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arxiv: 1906.09558 · v1 · pith:HSQUKY26new · submitted 2019-06-23 · 🧮 math.OC

New sharp necessary optimality conditions for mathematical programs with equilibrium constraints

Helmut Gfrerer , Jane J. Ye This is my paper

Pith reviewed 2026-05-25 18:09 UTC · model grok-4.3

classification 🧮 math.OC
keywords mathematical programs with equilibrium constraintsnecessary optimality conditionsM-stationarityregular normal coneMPCC reformulationgeneralized equations
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The pith

A new necessary optimality condition for MPECs is sharper than M-stationarity and holds without constraint qualifications on the MPCC reformulation.

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

The paper reformulates mathematical programs with equilibrium constraints as programs involving a parametric generalized equation with the regular normal cone. From this modeling choice it derives a necessary optimality condition that is strictly stronger than the usual M-stationary condition. The new condition remains valid precisely in cases where the standard MPCC reformulation possesses no constraint qualification at all. A sympathetic reader cares because the result enlarges the class of equilibrium problems for which local optimality can be certified without invoking auxiliary regularity assumptions that frequently fail in applications.

Core claim

We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with complementarity constraints (MPCC) reformulation.

What carries the argument

The reformulation of the MPEC as a mathematical program with a parametric generalized equation involving the regular normal cone.

If this is right

  • The new condition can certify optimality at points where the classical M-stationary condition is silent.
  • Analysis of equilibrium problems becomes possible without first verifying constraint qualifications on any complementarity reformulation.
  • The approach applies directly to the original equilibrium formulation rather than requiring an intermediate MPCC model.

Where Pith is reading between the lines

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

  • The same modeling device may yield sharper conditions for related classes of problems governed by variational inequalities.
  • Numerical schemes that solve the generalized-equation formulation could inherit the sharper stationarity test without extra qualification checks.
  • The result suggests examining whether other normal-cone choices produce intermediate-strength conditions between M- and S-stationarity.

Load-bearing premise

The equilibrium constraint must be expressible as a parametric generalized equation that uses the regular normal cone.

What would settle it

An explicit MPEC instance together with a feasible point that satisfies all the new stationarity conditions yet is not a local minimizer would falsify the necessity claim.

read the original abstract

In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with complementarity constraints (MPCC) reformulation.

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 paper studies mathematical programs with equilibrium constraints (MPECs) reformulated as mathematical programs with parametric generalized equations that involve the regular normal cone. It derives a new necessary optimality condition claimed to be strictly sharper than the standard M-stationary condition while remaining valid even in the absence of constraint qualifications for the corresponding MPCC reformulation.

Significance. If the derivation holds, the result strengthens the necessary optimality theory for MPECs by providing a tighter stationarity condition that does not rely on MPCC constraint qualifications. The modeling choice of the regular normal cone is explicitly used to achieve this improvement, which could have implications for both theoretical analysis and the design of numerical methods in variational analysis and equilibrium-constrained optimization.

minor comments (3)
  1. [Abstract] The abstract states that the new condition is 'sharper than the usual M-stationary condition,' but does not indicate the precise sense in which sharpness is measured (e.g., inclusion of multipliers or stricter complementarity). Adding one sentence clarifying this would help readers immediately grasp the improvement.
  2. [Introduction] Notation for the regular normal cone and the parametric generalized equation is introduced without an explicit forward reference to the section where the full definition and standing assumptions appear. A brief pointer in the introduction would improve readability.
  3. [Section 4] The manuscript would benefit from a short comparison table or paragraph contrasting the new condition with M-stationarity, S-stationarity, and C-stationarity under the same example, even if only to illustrate the claimed strict improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, which accurately summarizes our main contribution. The recommendation for minor revision is noted. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates the MPEC as a parametric generalized equation with the regular normal cone and derives a new necessary optimality condition claimed to be sharper than M-stationarity while holding without MPCC constraint qualifications. This modeling choice is stated explicitly as the basis for the derivation rather than being smuggled in or defined circularly. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the abstract or stated claims. The argument rests on standard variational-analysis properties of the regular normal cone and is self-contained; the central result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full text unavailable for detailed audit of assumptions.

axioms (1)
  • domain assumption The regular normal cone appears in the parametric generalized equation that defines the MPEC.
    This is the central modeling choice stated in the abstract that enables the new optimality condition.

pith-pipeline@v0.9.0 · 5573 in / 1184 out tokens · 28265 ms · 2026-05-25T18:09:11.790594+00:00 · methodology

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Reference graph

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