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arxiv: 2602.02394 · v2 · submitted 2026-02-02 · 🧮 math.OC · cs.SY· eess.SY

On the Practical Implementation of a Sequential Quadratic Programming Algorithm for Nonconvex Sum-of-squares Problems

Jan Olucak , Torbj{\o}rn Cunis This is my paper

Pith reviewed 2026-05-16 08:10 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords sum-of-squares optimizationnonconvex problemssequential quadratic programmingfilter line searchpolynomial nonnegativitynumerical benchmarkscontrol engineering
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The pith

A filter line search algorithm solves sequences of quadratic subproblems to handle nonconvex sum-of-squares optimization more efficiently.

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

Sum-of-squares optimization certifies polynomial nonnegativity and reduces to semidefinite programs when the instance is convex. Nonconvex cases require iterative procedures, yet efficient and reliable solvers remain scarce. The paper introduces a filter line search method that repeatedly solves quadratic subproblems derived from the original formulation. Benchmarks on nonconvex instances show markedly fewer iterations and lower total computation time than established approaches. This matters for control engineering and related fields where nonconvex SOS problems arise in stability and robustness analysis.

Core claim

The paper proposes a filter line search algorithm that solves a sequence of quadratic subproblems for nonconvex sum-of-squares optimization. Numerical benchmarks demonstrate that the algorithm can significantly reduce the number of iterations, resulting in a substantial decrease in computation time compared to established methods for nonconvex SOS programs.

What carries the argument

The filter line search algorithm applied to a transcribed sequence of quadratic subproblems.

If this is right

  • Nonconvex SOS problems become solvable with substantially fewer iterations in the tested cases.
  • Overall computation time for certifying polynomial nonnegativity drops compared with prior iterative solvers.
  • Tractable solutions open for nonconvex instances in control engineering that previously exceeded practical limits.
  • The method supplies a concrete implementation route for sequential quadratic programming on SOS constraints.

Where Pith is reading between the lines

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

  • The same quadratic-subproblem structure might extend to other classes of nonconvex polynomial programs that admit SOS relaxations.
  • Warm-starting the quadratic solves from previous iterates could produce further speed gains not explored in the benchmarks.
  • Convergence radius might be characterized by examining how the filter parameters interact with the degree of the polynomials.

Load-bearing premise

The filter line search and quadratic subproblem sequence remain reliable and convergent for the full range of nonconvex SOS instances encountered in practice.

What would settle it

A standard nonconvex SOS benchmark from control theory on which the algorithm either diverges or requires more iterations and time than current solvers would falsify the performance claim.

read the original abstract

Sum-of-squares (SOS) optimization provides a computationally tractable framework for certifying polynomial nonnegativity. If the considered problem is convex, the SOS problem can be transcribed into and solved by semi-definite programs. However, in case of nonconvex problems iterative procedures are needed. Yet tractable and efficient solution methods are still lacking, limiting their application, for instance, in control engineering. To address this gap, we propose a filter line search algorithm that solves a sequence of quadratic subproblems. Numerical benchmarks demonstrate that the algorithm can significantly reduce the number of iterations, resulting in a substantial decrease in computation time compared to established methods for nonconvex SOS programs

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 / 1 minor

Summary. The paper proposes a filter line search SQP algorithm for nonconvex sum-of-squares (SOS) optimization problems. It formulates the nonconvex SOS task as a sequence of quadratic subproblems solved iteratively, with a filter mechanism to enforce progress and convergence. Numerical benchmarks are presented to claim that the method substantially reduces iteration counts and runtime relative to existing solvers for nonconvex SOS programs.

Significance. If the performance claims are substantiated across representative instances, the work could supply a practical, convergent solver for nonconvex SOS problems that arise in control synthesis and polynomial optimization, where current methods remain limited.

major comments (1)
  1. [§5 / abstract] Numerical benchmarks (abstract and §5): the central performance claim—that the filter line search SQP reduces iterations and computation time versus established methods—lacks any description of the test set (problem sizes, polynomial degrees, degree of nonconvexity), baseline solvers, or statistical controls (e.g., multiple random instances, failure rates). Without this information it is impossible to determine whether the observed speedups are general or artifacts of a narrow, specially structured test collection.
minor comments (1)
  1. [Abstract] The abstract would be strengthened by a single sentence indicating the classes of test problems (e.g., control Lyapunov functions of degree 4–6) used to obtain the reported speedups.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the numerical benchmarks. We address the concern below and will revise the manuscript to provide the requested details.

read point-by-point responses

  1. Referee: [§5 / abstract] Numerical benchmarks (abstract and §5): the central performance claim—that the filter line search SQP reduces iterations and computation time versus established methods—lacks any description of the test set (problem sizes, polynomial degrees, degree of nonconvexity), baseline solvers, or statistical controls (e.g., multiple random instances, failure rates). Without this information it is impossible to determine whether the observed speedups are general or artifacts of a narrow, specially structured test collection.

    Authors: We agree that the current presentation of the numerical results in Section 5 does not provide sufficient detail on the test problems to allow readers to fully assess the generality of the reported speedups. In the revised manuscript we will expand Section 5 to include: (i) explicit characterization of the test set by problem dimension (number of variables), polynomial degree, and degree of nonconvexity (e.g., number of nonconvex terms and/or the spectrum of the Hessian at the computed solution); (ii) precise identification of all baseline solvers employed for comparison; and (iii) statistical controls consisting of averages and standard deviations computed over multiple randomly generated instances together with observed failure rates. These additions will make clear that the performance improvements are not limited to a narrow or specially structured collection of instances. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic proposal and empirical benchmarks are self-contained with no derivation chain reducing to inputs by construction.

full rationale

The paper proposes a filter line search SQP method for nonconvex sum-of-squares problems and supports its claims exclusively through numerical benchmarks comparing iteration counts and runtime against established solvers. No equations, derivations, fitted parameters, or self-citations appear in the abstract or described content that could create a self-definitional loop, a prediction forced by a prior fit, or an ansatz smuggled via citation. The central performance claims rest on external empirical evidence rather than any reduction of outputs to the algorithm's own inputs by construction. This is the expected honest outcome for an implementation-focused paper whose load-bearing steps are algorithmic description plus benchmarking, not mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The method implicitly relies on standard assumptions of SQP convergence and SOS feasibility that are not detailed here.

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