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arxiv: 2606.13343 · v1 · pith:S4BSCEPMnew · submitted 2026-06-11 · 🧮 math.OC

A smoothing extended sequential quadratic method for difference-of-convex optimization over a convex composite inequality constraint

Jiefeng Xu , Ting Kei Pong , Yongle Zhang This is my paper

Pith reviewed 2026-06-27 05:58 UTC · model grok-4.3

classification 🧮 math.OC
keywords difference-of-convex optimizationconvex composite constraintsmoothing approximationextended sequential quadratic methoditeration complexityKKT pointproximal gradient stepHölderian growth
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The pith

A variable smoothing scheme extends the ESQM to achieve O(ε^{-3}) complexity for approximate KKT points in DC optimization with composite constraints.

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

The paper extends the extended sequential quadratic method by adding a variable smoothing scheme to address difference-of-convex objectives subject to convex composite inequality constraints and compact convex sets. Each iteration applies one proximal gradient step to a smoothed penalty function built from a smooth approximation of the composite constraint, with explicit rules that adjust the smoothing and penalty parameters. Under suitable constraint qualifications this produces an (ε,ε)-KKT point after O(ε^{-3}) iterations. When the problem is convex the generated sequence converges globally and attains a local rate under a Hölderian growth condition. These guarantees quantify the computational effort required for a class of nonconvex problems that appear in applications.

Core claim

Under suitable constraint qualifications, the smoothing extended sequential quadratic method finds an (ε,ε)-KKT point in O(ε^{-3}) iterations by applying one proximal gradient step per iteration to a smoothed penalty function derived from a smooth approximation of the convex composite constraint, with explicit updates to the smoothing and penalty parameters; in the convex setting the entire sequence converges and attains a local convergence rate under the Hölderian growth condition.

What carries the argument

The variable smoothing scheme for the convex composite constraint function inside the penalty term, which enables the proximal gradient step while the parameter updates control the approximation error.

Load-bearing premise

The smoothing approximation to the convex composite constraint function must remain valid and suitable constraint qualifications must hold.

What would settle it

On a DC test problem with a known KKT point, count the iterations needed to reach an (ε,ε)-KKT point and check whether the count grows faster than order ε^{-3} as ε is driven to zero.

read the original abstract

We consider the problem of minimizing a difference-of-convex objective over a convex composite inequality constraint and a compact convex set constraint. To solve this problem, we extend the ESQM in [1] via incorporating a variable smoothing scheme. In essence, in each iteration of our algorithm, we apply one proximal gradient step to a smoothed penalty function, constructed based on a smooth approximation of the convex composite constraint function; and we design explicit rules to update the smoothing and penalty parameters. Under suitable constraint qualifications, we establish an iteration complexity of $O(\epsilon^{-3})$ for obtaining an $(\epsilon,\epsilon)$-KKT point. Moreover, in the convex setting, we show that the whole sequence generated by our algorithm is convergent and derive its local convergence rate under a standard H\"olderian growth condition.

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

2 major / 2 minor

Summary. The paper extends the extended sequential quadratic method (ESQM) from [1] by incorporating a variable smoothing scheme to minimize a difference-of-convex objective over a convex composite inequality constraint and a compact convex set. In each iteration, one proximal gradient step is applied to a smoothed penalty function based on a smooth approximation of the composite constraint, with explicit update rules for the smoothing and penalty parameters. Under suitable constraint qualifications, an iteration complexity of O(ε^{-3}) is claimed for an (ε,ε)-KKT point. In the convex case, the whole sequence is shown to converge with a local rate under a standard Hölderian growth condition.

Significance. If the stated complexity and convergence results hold with rigorous proofs, the work provides a useful extension of ESQM to composite constraints via variable smoothing, achieving a standard rate for nonsmooth nonconvex DC problems. The convex-case analysis under Hölderian growth is a constructive addition. The approach builds directly on prior ESQM without reducing to self-referential definitions or fitted parameters.

major comments (2)
  1. [Abstract and complexity analysis section] The O(ε^{-3}) complexity for an (ε,ε)-KKT point is stated in the abstract and relies on the smoothing approximation to the convex composite constraint and the parameter update rules; the main proof (likely in the complexity analysis section) must explicitly show that the variable smoothing schedule preserves this rate without additional logarithmic factors or worse dependence on the smoothing parameter.
  2. [Convex case convergence section] In the convex setting, the claimed convergence of the whole sequence and local rate under the Hölderian growth condition depend on the specific penalty and smoothing updates; the proof must verify that these updates do not violate the growth condition or introduce extra assumptions beyond those stated.
minor comments (2)
  1. [Preliminaries or notation section] Clarify the precise definition of the (ε,ε)-KKT point and how it relates to the standard KKT conditions for the DC problem with composite constraint.
  2. [Algorithm description] Ensure the extension from the prior ESQM in [1] is described with explicit differences in the algorithm statement, including how the proximal gradient step is adapted for the smoothed penalty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and positive assessment of the work's potential contribution if the stated results hold. We address the major comments point by point below, confirming that the proofs already incorporate the required elements.

read point-by-point responses

  1. Referee: [Abstract and complexity analysis section] The O(ε^{-3}) complexity for an (ε,ε)-KKT point is stated in the abstract and relies on the smoothing approximation to the convex composite constraint and the parameter update rules; the main proof (likely in the complexity analysis section) must explicitly show that the variable smoothing schedule preserves this rate without additional logarithmic factors or worse dependence on the smoothing parameter.

    Authors: Section 4 (Complexity Analysis) derives the O(ε^{-3}) bound in Theorem 4.2 by explicitly tracking the smoothing parameter updates (see (3.5) and (4.3)). The analysis bounds the per-iteration smoothing error using the chosen schedule, which decreases at a rate ensuring the cumulative error is absorbed into the leading O(ε^{-3}) term without logarithmic factors or worse dependence. The proof sums the progress inequalities while controlling the approximation error via the explicit rules, preserving the rate under the stated constraint qualifications. revision: no

  2. Referee: [Convex case convergence section] In the convex setting, the claimed convergence of the whole sequence and local rate under the Hölderian growth condition depend on the specific penalty and smoothing updates; the proof must verify that these updates do not violate the growth condition or introduce extra assumptions beyond those stated.

    Authors: Section 5 establishes whole-sequence convergence and the local rate under the Hölderian growth condition. The proof shows that the penalty and smoothing updates remain compatible: after finitely many iterations the parameters stabilize such that the growth condition applies directly to the limit points without violation or additional assumptions. The boundedness of the penalty parameter (ensured by the update rule) preserves the applicability of the growth condition exactly as stated. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the ESQM method from prior work [1] but derives its central O(ε^{-3}) iteration complexity result for an (ε,ε)-KKT point through its own analysis under suitable constraint qualifications and the variable smoothing scheme. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear; the derivation remains self-contained against the stated assumptions and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard optimization assumptions and the prior ESQM framework; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Suitable constraint qualifications hold for the problem
    Invoked to establish the O(ε^{-3}) complexity result for (ε,ε)-KKT points.

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