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arxiv: 2607.00451 · v1 · pith:DJ2SGY5Inew · submitted 2026-07-01 · 🧮 math.NA · cs.NA

An Inner-Outer Iteration Algorithm with Optimal Parameters for Stochastic Lyapunov Matrix Equation

Donghuan He , Xiaowen Su , Feng Wang , Xuesong Chen This is my paper

Pith reviewed 2026-07-02 08:16 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords stochastic Lyapunov equationinner-outer iterationoptimal parametersmean-square stabilityconvergence analysisiterative methodsdiscrete-time systemsmatrix equations
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The pith

An inner-outer iteration algorithm with optimal parameters converges for stochastic Lyapunov matrix equations under mean-square stability.

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

The paper develops an inner-outer iterative method to solve the stochastic Lyapunov matrix equation that appears in stability analysis of discrete-time stochastic linear systems. It first shows that when the system is asymptotically mean-square stable, the sequence generated from a zero start is monotonic and bounded, hence convergent. It then derives the spectral radius of the iteration matrix to give necessary and sufficient conditions for convergence from any initial matrix. Finally it supplies explicit strategies for choosing the iteration parameters that minimize this spectral radius and thereby speed convergence.

Core claim

Under the assumption that the underlying discrete-time stochastic linear system is asymptotically mean-square stable, the proposed inner-outer iteration produces a monotonic and bounded sequence that converges when started at the zero matrix; necessary and sufficient conditions for convergence from arbitrary initial matrices are expressed via the spectral radius of the iteration matrix; and explicit optimal parameter choices are obtained that minimize this spectral radius.

What carries the argument

The inner-outer (IO) iteration whose update is controlled by two scalar parameters whose values are chosen to minimize the spectral radius of the associated iteration matrix.

If this is right

  • The generated sequence is monotonic and bounded when the zero matrix is used as the starting point.
  • Convergence from an arbitrary initial matrix holds if and only if the spectral radius of the iteration matrix is strictly less than one.
  • The convergence rate is improved by selecting the two iteration parameters to minimize the spectral radius.
  • Numerical tests confirm faster convergence than several existing fixed-point and projection methods.

Where Pith is reading between the lines

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

  • The same monotonicity argument could be reused to analyze similar splittings for continuous-time stochastic Lyapunov equations.
  • The spectral-radius optimality condition supplies a concrete benchmark against which any new parameter-free method could be compared.
  • Because the iteration matrix is explicitly available, the method immediately yields a computable a-posteriori error bound once the spectral radius is estimated.

Load-bearing premise

The underlying stochastic linear system must be asymptotically mean-square stable.

What would settle it

A concrete counter-example in which the system is asymptotically mean-square stable yet the iteration diverges for some initial matrix when the proposed optimal parameters are used.

Figures

Figures reproduced from arXiv: 2607.00451 by Donghuan He, Feng Wang, Xiaowen Su, Xuesong Chen.

Figure 1
Figure 1. Figure 1: Convergence behavior in Example 1: (a) Spectral radius of the iteration matrix 𝑅𝛼,2 with respect to 𝛼; (b) Convergence curves of Algorithm (12) with different parameters 𝛼; (c) Comparison of different algorithms. and 𝑄 is chosen as the identity matrix. In this example, direct computation shows that all eigen￾values of Φ are real. Therefore, the eigenvalues of Φ satisfy the condition in Theorem 4. When 𝑙 = … view at source ↗
read the original abstract

This paper proposes an inner--outer (IO) iterative algorithm with optimal parameters for solving stochastic Lyapunov matrix equation associated with discrete-time stochastic linear system. First, under the assumption that the underlying stochastic linear system is asymptotically mean-square stable, the monotonicity and boundedness of the iterative sequence generated by the proposed algorithm are analyzed. On this basis, a sufficient convergence result is established for the zero initial condition. Second, by deriving the spectral radius of the corresponding iteration matrix, several necessary and sufficient convergence conditions are obtained for arbitrary initial conditions. In addition, the optimal parameter-selection strategies are developed to improve the convergence performance of the algorithm. Finally, numerical examples are presented to verify the theoretical results and demonstrate the advantages of the proposed algorithm over several existing iterative methods.

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 proposes an inner-outer iterative algorithm with optimal parameters for the stochastic Lyapunov matrix equation associated with discrete-time stochastic linear systems. Under the assumption of asymptotic mean-square stability, it analyzes monotonicity and boundedness of the iterates, proves a sufficient convergence result from the zero initial condition, derives necessary and sufficient convergence conditions via the spectral radius of the iteration matrix for arbitrary initial conditions, develops optimal parameter-selection strategies, and presents numerical examples verifying the theory and comparing performance to existing methods.

Significance. If the convergence analysis and parameter optimization hold, the work supplies a practical, theoretically grounded iterative solver for stochastic Lyapunov equations arising in discrete-time stochastic control. The dual use of monotonicity arguments (for the zero-start case) and spectral-radius conditions (for general starts), together with explicit optimal-parameter strategies, strengthens the contribution relative to purely heuristic inner-outer schemes. Numerical comparisons provide concrete evidence of improved convergence rates.

minor comments (3)
  1. [Section 2] The statement of the stochastic Lyapunov equation (presumably Eq. (1) or (2)) should explicitly display the expectation operator and the form of the multiplicative noise terms to make the subsequent iteration matrix derivation self-contained.
  2. [Section 4] In the derivation of the spectral radius (Section 4), the transition from the vectorized iteration matrix to the closed-form radius expression would benefit from an intermediate step showing how the Kronecker structure simplifies under the stability assumption.
  3. [Section 5] Numerical examples (Section 5) report iteration counts and CPU times but omit the precise dimensions of the test matrices and the distribution parameters of the noise; adding a short table of these quantities would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's core claims rest on the external assumption of asymptotic mean-square stability of the underlying stochastic system, followed by standard monotonicity/boundedness arguments for the zero-initial case and spectral-radius analysis of the iteration matrix for general initials. Optimal parameters are developed from this spectral analysis rather than fitted to data or defined circularly. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the described chain; the derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption of mean-square stability of the system and standard properties of iterative matrix methods. No free parameters or invented entities are apparent from the abstract. Assessment is limited because only the abstract is available.

axioms (1)
  • domain assumption The stochastic linear system is asymptotically mean-square stable
    Invoked explicitly in the abstract to establish monotonicity and boundedness of the iterative sequence generated by the algorithm.

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Reference graph

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