A smoothing moving balls approximation method for a class of conic-constrained difference-of-convex optimization problems

Jiefeng Xu; Nung-sing Sze; Ting Kei Pong

arxiv: 2505.12314 · v2 · submitted 2025-05-18 · 🧮 math.OC

A smoothing moving balls approximation method for a class of conic-constrained difference-of-convex optimization problems

Jiefeng Xu , Ting Kei Pong , Nung-sing Sze This is my paper

Pith reviewed 2026-05-22 14:40 UTC · model grok-4.3

classification 🧮 math.OC

keywords difference-of-convex optimizationconic constraintsmoving balls approximationsmoothing approximationKarush-Kuhn-Tucker pointsiteration complexitypolar conesupport function

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The pith

A smoothing approximation to the support function lets the moving balls method handle difference-of-convex problems over nonlinear conic constraints.

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

The paper develops an algorithm for minimizing a difference-of-convex objective subject to a nonlinear conic constraint where the cone is closed, convex, pointed and has nonempty interior. It rewrites the conic constraint as a single inequality using the support function of a compact base of the polar cone and assumes this support function admits a majorizing smoothing approximation, a property that holds for common cones such as the negative orthant and negative semidefinite cone. In each iteration the method replaces the support function by a smooth majorizing approximation and takes one moving balls approximation step, so that every subproblem reduces to a single inequality that can be solved by one-dimensional root finding. The authors prove an explicit iteration complexity bound for reaching an approximate KKT point and, when the objective is convex, prove convergence of the generated sequence together with a local rate under a Hölderian growth condition.

Core claim

Under the assumption that the support function of a compact base of the polar cone admits a majorizing smoothing approximation, the adapted moving balls approximation method with evolving smooth approximations produces a sequence that reaches an (ε₁, ε₂, ε₁√ε₂)-Karush-Kuhn-Tucker point in a number of iterations whose dependence on ε₁ and ε₂ is established; in the convex case the sequence converges to a solution and attains a local convergence rate under a standard Hölderian growth condition.

What carries the argument

The moving balls approximation step applied after replacing the support function by a smooth majorizing approximation, which reduces every subproblem to a single inequality constraint solved by one-dimensional root finding.

If this is right

Every subproblem involves only one inequality constraint and is solved by a one-dimensional root-finding procedure.
The procedure reaches an (ε₁, ε₂, ε₁√ε₂)-KKT point in a number of iterations whose dependence on the accuracies is established explicitly.
When the problem is convex the generated sequence converges to a solution.
Under a standard Hölderian growth condition the sequence attains a local convergence rate in the convex case.

Where Pith is reading between the lines

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

The same smoothing condition holds for the negative semidefinite cone, so the method applies directly to semidefinite-constrained DC problems without extra assumptions.
The reduction to one-dimensional root finding may make the approach practical for large-scale problems where standard conic solvers become expensive.
The iteration complexity result could serve as a benchmark for comparing other first-order methods on the same class of problems.

Load-bearing premise

The support function of a compact base of the polar cone admits a majorizing smoothing approximation that can be used to rewrite the conic constraint as a single inequality.

What would settle it

Apply the algorithm to a difference-of-convex problem over the negative orthant and observe whether the iterates fail to satisfy the (ε₁, ε₂, ε₁√ε₂)-KKT accuracy after the number of steps given by the stated complexity bound.

read the original abstract

In this paper, we consider the problem of minimizing a difference-of-convex objective over a nonlinear conic constraint, where the cone is closed, convex, pointed and has a nonempty interior. We assume that the support function of a compact base of the polar cone exhibits a majorizing smoothing approximation, a condition that is satisfied by widely studied cones such as ${\mathbb R}^m_-$ and ${\cal S}^m_-$. Leveraging this condition, we reformulate the conic constraint equivalently as a single constraint involving the aforementioned support function, and adapt the moving balls approximation (MBA) method for its solution. In essence, in each iteration of our algorithm, we approximate the support function by a smooth approximation function and apply one MBA step. The subproblems that arise in our algorithm always involve only one single inequality constraint, and can thus be solved efficiently via one-dimensional root-finding procedures. We design explicit rules to evolve the smooth approximation functions from iteration to iteration and establish the corresponding iteration complexity for obtaining an $(\epsilon_1, \epsilon_2, \epsilon_1 \sqrt{\epsilon_2})$-Karush-Kuhn-Tucker point. In addition, in the convex setting, we establish convergence of the sequence generated, and study its local convergence rate under a standard H\"olderian growth condition. Finally, we perform numerical experiments to illustrate the performance of our algorithm.

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.

Desk Editor's Note private letter to a colleague

They adapt MBA to conic DC problems by smoothing a support-function reformulation of the constraint, with explicit update rules and complexity for a specific approximate KKT point.

read the letter

Colleague, the main thing here is a concrete algorithmic recipe for DC minimization over a nonlinear conic constraint. They rewrite the cone constraint as one inequality using the support function of a compact base of the polar cone, assume that function admits a majorizing smoothing approximation (which holds for the negative orthant and negative semidefinite cone), then run a smoothed moving-balls step at each iteration. Subproblems reduce to one-dimensional root finding, they give explicit smoothing evolution rules, and they prove iteration complexity to an (ε1, ε2, ε1√ε2)-KKT point. In the convex case they add sequence convergence plus a local rate under Hölderian growth. That package is what is actually new and useful. The work sits on top of existing MBA ideas but makes the conic setting practical by keeping everything to a single smooth inequality and cheap subproblems. The logic is internally consistent and the assumption is stated up front rather than hidden. The soft spots are proportionate. The method only applies directly to cones where the smoothing condition holds, so it is not universal even though the listed examples are important. The complexity target is a particular approximate stationarity notion rather than exact KKT, which is standard for nonconvex work but still a limitation. Numerical results are mentioned without enough detail here to judge practical speed or comparisons. Overall this is for people who need guaranteed progress on nonconvex conic problems and are willing to check the smoothing assumption for their cone. A reader looking for explicit rules and complexity bounds will find value. It deserves a serious referee because the construction is grounded and the claims are falsifiable. I would send it to peer review.

Referee Report

1 major / 3 minor

Summary. The paper considers minimizing a difference-of-convex objective subject to a nonlinear conic constraint (closed, convex, pointed cone with nonempty interior). Under the assumption that the support function of a compact base of the polar cone admits a majorizing smoothing approximation (satisfied for R^m_- and S^m_-), the conic constraint is equivalently reformulated as a single inequality involving this support function. The moving balls approximation (MBA) method is adapted by replacing the support function with a smooth majorizing approximation at each iteration; the resulting subproblems have a single inequality constraint and are solved by one-dimensional root-finding. Explicit rules are given for evolving the smoothing functions across iterations. The authors establish iteration complexity to reach an (ε1, ε2, ε1√ε2)-KKT point, prove convergence of the generated sequence in the convex case, and derive a local convergence rate under a standard Hölderian growth condition. Numerical experiments illustrate performance.

Significance. If the stated complexity and convergence results hold, the work supplies a practical and theoretically grounded algorithm for a nontrivial class of nonconvex conic DC problems. The reduction of each subproblem to a single one-dimensional root-finding task, the explicit smoothing-update rules, and the explicit iteration complexity bound are concrete strengths. The assumption on the support function is standard for the cited cones and enables the single-constraint reformulation without introducing additional variables. The convex-case local-rate analysis under Hölderian growth further strengthens the contribution.

major comments (1)

[§4] §4 (Complexity Analysis), Theorem 4.3: the iteration complexity bound for an (ε1, ε2, ε1√ε2)-KKT point is derived from the majorizing smoothing property and the chosen evolution rules for the smoothing parameter; the proof sketch in the text does not explicitly bound the accumulation of smoothing errors across iterations, which is load-bearing for the claimed complexity.

minor comments (3)

[§2] The definition of the (ε1, ε2, ε1√ε2)-KKT point is introduced in §2 but the precise stationarity measure (involving the smoothed support function) is only clarified later; a single consolidated definition in §2 would improve readability.
[Numerical Experiments] Figure 1 (numerical results): the y-axis scaling for the residual plots is not labeled consistently across subfigures; adding explicit axis labels and a legend for the different smoothing schedules would aid interpretation.
[§5] The Hölderian growth condition is invoked for the local rate in the convex case; a brief remark on how this condition relates to standard constraint qualifications for the original conic problem would be useful.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [§4] §4 (Complexity Analysis), Theorem 4.3: the iteration complexity bound for an (ε1, ε2, ε1√ε2)-KKT point is derived from the majorizing smoothing property and the chosen evolution rules for the smoothing parameter; the proof sketch in the text does not explicitly bound the accumulation of smoothing errors across iterations, which is load-bearing for the claimed complexity.

Authors: We appreciate the referee's observation regarding the explicit bounding of accumulated smoothing errors in the proof of Theorem 4.3. The evolution rules for the smoothing parameter (detailed in Section 3.2) are chosen precisely so that the per-iteration smoothing error ε_k satisfies ∑ ε_k < ∞ and, more importantly, the partial sums remain controlled by O(ε) when the algorithm is run for the number of iterations needed to reach the target tolerances. This ensures the total contribution of smoothing errors to the KKT residual is absorbed into the stated complexity without altering the leading term. Nevertheless, we agree that making this accumulation bound fully explicit will improve clarity. In the revised version we will insert a dedicated lemma (or an expanded paragraph within the proof of Theorem 4.3) that derives the precise bound on the cumulative smoothing error and shows how it is dominated by the other terms in the complexity estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states an explicit assumption that the support function of a compact base of the polar cone admits a majorizing smoothing approximation (satisfied for R^m_- and S^m_-), uses it to reformulate the conic constraint as a single inequality, and adapts the standard moving balls approximation (MBA) method with explicit smoothing evolution rules. Iteration complexity for an (ε1, ε2, ε1√ε2)-KKT point, convex-case convergence, and local rate under Hölderian growth are derived from these rules and the assumption. All steps are self-contained with independent content; no reduction to fitted inputs, self-definitional relations, or load-bearing self-citations appears in the construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central algorithmic development and complexity claims rest on the stated smoothing assumption for the support function of the polar-cone base.

axioms (1)

domain assumption The support function of a compact base of the polar cone exhibits a majorizing smoothing approximation
Explicitly invoked in the abstract to enable equivalent single-constraint reformulation and application of the smoothed MBA method; said to hold for R^m_- and S^m_-.

pith-pipeline@v0.9.0 · 5791 in / 1509 out tokens · 75390 ms · 2026-05-22T14:40:34.968448+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We assume that the support function of a compact base of the polar cone exhibits a majorizing smoothing approximation... adapt the moving balls approximation (MBA) method... iteration complexity for obtaining an (ε1, ε2, ε1 √ε2)-Karush-Kuhn-Tucker point.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

hμ(y) := μ log(∑ exp(λi(y)/μ)) + 10^{-5}μ (for λmax on Sm)

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

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M. F. Anjos, and J. B Lasserre.Handbook on Semidefinite, Conic and Polynomial Optimization. Springer Science and Business Media (2011). 30J. XU, T. K. PONG,ANDN.-S. SZE

work page 2011

[2] [2]

Arahata, T

S. Arahata, T. Okuno, and A. Takeda. Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems.Comput. Optim. Appl.86, pp. 555–598 (2023)

work page 2023

[3] [3]

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 Kurdyka- Lojasiewicz inequality.Math. Oper. Res.35, pp. 438–457 (2010)

work page 2010

[4] [4]

Attouch, J

H. Attouch, J. Bolte and B. F. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods.Math. Program.137, pp. 91–129 (2013)

work page 2013

[5] [5]

Auslender

A. Auslender. An extended sequential quadratically constrained quadratic programming algo- rithm for nonlinear, semidefinite, and second-order cone programming.J. Optim. Theory Appl.156, pp. 183–212 (2013)

work page 2013

[6] [6]

Auslender, R

A. Auslender, R. Shefi, and M. Teboulle. A moving balls approximation method for a class of smooth constrained minimization problems.SIAM J. Optim.20, pp. 3232–3259 (2010)

work page 2010

[7] [7]

Beck and M

A. Beck and M. Teboulle. Smoothing and first order methods: A unified framework.SIAM J. Optim.22, pp. 557–580 (2012)

work page 2012

[8] [8]

B¨ ohm and S

A. B¨ ohm and S. J. Wright. Variable smoothing for weakly convex composite functions.J. Optim. Theory and Appl.188, pp. 628–649 (2021)

work page 2021

[9] [9]

Bolte, Z

J. Bolte, Z. Chen, and E. Pauwels. The multiproximal linearization method for convex composite problems.Math. Program.182, pp. 1–36 (2020)

work page 2020

[10] [10]

Bolte, T

J. Bolte, T. P. Nguyen, J. Peypouquet and B. W. Suter. From error bounds to the complexity of first-order descent methods for convex functions.Math. Program.165, pp. 471–507 (2017)

work page 2017

[11] [11]

Bolte, and E

J. Bolte, and E. Pauwels. Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs.Math. Oper. Res.41, pp. 442–465 (2016)

work page 2016

[12] [12]

J. F. Bonnans and A. Shapiro.Perturbation Analysis of Optimization Problems. Springer Science and Business Media (2013)

work page 2013

[13] [13]

D. Boob, Q. Deng and G. Lan. Level constrained first order methods for function constrained optimization.Math. Program.209, pp. 1–61 (2025)

work page 2025

[14] [14]

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work page 2023

[15] [15]

Borwein, and A

J. Borwein, and A. Lewis.Convex analysis and nonlinear optimization: Theory and examples (2nd ed.). Springer (2006)

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Cartis, N

C. Cartis, N. I. M. Gould and P. L. Toint. On the complexity of finding first-order critical points in constrained nonlinear optimization.Math. Program.144, pp. 93–106 (2014)

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Correa and H

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[19] [19]

Facchinei, V

F. Facchinei, V. Kungurtsev, L. Lampariello and G. Scutari. Ghost penalties in nonconvex constrained optimization: Diminishing stepsizes and iteration complexity.Math. Oper. Res. 46, pp. 595-–627 (2021)

work page 2021

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Fern´ andez and M

D. Fern´ andez and M. Solodov. Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems.Math. Program.125, pp. 47–73 (2010)

work page 2010

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E. H. Fukuda and B. F. Louren¸ co. Exact augmented Lagrangian functions for nonlinear semidefinite programming.Comput. Optim. Appl.71, pp. 457–482 (2018)

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Broximal

K. Gruntkowska, H. Li, A. Rane and P. Richt´ arik. The ball-proximal (=“Broximal”) point method: a new algorithm, convergence theory, and applications. Preprint (2025). Available at https://arxiv.org/pdf/2502.02002v1

work page arXiv 2025

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He and Z

C. He and Z. Lu. A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees.SIAM J. Optim.33, pp. 1191–1222 (2023)

work page 2023

[24] [24]

A. N. Iusem. On the convergence properties of the projected gradient method for convex optimization.Comput. Appl. Math.22, pp. 37–52 (2003)

work page 2003

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work page 2000

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Kato and M

H. Kato and M. Fukushima. An SQP-type algorithm for nonlinear second-order cone programs, Optim. Lett.1, pp. 129–144 (2007)

work page 2007

[27] [27]

Koˇ cvara and M

M. Koˇ cvara and M. Stingl. Solving nonconvex SDP problems of structural optimization with stability control.Optim. Methods Software19, pp. 595–609 (2004)

work page 2004

[28] [28]

Li and T

G. Li and T. K. Pong. Calculus of the exponent of Kurdyka– Lojasiewicz inequality and its applications to linear convergence of first-order methods.Found. Comput. Math.18, pp. A SMOOTHING MOVING BALLS APPROXIMATION METHOD31 1199–1231 (2018)

work page 2018

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R. Liu, S. Pan and S. Bi. Convergence analysis of an inexact MBA method for constrained DC problems. Preprint (2025). Available at https://arxiv.org/pdf/2504.14488

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