard convex analysis, constructs an inexact oracle via explicit proximal-point approximations with controlled error bounds, and feeds those bounds into a standard inexact-oracle convergence framework. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The central claims remain self-contained against external benchmarks in convex optimization.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities beyond the newly named HOME-DC object are stated. The work relies on standard domain assumptions of DC programming and proximal-operator theory.

axioms (1)
  • domain assumption The objective is a difference of two convex functions whose proximal operators exist and can be approximated
    Implicit in the construction of the inexact oracle for HOME-DC
invented entities (1)
  • HOME-DC (difference of high-order Moreau envelopes) no independent evidence
    purpose: Smoothing operator for DC functions that admits an inexact first-order oracle
    Newly defined object whose properties are established in the paper

pith-pipeline@v0.9.1-grok · 5658 in / 1231 out tokens · 36427 ms · 2026-07-01T01:43:51.431707+00:00 · methodology

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read the original abstract

This paper studies difference-of-convex (DC) optimization problems through smoothing descent techniques. In particular, we introduce the difference of high-order Moreau envelopes (HOME-DC) and establish its fundamental and differential properties. Approximating the underlying proximal points, we generate an inexact first-order oracle for HOME-DC and characterize its accuracy guarantees. Building upon this oracle, we propose a class of inexact descent methods for minimizing DC functions and provide a convergence analysis. The proposed framework extends the applicability of envelope-based optimization techniques to a broad class of structured nonconvex problems while accommodating inexact solutions to subproblems. Preliminary numerical experiments on a sparse clustering problem demonstrate the approach's practical potential and support the theoretical findings.

Figures

Figures reproduced from arXiv: 2606.30991 by Alireza Kabgani, Masoud Ahookhosh, Moslem Zamani.

Figure 1
Figure 1. Figure 1: Starting from a DC-stationary point x of the original objective, each common subgradient ξ ∈ ∂g(x) ∩ ∂h(x) generates a lifted center s = x+γ q−1Jq(ξ), which is a critical point of the HOME-DC function. In contrast, every critical point of the envelope is projected back to a DC-stationary point of the original problem through the common proximal point x = u(s) = v(s) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Schematic lift/projection relation between DC-stationary points of the original objective and critical points [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A single stationary point of the original DC function may lift to a continuum of critical points of the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance profiles comparing DCA and IDEA with LBFGS directions in terms of CPU Time. [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

discussion (0)

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Works this paper leans on

43 extracted references · 43 canonical work pages · 2 internal anchors

  1. [1]

    Journal of Optimization Theory and Applications 202:475–496 24 A

    Abbaszadehpeivasti H, de Klerk E, Zamani M (2024) On the rate of convergence of the difference-of-convex algorithm (DCA). Journal of Optimization Theory and Applications 202:475–496 24 A. Kabgani, M. Zamani, M. Ahookhosh

    work page 2024

  2. [2]

    Applied Mathemat- ical Modelling 43:170–190

    Ahookhosh M, Ghaderi S (2017) On efficiency of nonmonotone Armijo-type line searches. Applied Mathemat- ical Modelling 43:170–190

    work page 2017

  3. [3]

    Mathematical Programming 208:365–407

    Ahookhosh M, Nesterov Y (2024) High-order methods beyond the classical complexity bounds: inexact high- order proximal-point methods. Mathematical Programming 208:365–407

    work page 2024

  4. [4]

    Optimization 61(4):387–404

    Ahookhosh M, Amini K, Bahrami S (2012) A class of nonmonotone Armijo-type line search method for unconstrained optimization. Optimization 61(4):387–404

    work page 2012

  5. [5]

    SIAM Journal on Optimization (revised positively) URLhttps://doi

    Ahookhosh M, Iusem A, Kabgani A, Lara F (2025) Asymptotic convergence analysis of high-order proximal- point methods beyond sublinear rates. SIAM Journal on Optimization (revised positively) URLhttps://doi. org/10.48550/arXiv.2505.20484

    work page internal anchor Pith review doi:10.48550/arxiv.2505.20484 2025

  6. [6]

    Numerical Algorithms 66(1):49–78

    Amini K, Ahookhosh M, Nosratipour H (2014) An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numerical Algorithms 66(1):49–78

    work page 2014

  7. [7]

    SIAM Journal on Optimization 30(1):980–1006

    Aragon Artacho FJ, Vuong PT (2020) The boosted difference of convex functions algorithm for nonsmooth functions. SIAM Journal on Optimization 30(1):980–1006

    work page 2020

  8. [8]

    Mathematical Programming 169:95–118

    Arag´ on-Artacho FJ, Fleming RMT, Vuong PT (2018) Accelerating the DC algorithm for smooth functions. Mathematical Programming 169:95–118

    work page 2018

  9. [9]

    Set-Valued and Variational Analysis 30(4):1265–1289

    Arag´ on-Artacho FJ, Campoy R, Vuong PT (2022) The boosted dc algorithm for linearly constrained dc programming. Set-Valued and Variational Analysis 30(4):1265–1289

    work page 2022

  10. [10]

    Optimization methods and software 33(1):194–219

    Bagirov AM, Ugon J (2018) Nonsmooth dc programming approach to clusterwise linear regression: optimality conditions and algorithms. Optimization methods and software 33(1):194–219

    work page 2018

  11. [11]

    Optimization Methods and Software 37(1):338–360

    Bajaj A, Mordukhovich BS, Nam NM, Tran T (2022) Solving a continuous multifacility location problem by dc algorithms. Optimization Methods and Software 37(1):338–360

    work page 2022

  12. [12]

    Mathematical programming 178(1):301–326

    Banert S, Bot , RI (2019) A general double-proximal gradient algorithm for dc programming. Mathematical programming 178(1):301–326

    work page 2019

  13. [13]

    Springer International Publishing

    Bauschke H, Combettes P (2017) Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer International Publishing

    work page 2017

  14. [14]

    La Sapienza

    Bock HH (1998) Clustering and neural networks. In: Advances in Data Science and Classification: Proceedings of the 6th Conference of the International Federation of Classification Societies (IFCS-98) Universit` a “La Sapienza”, Rome, 21–24 July, 1998, Springer, pp 265–277

    work page 1998

  15. [15]

    Society for Industrial and Applied Mathematics

    Clarke FH (1990) Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics

    work page 1990

  16. [16]

    Journal of Optimization Theory and Applications 208(2):71

    Ferreira OP, Mordukhovich BS, Santos W, de O Souza JC, et al (2026) An inexact boosted difference of convex algorithm for nondifferentiable functions. Journal of Optimization Theory and Applications 208(2):71

    work page 2026

  17. [17]

    SIAM journal on Numerical Analysis 23(4):707–716

    Grippo L, Lampariello F, Lucidi S (1986) A nonmonotone line search technique for Newton’s method. SIAM journal on Numerical Analysis 23(4):707–716

    work page 1986

  18. [18]

    In: Ponstein J (ed) Convexity and Duality in Optimization, Lecture Notes in Economics and Mathematical Systems, vol 256, Springer, Berlin, pp 37–70

    Hiriart-Urruty JB (1985) Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Ponstein J (ed) Convexity and Duality in Optimization, Lecture Notes in Economics and Mathematical Systems, vol 256, Springer, Berlin, pp 37–70

    work page 1985

  19. [19]

    Journal of mathematical analysis and applications 162(1):196–209

    Hiriart-Urruty JB (1991) How to regularize a difference of convex functions. Journal of mathematical analysis and applications 162(1):196–209

    work page 1991

  20. [20]

    Journal of Optimization Theory and Applications 103(1):1–43

    Horst R, Thoai NV (1999) DC programming: overview. Journal of Optimization Theory and Applications 103(1):1–43

    work page 1999

  21. [21]

    SIAM Journal on Optimization (revised positively) URLhttps: //arxiv.org/abs/2410.19928

    Kabgani A, Ahookhosh M (2024) ItsOPT: An inexact two-level smoothing framework for nonconvex opti- mization via high-order Moreau envelope. SIAM Journal on Optimization (revised positively) URLhttps: //arxiv.org/abs/2410.19928

    work page internal anchor Pith review arXiv 2024

  22. [22]

    Computational Optimization and Applications (revised positively) URLhttps://doi.org/10

    Kabgani A, Ahookhosh M (2025) ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization. Computational Optimization and Applications (revised positively) URLhttps://doi.org/10. 48550/arXiv.2501.02155

    work page arXiv 2025

  23. [23]

    Set-Valued and Variational Analysis 33, 47

    Kabgani A, Ahookhosh M (2025) Moreau envelope and proximal-point methods under the lens of high-order regularization. Set-Valued and Variational Analysis 33, 47

    work page 2025

  24. [24]

    Journal of Optimization Theory and Applications 210, 14

    Kabgani A, Ahookhosh M (2026) On fundamental properties of high-order forward-backward envelope. Journal of Optimization Theory and Applications 210, 14

    work page 2026

  25. [25]

    Nonlinear Analysis: Theory, Methods & Applications 127:157–181

    Kecis I, Thibault L (2015) Moreau envelopes ofs-lower regular functions. Nonlinear Analysis: Theory, Methods & Applications 127:157–181

    work page 2015

  26. [26]

    Annals of Operations Research 133:23–46, DOI 10.1007/s10479-004-5022-1

    Le Thi HA, Pham Dinh T (2005) The dc programming and dca revisited with dc models of real-world nonconvex optimization problems. Annals of Operations Research 133:23–46, DOI 10.1007/s10479-004-5022-1

    work page doi:10.1007/s10479-004-5022-1 2005

  27. [27]

    European Journal of Operational Research 183(3):1067–1085 Difference-of-Convex Optimization via HOME-DC 25

    Le Thi HA, Pham Dinh T (2007) Optimization based dc programming and dca for hierarchical clustering. European Journal of Operational Research 183(3):1067–1085 Difference-of-Convex Optimization via HOME-DC 25

    work page 2007

  28. [28]

    Mathematical Programming 169:5–68, DOI 10.1007/s10107-018-1235-y

    Le Thi HA, Pham Dinh T (2018) DC programming and DCA: thirty years of developments. Mathematical Programming 169:5–68, DOI 10.1007/s10107-018-1235-y

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

  29. [29]

    Advances in Data Analysis and Classification 2(3):259–278

    Le Thi HA, Vo XT, Nguyen VN, Pham Dinh T (2008) Dc programming approach for feature selection in support vector machines learning. Advances in Data Analysis and Classification 2(3):259–278

    work page 2008

  30. [30]

    European Journal of Operational Research 244(1):26–46, DOI 10.1016/j.ejor.2015.01.032

    Le Thi HA, Pham Dinh T, Le HM, Vo XT (2015) Dc approximation approaches for sparse optimization. European Journal of Operational Research 244(1):26–46, DOI 10.1016/j.ejor.2015.01.032

    work page doi:10.1016/j.ejor.2015.01.032 2015

  31. [31]

    Bulletin de la Soci´ et´ e Math´ ematique de France 93:273–299

    Moreau JJ (1965) Proximit´ e et dualit´ e dans un espace Hilbertien. Bulletin de la Soci´ et´ e Math´ ematique de France 93:273–299

    work page 1965

  32. [32]

    Springer Cham

    Nesterov Y (2018) Lectures on Convex Optimization, 2nd edn. Springer Cham

    work page 2018

  33. [33]

    Journal of Global Optimization 61(2):341–361

    Ordin B, Bagirov AM (2015) A heuristic algorithm for solving the minimum sum-of-squares clustering prob- lems. Journal of Global Optimization 61(2):341–361

    work page 2015

  34. [34]

    the Journal of machine Learning research 12:2825–2830

    Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, et al (2011) Scikit-learn: Machine learning in python. the Journal of machine Learning research 12:2825–2830

    work page 2011

  35. [35]

    Acta Mathematica Vietnamica 22(1):289–355

    Pham Dinh T, Le Thi HA (1997) Convex analysis approach to DC programming: theory, algorithms and applications. Acta Mathematica Vietnamica 22(1):289–355

    work page 1997

  36. [36]

    Journal of Optimization Theory and Applications 185:303–326

    Rodomanov A, Nesterov Y (2020) Smoothness parameter of power of Euclidean norm. Journal of Optimization Theory and Applications 185:303–326

    work page 2020

  37. [37]

    arXiv:250304486

    Rotaru T, Patrinos P, Glineur F (2025) Tight analysis of difference-of-convex algorithm (dca) improves con- vergence rates for proximal gradient descent. arXiv:250304486

    work page 2025

  38. [38]

    INFORMS Journal on Optimization 5(4):321–339

    Sun K, Sun XA (2023) Algorithms for difference-of-convex programs based on difference-of-moreau-envelopes smoothing. INFORMS Journal on Optimization 5(4):321–339

    work page 2023

  39. [39]

    Journal of Optimization Theory and Applications 208(1):56

    Sun S (2026) Equivalence of the Polyak- Lojasiewicz-kurdyka exponent via difference-of-moreau-envelope smoothing. Journal of Optimization Theory and Applications 208(1):56

    work page 2026

  40. [40]

    arXiv:240213461

    Tang Y, Zhang S (2024) Approximation analysis for the minimization problem of difference-of-convex functions with moreau envelopes. arXiv:240213461

    work page 2024

  41. [41]

    59th IEEE Conference on Decision and Control (CDC) pp 4967–4702

    Themelis A, Hermans B, Patrinos P (2020) A new envelope function for nonsmooth DC optimization. 59th IEEE Conference on Decision and Control (CDC) pp 4967–4702

    work page 2020

  42. [42]

    Journal of Mathematical Analysis and Applications 157:189–210

    Xu ZB, Roach GF (1991) Characteristic inequalities of uniformly convex and uniformly smooth banach spaces. Journal of Mathematical Analysis and Applications 157:189–210

    work page 1991

  43. [43]

    SIAM Journal on Optimization 14(4):1043–1056

    Zhang H, Hager WW (2004) A nonmonotone line search technique and its application to unconstrained optimization. SIAM Journal on Optimization 14(4):1043–1056

    work page 2004