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arxiv: 2604.15624 · v1 · submitted 2026-04-17 · 📡 eess.SY · cs.SY

A Common Lyapunov Matrix Approach to the Exponential Stability of Augmented Primal-Dual Gradient Flow as LPV Systems

Mengmou Li , Lijun Zhu , Masaaki Nagahara This is my paper

Pith reviewed 2026-05-10 09:09 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords common Lyapunov matrixexponential stabilityprimal-dual gradient flowLPV systemsconstrained optimizationactive constraintsintegral quadratic constraintsHurwitz matrices
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The pith

A common Lyapunov matrix exists for two Hurwitz matrices exactly when strict and non-strict Lyapunov matrix sets intersect, and this certifies exponential stability for augmented primal-dual gradient flows under relaxed strong convexity.

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

The paper shows that a common Lyapunov matrix for the convex combination of two Hurwitz matrices exists if and only if the intersection of strict Lyapunov matrices for one and non-strict Lyapunov matrices for the other is nonempty. This condition is applied to the augmented primal-dual gradient flow for constrained optimization with affine inequality constraints, which is represented exactly as a polytopic linear parameter-varying system whose parameter is the active-constraint selector. With a relaxed strong convexity condition on the objective, the common matrix establishes exponential convergence of the flow to the optimum. The same approach extends to the integral quadratic constraints framework, allowing numerical search for the convergence rate.

Core claim

We show that a common Lyapunov matrix exists for the convex combination of two Hurwitz matrices if and only if the intersection of the set of strict Lyapunov matrices for one matrix and the set of non-strict Lyapunov matrices for the other is nonempty. This simple relaxation is useful for the convergence analysis of the augmented primal-dual gradient flow for constrained optimization problems with affine inequality constraints, which can be viewed as a polytopic linear parameter-varying (LPV) system driven by the active-constraint selector. Under a relaxed strong convexity condition, exponential convergence is proved for the LPV system. The analysis can further be extended to the integral-ic

What carries the argument

The intersection condition between the set of strict Lyapunov matrices for one Hurwitz matrix and the set of non-strict Lyapunov matrices for the other, which is necessary and sufficient for the existence of a common Lyapunov matrix certifying stability of their convex combination.

If this is right

  • Exponential convergence holds for the augmented primal-dual gradient flow solving constrained optimization problems with affine inequality constraints.
  • The convergence rate of the LPV system can be bounded numerically via the integral quadratic constraints framework.
  • Stability certificates for any system that switches between two Hurwitz modes become checkable by testing the intersection condition on Lyapunov matrix sets.
  • The approach applies directly to polytopic LPV representations where the parameter is a selector among active sets.

Where Pith is reading between the lines

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

  • The intersection condition may simplify stability proofs for other primal-dual or switching optimization algorithms that activate different constraint sets.
  • Weakening the usual strong convexity assumption could extend convergence guarantees to a larger family of convex but not strongly convex objectives.
  • Software implementations of the IQC search could produce explicit rate estimates for concrete problem instances without deriving closed-form expressions.

Load-bearing premise

The augmented primal-dual gradient flow can be represented exactly as a polytopic LPV system parameterized by the active-constraint selector, and the relaxed strong convexity condition is sufficient for the common Lyapunov matrix to certify exponential decay.

What would settle it

A pair of Hurwitz matrices for which the intersection of strict and non-strict Lyapunov matrix sets is nonempty yet no common Lyapunov matrix exists, or a numerical simulation of the augmented primal-dual gradient flow under the relaxed strong convexity condition that fails to converge exponentially.

Figures

Figures reproduced from arXiv: 2604.15624 by Lijun Zhu, Masaaki Nagahara, Mengmou Li.

Figure 1
Figure 1. Figure 1: Trajectories of x over time. 0 10 20 30 40 50 60 70 10-12 10-10 10-8 10-6 10-4 10-2 100 102 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trajectory of the error ∥x − x ∗∥ over time. observed that the augmented primal-dual gradient flow in￾deed achieves exponential convergence even without global strong convexity, as suggested by Theorem 10. As ∇f(x) is piecewise continuous and the system matrices change depending on whether the inequality constraint is active or inactive, the resulting dynamics yield the following three matrix inequalities,… view at source ↗
read the original abstract

We show that a common Lyapunov matrix exists for the convex combination of two Hurwitz matrices if and only if the intersection of the set of strict Lyapunov matrices for one matrix and the set of non-strict Lyapunov matrices for the other is nonempty. This simple relaxation is useful for the convergence analysis of the augmented primal-dual gradient flow for constrained optimization problems with affine inequality constraints, which can be viewed as a polytopic linear parameter-varying (LPV) system driven by the active-constraint selector. Under a relaxed strong convexity condition, exponential convergence is proved for the LPV system. The analysis can further be extended to the integral quadratic constraints (IQCs) framework for LPV systems to facilitate numerical search of the convergence rate.

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

1 major / 2 minor

Summary. The manuscript proves that a common Lyapunov matrix exists for the convex combination of two Hurwitz matrices if and only if the intersection of the set of strict Lyapunov matrices for one matrix and the set of non-strict Lyapunov matrices for the other is nonempty. It applies this result to establish exponential convergence of the augmented primal-dual gradient flow for constrained optimization problems with affine inequalities, by modeling the dynamics as a polytopic LPV system whose parameter is the active-constraint selector. Under a relaxed strong-convexity condition, the common Lyapunov matrix certifies stability; the approach is further extended to the IQC framework to enable numerical search for convergence rates.

Significance. If the central claims hold, the relaxed common-Lyapunov condition offers a practical tool for stability analysis of two-vertex polytopic LPV systems that arise in primal-dual optimization flows. The explicit connection to IQCs supplies a computational pathway for rate estimation, which is valuable for systems-and-control applications in optimization. The iff characterization itself is a clean contribution to Lyapunov theory for switched systems.

major comments (1)
  1. [LPV modeling and exponential-stability proof] The proof that the common Lyapunov matrix certifies uniform exponential decay of the LPV system (abstract and the LPV stability section): the non-strict inequality BᵀP + PB ≤ 0 only guarantees a semi-definite derivative when the active-constraint selector θ(t) is constantly at the corresponding vertex. The relaxed strong-convexity assumption is invoked to obtain this non-strict case, yet no argument is given showing that a uniform α > 0 exists such that xᵀ(A(θ)ᵀP + PA(θ))x ≤ −α xᵀPx for all admissible θ(·) and all x. Constant trajectories at the non-strict vertex are admissible in the polytopic LPV model, so individual Hurwitzness of the vertices does not automatically yield a common exponential rate. This point is load-bearing for the exponential-convergence claim.
minor comments (2)
  1. [Modeling section] The precise definition of the active-constraint selector and the resulting two-vertex polytopic representation of the augmented primal-dual flow should be stated explicitly with the state-space matrices A and B.
  2. [IQC extension] Notation for the relaxed strong-convexity parameter and its relation to the non-strict Lyapunov inequality could be clarified to avoid ambiguity when extending to the IQC framework.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Dear Editor, We thank the referee for the positive evaluation of our contribution and for the constructive comment regarding the exponential stability proof for the LPV system. We address the major comment point by point below and will revise the manuscript accordingly to strengthen the relevant section.

read point-by-point responses

  1. Referee: The proof that the common Lyapunov matrix certifies uniform exponential decay of the LPV system (abstract and the LPV stability section): the non-strict inequality BᵀP + PB ≤ 0 only guarantees a semi-definite derivative when the active-constraint selector θ(t) is constantly at the corresponding vertex. The relaxed strong-convexity assumption is invoked to obtain this non-strict case, yet no argument is given showing that a uniform α > 0 exists such that xᵀ(A(θ)ᵀP + PA(θ))x ≤ −α xᵀPx for all admissible θ(·) and all x. Constant trajectories at the non-strict vertex are admissible in the polytopic LPV model, so individual Hurwitzness of the vertices does not automatically yield a common exponential rate. This point is load-bearing for the exponential-convergence claim.

    Authors: We thank the referee for highlighting this subtlety. The manuscript shows existence of a common P>0 such that the quadratic form satisfies A(θ)ᵀP + PA(θ) ≤ 0 for all θ in the polytope (strict at one vertex) under the relaxed strong-convexity condition, and claims this certifies exponential stability of the LPV system. However, the current text does not explicitly construct or verify a uniform α>0 in the differential inequality that holds for every admissible θ(·), including constant trajectories at the non-strict vertex. We agree that individual Hurwitzness of the vertices together with a common P yielding a non-positive derivative does not automatically deliver a uniform exponential rate for arbitrary switching. In the revised manuscript we will expand the LPV stability section with a complete argument establishing uniform exponential stability. The added proof will combine (i) the strict negative-definiteness of the form for all θ>0, (ii) the exponential stability of the LTI system at the non-strict vertex, and (iii) a comparison argument that produces a positive lower bound on the decay rate independent of the particular measurable θ(·). We will also clarify the relation to the state-dependent active-set selector arising in the optimization dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct Lyapunov and LPV arguments without self-definition or fitted-input reductions.

full rationale

The paper's core result is an iff statement on existence of a common Lyapunov matrix for convex combinations of two Hurwitz matrices, derived from standard matrix inequalities (strict for one vertex, non-strict for the other). This is then applied to represent the augmented primal-dual flow as a polytopic LPV system whose parameter is the active-constraint selector. The relaxed strong-convexity assumption is used to certify the non-strict case, after which exponential convergence is asserted for the LPV trajectories. No step renames a fitted parameter as a prediction, defines a quantity in terms of its own output, or relies on a load-bearing self-citation whose content is unverified. The derivation chain remains self-contained against external Lyapunov theory and LPV stability criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard facts from linear systems theory: every Hurwitz matrix admits a strict quadratic Lyapunov function, convex combinations preserve the polytopic structure, and the active-constraint selector produces a finite number of vertex matrices. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Every Hurwitz matrix admits a positive-definite solution to the Lyapunov equation for any positive-definite right-hand side.
    Invoked to define the sets of strict and non-strict Lyapunov matrices.
  • domain assumption The active-constraint selector produces a polytopic LPV system whose vertices are the dynamics corresponding to each possible active-set combination.
    Central modeling step that allows the common-Lyapunov test to be applied.

pith-pipeline@v0.9.0 · 5429 in / 1501 out tokens · 45889 ms · 2026-05-10T09:09:33.816698+00:00 · methodology

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Reference graph

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