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arxiv: 2604.18858 · v1 · submitted 2026-04-20 · 🧮 math.OC

A semi-smooth Newton method for the nonlinear conic problem with generalized simplicial cones

Nicolas F. Armijo , Yunier Bello Cruz , Gabriel Haeser This is my paper

Pith reviewed 2026-05-10 03:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords semi-smooth Newton methodgeneralized simplicial conesnonlinear conic programmingprojection operatorquadratic convergenceKKT conditionsconic projection equations
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The pith

A semi-smooth Newton method solves the conic projection equations for nonlinear programs over generalized simplicial cones with local quadratic convergence.

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

The paper develops a semi-smooth Newton method for nonlinear conic programming problems restricted to generalized simplicial cones, which are linear images of symmetric cones. It rewrites the KKT conditions as conic projection equations that generalize Robinson's normal map. Strong semi-smoothness of the projection operator onto these cones is established, which directly supplies the local quadratic convergence of the Newton iteration. Numerical tests on circular-cone programs and low-rank matrix completion illustrate practical performance against smoothing alternatives.

Core claim

The projection operator onto a generalized simplicial cone is strongly semi-smooth. Consequently, the semi-smooth Newton iteration applied to the conic projection equations, which are equivalent to the KKT system of the nonlinear conic program, converges quadratically to a solution in a neighborhood of that solution.

What carries the argument

The conic projection equations (generalization of Robinson's normal equations) together with the strong semi-smoothness property of the projection operator onto generalized simplicial cones.

If this is right

  • The iteration converges quadratically near a solution whenever the projection is strongly semi-smooth.
  • The same framework applies directly to circular cone programming.
  • The method extends to low-rank matrix completion problems cast as nonlinear conic programs.
  • The approach supplies an alternative to smoothing Newton methods whose local rate is also quadratic but whose smoothing parameter must be driven to zero.

Where Pith is reading between the lines

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

  • If many applied cones can be represented as generalized simplicial cones, the same proof technique may transfer to other structured cones whose projections admit explicit descriptions.
  • Global convergence safeguards or merit-function line searches could be added without altering the local quadratic rate already established.
  • The exact equivalence to KKT conditions means any limit point satisfies first-order optimality for the original conic program.

Load-bearing premise

The projection operator onto generalized simplicial cones is strongly semi-smooth.

What would settle it

A concrete point in a generalized simplicial cone at which the projection operator fails the strong semi-smoothness inequality would invalidate the quadratic-convergence claim.

Figures

Figures reproduced from arXiv: 2604.18858 by Gabriel Haeser, Nicolas F. Armijo, Yunier Bello Cruz.

Figure 1
Figure 1. Figure 1: Stationary point for θ(x, λ) with a non optimal and a optimal multiplier. We conclude this subsection with a remark on the role of strongly stationary points [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
read the original abstract

In this work we develop and analyze a semi-smooth Newton method for the general nonlinear conic programming problem. In particular, we study the problem with a generalized simplicial cone, i.e., the image of a symmetric cone under a linear mapping. We generalize Robinson's normal equations to a conic setting, yielding what we call the conic projection equations. The resulting system is equivalent to the KKT conditions associated with the nonlinear conic programming problem. A semi-smooth Newton iteration is proposed for solving it, and local quadratic convergence is established. We study properties of generalized simplicial cones and prove strong semi-smoothness of the projection operator onto them. Numerical experiments compare the method against a recent smoothing Newton approach on the circular cone programming problem, and we also apply it to the low-rank matrix completion problem.

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 / 3 minor

Summary. The manuscript develops a semi-smooth Newton method for the nonlinear conic programming problem with generalized simplicial cones (linear images of symmetric cones). It generalizes Robinson's normal equations to conic projection equations, establishes their equivalence to the KKT conditions, proposes a semi-smooth Newton iteration for solving the system, proves local quadratic convergence, studies properties of generalized simplicial cones, and establishes strong semi-smoothness of the associated projection operator. Numerical experiments compare the method to a smoothing Newton approach on circular cone programs and apply it to low-rank matrix completion.

Significance. If the central proof of strong semi-smoothness holds, the work extends semi-smooth Newton methods to a useful class of non-self-dual cones while preserving quadratic local convergence. The equivalence to KKT systems and the numerical validation on circular cones and matrix completion add practical value. The generalization of Robinson's framework and the explicit semi-smoothness result are the primary contributions.

minor comments (3)
  1. [Abstract] Abstract: the statement that the method is applied to the low-rank matrix completion problem would benefit from a brief indication of the cone used and the scale of the instances solved.
  2. [Section 5] Section 5 (numerical experiments): the comparison tables would be strengthened by reporting iteration counts, CPU times, and final residuals for both methods on identical instances and tolerances.
  3. [Preliminaries] Notation: the definition of the linear mapping that generates the generalized simplicial cone should be stated explicitly once in the preliminaries and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We appreciate the recognition of the generalization of Robinson's framework to conic projection equations, the equivalence to KKT conditions, the local quadratic convergence result, and the numerical applications to circular cone programs and low-rank matrix completion. Since no specific major comments were provided in the report, we have no point-by-point responses to address. We will incorporate minor improvements to the presentation and clarity in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation generalizes Robinson's normal equations (an external reference) to conic projection equations, establishes equivalence to KKT conditions, proposes the semi-smooth Newton iteration, and proves strong semi-smoothness of the projection onto generalized simplicial cones via their structural properties. None of these steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the quadratic convergence claim rests on an independent proof of the semi-smoothness property.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard optimization theory for KKT conditions and semi-smooth Newton methods, plus the specific structural properties of generalized simplicial cones.

axioms (2)
  • domain assumption The conic projection equations are equivalent to the KKT conditions of the nonlinear conic programming problem.
    This equivalence is the foundation for applying the Newton method to the reformulated system.
  • domain assumption The projection operator onto generalized simplicial cones is strongly semi-smooth.
    This property is stated as proven and is required for establishing local quadratic convergence.

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