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arxiv: 2603.23389 · v2 · submitted 2026-03-24 · 🧮 math.OC

Piecewise M-Stationarity and Related Algorithms for Mathematical Programs with Complementarity Constraints

Kexin Wang , Lorenz T. Biegler This is my paper

Pith reviewed 2026-05-15 00:29 UTC · model grok-4.3

classification 🧮 math.OC
keywords piecewise M-stationarityB-stationarityMPCCmathematical programs with complementarity constraintsNCP reformulationMPCC-ACQbounding algorithms
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The pith

Piecewise M-stationarity is equivalent to B-stationarity for MPCCs under the MPCC-ACQ condition.

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

This paper introduces piecewise M-stationarity as a way to characterize stationarity in mathematical programs with complementarity constraints, especially when some complementarities are biactive. It proves that this new notion is equivalent to the standard B-stationarity condition as long as the MPCC-ACQ regularity assumption holds. The authors then use the equivalence to study convergence of their earlier NCP-based bounding methods without needing the stronger MPCC-LICQ condition. The same link produces a cheaper way to verify B-stationarity in practice and explains why the methods avoid unbounded multipliers at non-strongly stationary points.

Core claim

We propose the concept of piecewise M-stationarity and prove its equivalence to B-stationarity under MPCC-ACQ. We investigate convergence properties of the NCP-based bounding methods we proposed in [31], without requiring MPCC-LICQ; an interpretation of the algorithm's behavior together with the concept of piecewise M-stationarity leads to a cost reduction in B-stationarity verification. In addition, practical issues related to convergence to non-strongly stationary solutions are discussed, which shows that the NCP-based complementarity reformulations have an advantage in avoiding unbounded multipliers near these solutions.

What carries the argument

Piecewise M-stationarity, obtained by partitioning biactive constraints into strongly and weakly active sets and requiring stationarity on each piece.

If this is right

  • B-stationarity verification reduces to checking piecewise M-stationarity on each active piece.
  • NCP-based bounding algorithms converge to B-stationary points even when MPCC-LICQ does not hold.
  • The methods remain well-behaved near non-strongly stationary solutions because multipliers stay bounded.
  • Algorithms can handle biactive complementarity sets without additional regularization.

Where Pith is reading between the lines

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

  • The equivalence may allow reuse of existing nonlinear-programming solvers for MPCC verification tasks.
  • Similar piecewise ideas could be tested on equilibrium problems with equilibrium constraints that lack strong regularity.
  • Numerical tests on contact or traffic models could quantify the claimed reduction in verification cost.

Load-bearing premise

The MPCC-ACQ condition must hold for piecewise M-stationarity to coincide with B-stationarity.

What would settle it

An explicit MPCC example where MPCC-ACQ fails yet a point satisfies piecewise M-stationarity but not B-stationarity.

read the original abstract

This study explores B-stationarity of mathematical programs with complementarity constraints (MPCCs) and convergence behavior of MPCC algorithms. Special attention is given to the cases with biactive complementarity constraints. First, we propose the concept of piecewise M-stationarity and prove its equivalence to B-stationarity under MPCC-ACQ. Then, we investigate convergence properties of the NCP-based bounding methods we proposed in [31], without requiring MPCC-LICQ; an interpretation of the algorithm's behavior together with the concept of piecewise M-stationarity leads to a cost reduction in B-stationarity verification. In addition, practical issues related to convergence to non-strongly stationary solutions are discussed, which shows that the NCP-based complementarity reformulations have an advantage in avoiding unbounded multipliers near these solutions.

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 manuscript proposes the concept of piecewise M-stationarity for mathematical programs with complementarity constraints (MPCCs), proves its equivalence to B-stationarity under the MPCC-Abadie constraint qualification (MPCC-ACQ), and analyzes the convergence of NCP-based bounding methods from the authors' prior work [31] without requiring MPCC-LICQ. It argues that this combination yields a cost reduction in B-stationarity verification and discusses practical advantages of the reformulations in handling non-strongly stationary solutions with biactive constraints.

Significance. If the equivalence and convergence results hold, the work offers a practical tool for verifying B-stationarity in MPCCs via a weaker notion that aligns with standard CQ assumptions, potentially reducing computational effort in cases where MPCC-LICQ fails. The emphasis on biactive sets and non-strong stationarity addresses recurring challenges in the MPCC literature.

major comments (2)
  1. [§3] §3 (equivalence theorem): The proof establishes piecewise M-stationarity ⇔ B-stationarity under MPCC-ACQ, but does not explicitly verify that the piecewise active-set linearization remains valid when biactive indices produce directions that are feasible for one piece but infeasible for the full MPCC tangent cone; this case is load-bearing for the claimed equivalence at points with nonempty biactive sets.
  2. [§4] §4 (convergence analysis): The NCP-based bounding methods are shown to converge without MPCC-LICQ, yet the argument does not establish that accumulation points satisfy MPCC-ACQ. Because the cost-reduction claim for B-stationarity verification (abstract and §5) rests on the equivalence holding at these limits, the interpretive step does not follow in general.
minor comments (2)
  1. [Throughout] Notation for active, inactive, and biactive index sets should be introduced once with a single table or definition block rather than redefined in each section.
  2. [Abstract] The abstract states that the interpretation 'leads to a cost reduction'; this interpretive claim would be clearer if tied to a specific corollary or numerical illustration in §5.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address the major comments point by point below, clarifying the proofs and indicating revisions to improve the exposition of the equivalence and its application to the convergence results.

read point-by-point responses

  1. Referee: [§3] §3 (equivalence theorem): The proof establishes piecewise M-stationarity ⇔ B-stationarity under MPCC-ACQ, but does not explicitly verify that the piecewise active-set linearization remains valid when biactive indices produce directions that are feasible for one piece but infeasible for the full MPCC tangent cone; this case is load-bearing for the claimed equivalence at points with nonempty biactive sets.

    Authors: We appreciate the referee's careful attention to this detail in the proof of Theorem 3.1. The argument proceeds by showing that, under MPCC-ACQ, any B-stationary point satisfies the piecewise M-stationarity conditions for each active-set piece, and conversely. For directions generated by biactive indices, feasibility in one linearized piece but not in the full MPCC tangent cone is precluded because such a direction would violate the complementarity constraint in the limit; the MPCC-ACQ ensures the tangent cone is exactly the union of the piecewise linearized cones, so no extraneous directions are admitted. To make this verification fully explicit, we will insert an additional remark immediately after the proof of Theorem 3.1 that isolates the biactive case and confirms the linearization remains valid. This revision will be incorporated in the next version. revision: yes

  2. Referee: [§4] §4 (convergence analysis): The NCP-based bounding methods are shown to converge without MPCC-LICQ, yet the argument does not establish that accumulation points satisfy MPCC-ACQ. Because the cost-reduction claim for B-stationarity verification (abstract and §5) rests on the equivalence holding at these limits, the interpretive step does not follow in general.

    Authors: We agree that the convergence theorem in Section 4 establishes only that accumulation points are piecewise M-stationary and does not prove MPCC-ACQ holds at those points. Consequently, the claimed cost reduction in B-stationarity verification (via the equivalence) applies only when the limit point additionally satisfies MPCC-ACQ. We will revise the abstract, the final paragraph of Section 5, and the concluding remarks to state this conditional explicitly, rather than leaving the impression that the reduction holds unconditionally. With this clarification the interpretive step follows under the stated hypothesis. Note that MPCC-ACQ remains a substantially weaker requirement than MPCC-LICQ, consistent with the paper's emphasis on relaxing the latter. revision: partial

Circularity Check

0 steps flagged

No significant circularity; new stationarity concept derived independently

full rationale

The paper introduces the definition of piecewise M-stationarity and provides a proof of its equivalence to B-stationarity under the standard MPCC-ACQ condition. Convergence properties of the NCP-based bounding methods are analyzed using the new concept, with reference to prior work [31] only for algorithm definition. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from authors, ansatz smuggling, or renaming of known results occur. The derivation chain relies on standard constraint qualification arguments and independent analysis rather than circular reductions to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard mathematical assumptions from nonlinear programming and MPCC theory; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption MPCC-ACQ (MPCC Abadie constraint qualification)
    Invoked to establish equivalence between piecewise M-stationarity and B-stationarity.

pith-pipeline@v0.9.0 · 5431 in / 1232 out tokens · 23026 ms · 2026-05-15T00:29:35.975297+00:00 · methodology

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Reference graph

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