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arxiv: 2605.24954 · v1 · pith:D7RYLPHUnew · submitted 2026-05-24 · 🧮 math.OC

An Infeasible Method with Feasibility Safeguard for Nonsmooth Composite Optimization Over Manifolds

Xiyua Xie , Qia Li This is my paper

Pith reviewed 2026-06-29 23:42 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonsmooth composite optimizationmanifold constraintsproximal linearized methodfeasibility safeguarditeration complexityKurdyka-Łojasiewicz propertysparse PCAconvergence analysis
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The pith

FSIPL method delivers O(ε^{-2}) complexity for nonsmooth optimization on manifolds with built-in feasibility control

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

This paper introduces the feasibility-safeguarded inexact proximal linearized (FSIPL) method for nonsmooth composite optimization problems constrained to compact embedded submanifolds. The approach permits iterates that violate the manifold constraints to a limited degree while enforcing a bound on how far they stray through safeguards in the correction step and line search. It establishes that backtracking terminates after finitely many steps, that limit points are stationary, and that the number of outer iterations needed to reach epsilon accuracy is bounded by a constant times one over epsilon squared. An additional Kurdyka-Łojasiewicz property on an auxiliary function yields convergence of the entire sequence. These results apply directly to tasks such as sparse principal component analysis and spectral clustering.

Core claim

The FSIPL method achieves an O(ε^{-2}) outer iteration complexity bound, finite termination of backtracking, subsequential convergence to stationary points, and full-sequence convergence under a Kurdyka-Łojasiewicz assumption on a suitable auxiliary function, with the feasibility safeguard making boundedness of iterates a consequence of the design.

What carries the argument

The feasibility-safeguarded inexact proximal linearized (FSIPL) method, which approximately solves strongly convex proximal linearized subproblems, applies a correction step, and uses merit-function-based nonmonotone backtracking line search with safeguards to control infeasibility.

Load-bearing premise

The feasibility safeguard incorporated into the correction step and line search is sufficient to make boundedness of iterates a consequence of the algorithmic design rather than an a priori assumption.

What would settle it

Running the algorithm on a specific instance and checking if the distance of iterates to the manifold remains below the prescribed bound throughout, or verifying finite termination by counting backtracking steps until acceptance.

read the original abstract

In this paper, we consider nonsmooth composite optimization over compact embedded submanifolds defined by nonlinear equality constraints. We propose a feasibility-safeguarded inexact proximal linearized method (FSIPL), which allows infeasible iterates while keeping them within a prescribed bounded neighborhood of the manifold. Each iteration approximately solves a strongly convex proximal linearized subproblem, performs a correction step to reduce constraint violation, and uses a merit-function-based nonmonotone backtracking line search to select stepsizes and accept trial iterates. The feasibility safeguard, incorporated into both correction and line search, controls infeasibility and makes the boundedness needed in the analysis a consequence of the algorithmic design. We prove finite termination of backtracking, subsequential convergence to stationary points, and an $O(\varepsilon^{-2})$ outer iteration complexity bound. Under a Kurdyka--\L{}ojasiewicz assumption on a suitable auxiliary function, we further establish full-sequence convergence. Numerical results on sparse PCA and sparse spectral clustering illustrate its efficiency.

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

0 major / 2 minor

Summary. The manuscript proposes the feasibility-safeguarded inexact proximal linearized (FSIPL) method for nonsmooth composite optimization over compact embedded submanifolds defined by nonlinear equality constraints. The algorithm permits infeasible iterates within a prescribed bounded neighborhood of the manifold, approximately solves a strongly convex proximal linearized subproblem at each step, applies a correction step to reduce constraint violation, and selects stepsizes via a merit-function-based nonmonotone backtracking line search that incorporates the feasibility safeguard. The analysis establishes finite termination of the backtracking procedure, subsequential convergence to stationary points, an O(ε^{-2}) outer iteration complexity bound, and full-sequence convergence under a Kurdyka-Łojasiewicz assumption on a suitable auxiliary function. Numerical results on sparse PCA and sparse spectral clustering are included.

Significance. If the derivations hold, the contribution lies in extending proximal linearized methods to manifold settings while relaxing strict feasibility at every iterate; the feasibility safeguard is shown to make boundedness a direct algorithmic consequence rather than an a priori assumption. The O(ε^{-2}) complexity and KL-based full-sequence result are obtained via standard arguments, and the explicit constructions for the correction step and safeguarded line search close the relevant gaps. The numerical illustrations on sparse PCA and clustering provide concrete evidence of practical performance.

minor comments (2)
  1. A short remark in the numerical experiments section on how the radius of the prescribed neighborhood is selected in practice would improve reproducibility.
  2. The abstract contains a minor LaTeX formatting inconsistency in the rendering of 'Kurdyka--Łojasiewicz' that should be standardized for the final version.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. The report accurately captures the FSIPL method, its feasibility safeguard, complexity bound, and convergence results.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its O(ε^{-2}) complexity, finite backtracking termination, subsequential convergence, and KL-based full-sequence convergence through explicit constructions of the proximal linearized subproblem, correction step, and merit-function line search incorporating the feasibility safeguard. Boundedness of iterates follows directly as an algorithmic consequence from the prescribed neighborhood control and line-search acceptance criteria, without any reduction to fitted inputs, self-definitional quantities, or load-bearing self-citations. All steps rely on standard proximal and line-search arguments plus the external KL assumption on an auxiliary function, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the problem setting being a compact embedded submanifold and on the KL property for the stronger convergence result; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The feasible set is a compact embedded submanifold defined by nonlinear equality constraints.
    Explicitly stated as the problem domain in the abstract.
  • domain assumption Kurdyka-Łojasiewicz property holds for a suitable auxiliary function.
    Invoked to obtain full-sequence convergence.

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