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arxiv: 2511.18998 · v3 · submitted 2025-11-24 · 🧮 math.OC · cs.NA· math.NA

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A trust-region funnel algorithm for gray-box optimization

Gul Hameed , Tao Chen , Antonio del Rio Chanona , Lorenz T. Biegler , Michael Short
Authors on Pith no claims yet

Pith reviewed 2026-05-17 05:31 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords gray-box optimizationtrust-region methodsfunnel algorithmreduced modelsglobal convergencefirst-order critical pointblack-box modelsapproximation error bound
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The pith

A trust-region funnel algorithm maintains a monotonically decreasing upper bound on reduced model approximation error for gray-box optimization and proves global convergence to a first-order critical point.

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

The paper sets out to establish that trust-region methods for gray-box optimization can be simplified by swapping the usual filter for a uni-dimensional funnel that enforces a steadily tightening upper bound on the error of local reduced models for black-box parts. A sympathetic reader would care because gray-box problems mix known algebraic equations with derivative-free black-box simulations, and current trust-region approaches demand heavy parameter tuning that limits their use in engineering applications. If the funnel rules succeed in keeping the error bound decreasing while still ensuring objective progress, the result is a cleaner algorithm with a convergence guarantee that supports multiple reduced model types and both filter and funnel globalization options.

Core claim

The authors propose a trust-region funnel algorithm for gray-box optimization that replaces the filter acceptance criterion with a uni-dimensional funnel. This funnel maintains a monotonically decreasing upper bound on the approximation error of the local black-box reduced models. They prove global convergence of the algorithm to a first-order critical point. Benchmark tests indicate that the new method achieves performance comparable to or better than the classical trust-region filter method.

What carries the argument

The uni-dimensional funnel, which replaces the filter acceptance criterion and maintains a monotonically decreasing upper bound on the approximation error of local black-box reduced models via specific update rules.

If this is right

  • The algorithm reaches a first-order critical point from any starting point under the stated assumptions.
  • The method works with multiple forms of reduced models and both filter and funnel globalization strategies.
  • Benchmark results show performance that is comparable or improved relative to the trust-region filter method.
  • An open-source implementation supports direct use on gray-box problems mixing algebraic and black-box components.

Where Pith is reading between the lines

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

  • If the error bound holds reliably, the funnel structure could be adapted to other trust-region solvers that mix known models with black-box elements.
  • Applying the algorithm to problems with higher-dimensional or noisier black-box components would test whether the monotonic bound remains practical outside the reported benchmarks.
  • The uni-dimensional funnel may offer a template for simplifying acceptance mechanisms in related derivative-free or gray-box methods where error control is central.

Load-bearing premise

The funnel update rules can keep the upper bound on reduced model approximation error monotonically decreasing while still allowing sufficient objective decrease at each accepted step.

What would settle it

A specific gray-box test problem or iteration sequence in which the upper bound on reduced model error increases under the stated funnel rules, or in which the algorithm stops making progress without reaching a first-order critical point.

read the original abstract

Gray-box optimization, where parts of optimization problems are represented by algebraic models while others are treated as black-box models lacking analytic derivatives, remains a challenge. Trust-region (TR) methods provide a robust framework for gray-box problems through local reduced models (RMs) for black-box components, but they are complex and require extensive parameter tuning. Motivated by recent advances in funnel-based convergence theory for nonlinear optimization, we propose a novel TR funnel algorithm for gray-box optimization, replacing the filter acceptance criterion with a uni-dimensional funnel, maintaining a monotonically decreasing upper bound on approximation error of local black-box RMs. A global convergence proof to a first-order critical point is established. The algorithm, implemented open-source in Pyomo, supports multiple RM forms and globalization strategies (filter or funnel). Benchmark tests show the TR funnel algorithm achieves comparable and often improved performance relative to the classical TR filter method, thus providing a simpler, effective alternative for gray-box optimization.

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 a trust-region funnel algorithm for gray-box optimization problems, in which algebraic components are treated exactly while black-box components are approximated via local reduced models (RMs). The filter acceptance criterion is replaced by a uni-dimensional funnel that enforces a monotonically decreasing upper bound on RM approximation error. A global convergence proof to a first-order critical point is claimed, the method is implemented open-source in Pyomo, and benchmark comparisons indicate performance that is comparable to or better than the classical TR filter approach.

Significance. If the claimed convergence result is valid, the work supplies a simpler, lower-parameter alternative to filter-based TR methods for mixed algebraic/black-box problems while retaining global convergence guarantees. The open-source Pyomo implementation and empirical benchmarks constitute concrete strengths that would facilitate adoption and further testing.

major comments (2)
  1. [§4] §4 (Convergence Analysis), funnel update rules: the proof that every limit point is first-order critical rests on the uni-dimensional funnel simultaneously maintaining a strictly monotonically decreasing upper bound on RM approximation error and delivering sufficient objective decrease at accepted steps. It is not evident from the stated rules how this holds for black-box functions with high curvature without additional assumptions (e.g., uniform Lipschitz continuity of the black-box Hessian); a concrete counter-example or explicit bound derivation would be needed to close the argument.
  2. [§3.2] §3.2, Eq. (12)–(15): the funnel parameter update that replaces the filter is presented as independent of prior fitted quantities, yet the error-bound maintenance appears to depend on the specific choice of RM form and the radius update; clarification is required on whether the decrease inequality remains valid when the black-box curvature violates the implicit smoothness used to derive the funnel contraction.
minor comments (2)
  1. [Numerical Results] Table 1 and Figure 3: the benchmark exclusion criteria and the precise definition of 'improved performance' (e.g., function evaluations or final objective value) should be stated explicitly so that the cross-method comparison is reproducible.
  2. [Preliminaries] Notation: the symbol for the funnel upper bound is introduced without a dedicated definition paragraph; adding a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions made to strengthen the presentation of the convergence analysis.

read point-by-point responses

  1. Referee: [§4] §4 (Convergence Analysis), funnel update rules: the proof that every limit point is first-order critical rests on the uni-dimensional funnel simultaneously maintaining a strictly monotonically decreasing upper bound on RM approximation error and delivering sufficient objective decrease at accepted steps. It is not evident from the stated rules how this holds for black-box functions with high curvature without additional assumptions (e.g., uniform Lipschitz continuity of the black-box Hessian); a concrete counter-example or explicit bound derivation would be needed to close the argument.

    Authors: The convergence analysis in Section 4 establishes that the funnel maintains a monotonically decreasing upper bound on the RM error while ensuring sufficient model decrease at accepted steps. The proof proceeds by showing that the trust-region radius remains bounded away from zero only if the limit point satisfies the first-order criticality condition, with the funnel contraction factor chosen to dominate the local approximation error. We do not invoke a global Lipschitz assumption on the black-box Hessian; instead, the local quadratic or higher-order RM fitting within each trust region, combined with the radius update, ensures the error bound holds locally. To make this explicit, we have added a lemma in the revised Section 4 deriving the bound on the approximation error in terms of the current funnel parameter and trust-region radius, confirming that the decrease inequality is preserved without requiring uniform curvature bounds. revision: yes

  2. Referee: [§3.2] §3.2, Eq. (12)–(15): the funnel parameter update that replaces the filter is presented as independent of prior fitted quantities, yet the error-bound maintenance appears to depend on the specific choice of RM form and the radius update; clarification is required on whether the decrease inequality remains valid when the black-box curvature violates the implicit smoothness used to derive the funnel contraction.

    Authors: The funnel parameter updates in Equations (12)–(15) are intentionally formulated to depend only on the current objective value, model prediction, and the previous funnel level, preserving generality across RM forms. The error-bound maintenance is guaranteed by the standard trust-region assumption that the RM approximates the black-box function with error O(Δ²) inside the trust region of radius Δ. When curvature is high, the radius is reduced by the algorithm, which in turn tightens the funnel contraction to maintain the inequality. We have inserted a clarifying paragraph after Equation (15) that explicitly states the local smoothness implicit in the RM construction and verifies that the decrease inequality continues to hold under the algorithm's radius adjustment rule. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence proof rests on algorithm rules and standard TR analysis, not self-definition or fitted inputs

full rationale

The paper defines a uni-dimensional funnel to replace the filter, states update rules that enforce a monotonically decreasing upper bound on RM approximation error, and then proves global convergence to first-order criticality using the standard trust-region decrease condition plus the error bound vanishing at limit points. No equation reduces the claimed bound or the criticality result to a fitted parameter or prior self-citation by construction; the funnel mechanics and proof are presented as self-contained once the update rules are accepted. The abstract and method sections treat the bound maintenance as a consequence of the new acceptance criterion rather than an input assumption that is redefined as output.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from nonlinear optimization theory regarding the existence of first-order critical points and the ability to construct local reduced models with controllable approximation error for black-box components.

axioms (2)
  • domain assumption Black-box components admit local reduced models whose approximation error can be bounded and controlled monotonically.
    Invoked to support the funnel's monotonically decreasing upper bound on error.
  • standard math The objective function satisfies conditions allowing sufficient decrease within the trust region.
    Standard assumption for trust-region convergence proofs.

pith-pipeline@v0.9.0 · 5473 in / 1312 out tokens · 37562 ms · 2026-05-17T05:31:27.157681+00:00 · methodology

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Reference graph

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    From the update rule (A4), we have: 𝜙(𝑘+1) = (1 − 𝜅𝑓)𝜃(𝑘+1) + 𝜅𝑓𝜙(𝑘) ≤ (1 − 𝜅𝑓)𝜏𝜙(𝑘) + 𝜅𝑓𝜙(𝑘) ≤ (1 − (1 − 𝜏)(1 − 𝜅𝑓))𝜙(𝑘)

    Infinite 𝜃 − 𝑡𝑦𝑝𝑒 steps. From the update rule (A4), we have: 𝜙(𝑘+1) = (1 − 𝜅𝑓)𝜃(𝑘+1) + 𝜅𝑓𝜙(𝑘) ≤ (1 − 𝜅𝑓)𝜏𝜙(𝑘) + 𝜅𝑓𝜙(𝑘) ≤ (1 − (1 − 𝜏)(1 − 𝜅𝑓))𝜙(𝑘). Define 𝑞 ≝ 1 − (1 − 𝜏)(1 − 𝜅𝑓) ∈ (0,1). Then 𝜙(𝑘+1) ≤ 𝑞𝜙(𝑘) implies 𝜙(𝑘) → 0 for 𝑘 → ∞. Page 26 of 39

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  46. [46]

    Finite 𝜃 − 𝑡𝑦𝑝𝑒 steps. If only finitely many 𝜃 − 𝑡𝑦𝑝𝑒 steps occur, there exists an iteration index 𝑘′ such that for all 𝑘 ≥ 𝑘′, only 𝑓 − 𝑡𝑦𝑝𝑒 steps are taken, satisfying both the switching condition (14) and the Armijo sufficient decrease condition (15). Summing (A5) from 𝑘 = 𝑘′ to 𝑘 = 𝑘′ + 𝑁 gives ∑ (𝑓(𝑘) − 𝑓(𝑘+1)) 𝑘′+𝑁 𝑘=𝑘′ ≥ 𝜂𝛿 ∑ (𝜃(𝑘))𝛾𝑠 𝑘′+𝑁 𝑘=𝑘′ , w...

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