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arxiv: 2402.16439 · v3 · submitted 2024-02-26 · 🧮 math.OC

Solving Nonlinear Absolute Value Equations

Aris Daniilidis (VADOR) , Mounir Haddou (IRMAR) , Tri Minh Le (VADOR) , Olivier Ley (IRMAR) , Phi Hoang Tran (IRMAR) This is my paper

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classification 🧮 math.OC
keywords nonlinear absolute value equationsnonlinear complementarity problemssmoothing regularizationLojasiewicz inequalityerror boundsoptimizationdifferential equations
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The pith

Nonlinear absolute value equations can be reformulated as nonlinear complementarity problems and solved using smoothing regularization under mild assumptions.

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

The paper shows that problems naturally expressed as nonlinear absolute value equations can be rewritten as nonlinear complementarity problems. These reformulated problems are then solved numerically by smoothing regularization techniques. The approach is presented as the first direct numerical treatment of such equations. A common technical assumption in smoothing methods is proved equivalent to the classical Łojasiewicz inequality at infinity. Error estimates from complementarity solvers are extended to the original equations under weaker conditions, with examples in asymmetric ridge optimization and nonlinear ordinary differential equations.

Core claim

Nonlinear absolute value equations can be reformulated as nonlinear complementarity problems and efficiently solved using smoothing regularization techniques under mild assumptions. The technical assumption commonly used in smoothing is equivalent to the Łojasiewicz inequality at infinity. Established error estimates for NCP solvers extend to NAVE problems under weaker assumptions.

What carries the argument

Reformulation of nonlinear absolute value equations into nonlinear complementarity problems, solved via smoothing regularization.

If this is right

  • Several problems represented as nonlinear absolute value equations become solvable as nonlinear complementarity problems.
  • Error bounds for nonlinear complementarity problem solvers extend to nonlinear absolute value equations under weaker assumptions.
  • The method applies to asymmetric ridge optimization problems.
  • The method applies to nonlinear ordinary differential equations.

Where Pith is reading between the lines

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

  • Existing numerical solvers for complementarity problems could now be applied directly to a new class of absolute-value problems.
  • The equivalence proof may allow similar smoothing techniques to be justified in other contexts where the Łojasiewicz inequality appears.
  • Practical performance on ridge optimization suggests the reformulation could reduce computational cost in related absolute-value models.

Load-bearing premise

The reformulation of nonlinear absolute value equations into nonlinear complementarity problems is valid and the smoothing regularization applies under the mild assumptions stated.

What would settle it

A concrete nonlinear absolute value equation where the reformulation to a complementarity problem fails, or where the smoothing method does not converge under the stated mild assumptions, would disprove the central claim.

Figures

Figures reproduced from arXiv: 2402.16439 by Aris Daniilidis (VADOR), Mounir Haddou (IRMAR), Olivier Ley (IRMAR), Phi Hoang Tran (IRMAR), Tri Minh Le (VADOR).

Figure 1
Figure 1. Figure 1: Problem in dimension m = 20 and d = 40. 3.2 Nonlinear ordinary differential equations A NAVE problem also naturally arises when we deal with a discretization of a nonlinear ordinary differential equation (ODE, for short) involving rough velocity, for example ˙γ(t) = p |γ(t)| as well as an ODE of the form Φ(X(2k) , X(2k−1) , . . . , X˙ ) = |X| In this subsection we provide two examples (one being a stiff OD… view at source ↗
Figure 2
Figure 2. Figure 2: (b), we apply difference mesh sizes in the same time interval [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence rate O(h) for a boundary value problem Example 3.2. For a continuous function f : [0, +∞) → R, we consider an ODE ( x¨ + arctan(x) − |x| = f(t), t > 0, x(0) = x0 ∈ R, x˙(0) = 0. (3.8) Using a similar discretization as in Example 3.1, the unknown variable x ≈ x(t) solves a NAVE problem as follows Ax + arctan(x) − |x| = b, (3.9) where the matrix A is determined as in Example 3.1 and the vector b … view at source ↗
Figure 4
Figure 4. Figure 4: Solving equation 3.8. 3.3 Comparison of methods for NAVE Instead of smoothing procedure considered in Section 2, one can solve a NCP via other numerical methods. In this subsection we give examples to compare the efficiency of four methods • Newton–like method with smoothing functions θ1 and θ2; • approximating by Soft–Max function, in which the main idea is to approximate the comple￾mentarity condition vi… view at source ↗
Figure 5
Figure 5. Figure 5: Performance time. 4 Conclusion and discussion In this work, we applied smoothing techniques commonly used for Nonlinear Complementarity Problems (NCP) to Nonlinear Absolute Value Equations (NAVE). We first showed that a NAVE can be formulated as a NCP with an implicitly known corresponding mapping. We then established that a NAVE can be effectively addressed under a mild direct assumption on the NAVE funct… view at source ↗
read the original abstract

In this work, we show that several problems naturally represented as Nonlinear Absolute Value Equations (NAVE) can be reformulated as Nonlinear Complementarity Problems (NCP) and efficiently solved using smoothing regularization techniques under mild assumptions. As far as we know, this is the first numerical approach that directly deals with NAVE. We also identify a technical assumption commonly utilized in smoothing techniques and prove its equivalence to a classical __ojasiewicz inequality at infinity, validating its non-restrictive nature. Furthermore, we extend established error estimates for NCP solvers to derive error bounds for NAVE problems under weaker assumptions. We illustrate the effectiveness of our approach through applications including asymmetric ridge optimization and nonlinear ordinary differential equations.

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 paper claims that Nonlinear Absolute Value Equations (NAVE) can be reformulated as Nonlinear Complementarity Problems (NCP) and solved via smoothing regularization under mild assumptions, providing the first direct numerical method for NAVE. It proves equivalence between a standard smoothing assumption and the classical Łojasiewicz inequality at infinity, extends NCP error bounds to NAVE under weaker conditions, and demonstrates the approach on asymmetric ridge optimization and nonlinear ODEs.

Significance. If the reformulation preserves solutions and the equivalence holds, the work supplies a practical numerical framework for NAVE instances that arise in optimization and differential equations, together with validated assumptions and extended error bounds. The applications supply concrete evidence of utility.

minor comments (2)
  1. [Abstract] Abstract: the placeholder '__ojasiewicz' should be corrected to 'Łojasiewicz'.
  2. [Abstract] Abstract: the phrase 'mild assumptions' is used without enumeration; a parenthetical list of the key conditions would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the provided report, so we have no point-by-point responses to prepare. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core contribution is a reformulation of NAVE into NCP, followed by smoothing regularization whose key assumption is proven equivalent to the classical Łojasiewicz inequality at infinity, plus extension of existing NCP error bounds to NAVE under weaker conditions. These steps consist of explicit mathematical equivalences and proofs that stand independently of the target result; no parameter is fitted and then renamed as a prediction, no self-citation chain bears the central claim, and no ansatz or uniqueness statement is smuggled in via prior work by the same authors. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the NAVE-to-NCP reformulation under mild assumptions and the equivalence of the smoothing assumption to the Łojasiewicz inequality; these are presented as proven in the paper rather than fitted or invented ad hoc.

axioms (1)
  • domain assumption The technical assumption commonly used in smoothing techniques for NCP is equivalent to the classical Łojasiewicz inequality at infinity
    Identified and proven equivalent in the paper per the abstract, validating its non-restrictive nature.

pith-pipeline@v0.9.0 · 5661 in / 1187 out tokens · 21792 ms · 2026-05-24T04:18:17.277060+00:00 · methodology

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