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arxiv: 2605.17913 · v1 · pith:Q6YFQZTKnew · submitted 2026-05-18 · 🧮 math.OC

A Differentiable Interior-Point Method in Single Precision

Jon Arrizabalaga , Kevin Tracy , Zachary Manchester This is my paper

Pith reviewed 2026-05-20 09:56 UTC · model grok-4.3

classification 🧮 math.OC
keywords interior-point methodsdifferentiable optimizationsingle precisioncomplementarity conditionsconvex optimizationbilevel optimizationgradient computation
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The pith

Changing the complementarity representation keeps linear systems well-conditioned near the solution, enabling reliable single-precision interior-point methods and differentiation.

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

Primal-dual interior-point methods solve constrained convex optimization problems to high accuracy, and their solutions can be differentiated with respect to problem data using the implicit function theorem. Standard complementarity conditions cause the linear systems to become increasingly ill-conditioned near the solution, which creates arithmetic problems and inaccurate gradients when working in single precision. The paper introduces an alternative complementarity representation that keeps those linear systems spectrally bounded throughout the solve, even close to optimality. This modification preserves the convergence rate and final accuracy of the original method while removing the precision barrier. The result is that interior-point solvers can now be used to solve and differentiate optimization problems in single precision for bilevel and end-to-end learning tasks that previously required double precision.

Core claim

The central claim is that an alternative complementarity representation in primal-dual interior-point methods ensures the underlying linear systems remain spectrally bounded even near the solution. This boundedness supports accurate gradient computation through the implicit function theorem and prevents arithmetic exceptions in single-precision arithmetic, allowing the solvers to handle and differentiate constrained convex problems that were previously restricted to double precision.

What carries the argument

The alternative complementarity representation, which replaces the standard treatment of primal-dual complementarity conditions to control the spectral properties of the KKT linear systems.

If this is right

  • Interior-point solvers become reliable for single-precision arithmetic without arithmetic exceptions or loss of accuracy near the solution.
  • Gradients computed through the solver remain numerically stable and accurate even when the iterates approach optimality.
  • Differentiation through constrained convex optimization becomes practical in bilevel and end-to-end learning pipelines on low-precision hardware.
  • Problems that previously required double precision can now be solved and differentiated at lower memory and compute cost.

Where Pith is reading between the lines

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

  • The same complementarity change might be tested inside other interior-point variants or related methods such as sequential quadratic programming to see whether conditioning benefits transfer.
  • Single-precision support could open the door to faster execution on hardware accelerators that favor reduced-precision arithmetic in machine-learning training loops.
  • Further checks could examine whether the modified systems affect warm-start performance or the detection of infeasible problems.

Load-bearing premise

The alternative complementarity representation preserves the convergence rate and solution accuracy of the original primal-dual interior-point method.

What would settle it

Run both the new method and the standard interior-point method to convergence on the same collection of convex test problems in double precision, then compare final objective values and the gradients obtained via the implicit function theorem; a noticeable difference in either quantity would falsify the preservation claim.

Figures

Figures reproduced from arXiv: 2605.17913 by Jon Arrizabalaga, Kevin Tracy, Zachary Manchester.

Figure 1
Figure 1. Figure 1: Implicit complementarity enables accurate and robust single-precision gradients. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bilevel trajectory optimization. Implicit f32 remains stable at tight relaxations and matches [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: End-to-end learning of a multi-agent safety filter. Implicit f32 matches f64 training stability [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NaN tracing for operations in explicit f32. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bilevel optimization for fast (outer opt.) and safe (inner opt.) trajectories. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A subset of testing configurations for 1 agent. Only case where explicit f32 works. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A subset of testing configurations for 3 agents. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A subset of testing configurations for 5 agents. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A subset of testing configurations for 9 agents. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
read the original abstract

Primal-dual interior-point methods solve constrained convex optimization problems to tight tolerances with speed and robustness. Their solutions are also efficiently differentiable with respect to the problem data through the implicit function theorem. However, the standard treatment of primal-dual complementarity makes the underlying linear systems increasingly ill-conditioned near the solution. While this ill-conditioning is often benign in double precision, it can be catastrophic in single precision, preventing interior-point methods from fully exploiting the accelerated hardware that underpins modern machine learning. This paper introduces a differentiable interior-point method designed for low-precision arithmetic. By using an alternative complementarity representation, we ensure that the underlying linear systems remain spectrally bounded -- even near the solution -- a property that is essential for computing accurate gradients and avoiding arithmetic exceptions. As a result, our method enables interior-point solvers to reliably solve and differentiate optimization problems in single precision that were previously confined to double precision. We demonstrate the approach through an ablation study against the standard interior-point formulation and applications in bilevel and end-to-end learning settings where differentiating through constrained optimization is essential. The source code is available at https://github.com/qpax-solver/qpax.

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 / 3 minor

Summary. The paper introduces a differentiable primal-dual interior-point method for convex optimization problems that is designed to operate in single precision. The central innovation is an alternative algebraic representation of the complementarity condition that is intended to keep the Newton linear systems spectrally bounded even near optimality, thereby enabling stable implicit differentiation via the implicit function theorem and avoiding arithmetic exceptions. The approach is supported by an ablation study against the standard formulation and by demonstrations in bilevel and end-to-end learning settings; open-source code is provided.

Significance. If the spectral-boundedness property holds, the method would allow interior-point solvers to be used reliably in single precision for differentiable optimization layers, taking advantage of accelerated hardware common in machine learning. The open-source implementation and the empirical ablation plus application results are concrete strengths that support practical utility.

major comments (2)
  1. [§3] §3 (alternative complementarity formulation): the manuscript asserts that the new representation keeps the KKT/Newton systems spectrally bounded as the duality gap vanishes, yet supplies no theorem or uniform eigenvalue bound independent of problem data, conditioning, and dimension. The ablation study demonstrates practical improvement on selected instances but does not establish the claimed O(1) conditioning; this bound is load-bearing for the single-precision stability and gradient-accuracy claims.
  2. [§4] §4 (convergence and accuracy): the claim that the alternative representation preserves the original convergence rate and solution accuracy is implicit in the ablation results but is not accompanied by a formal rate analysis or proof that the modified Newton system retains the same local convergence properties as the standard primal-dual IPM.
minor comments (3)
  1. [Figure 2] Figure 2 (ablation plots): axis labels and legends would benefit from explicit mention of the duality-gap range and the precise metric being plotted (e.g., relative residual or gradient error).
  2. Notation: the mapping from the new complementarity variable to the standard slack variables is introduced without a side-by-side comparison to the classical form; a short table or equation block would improve clarity.
  3. References: recent literature on low-precision linear solvers and differentiable optimization (e.g., works on single-precision QP solvers) is lightly cited; adding 2–3 targeted references would strengthen the positioning.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Below we respond point by point to the major comments, indicating the changes we will make in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (alternative complementarity formulation): the manuscript asserts that the new representation keeps the KKT/Newton systems spectrally bounded as the duality gap vanishes, yet supplies no theorem or uniform eigenvalue bound independent of problem data, conditioning, and dimension. The ablation study demonstrates practical improvement on selected instances but does not establish the claimed O(1) conditioning; this bound is load-bearing for the single-precision stability and gradient-accuracy claims.

    Authors: We agree that a formal statement would strengthen the section. In the revision we will insert a theorem in §3 establishing that, under the alternative complementarity formulation, the eigenvalues of the Newton matrix (or its Schur complement) remain bounded away from zero and infinity as the barrier parameter μ → 0. The bound depends on standard problem quantities (Lipschitz constants of the objective and constraint functions, and the conditioning of the constraint Jacobian) but is independent of μ itself. This is the sense in which the systems are “spectrally bounded as the duality gap vanishes,” in contrast to the standard formulation whose condition number grows without bound as μ → 0. While a bound that is completely independent of all problem data and dimension is not claimed (and would require stronger assumptions than those in the problem class), the μ-independence is the property needed for single-precision stability and reliable implicit differentiation. We will also add a short remark clarifying this distinction and will reference the ablation results as empirical corroboration of the theoretical statement. revision: yes

  2. Referee: [§4] §4 (convergence and accuracy): the claim that the alternative representation preserves the original convergence rate and solution accuracy is implicit in the ablation results but is not accompanied by a formal rate analysis or proof that the modified Newton system retains the same local convergence properties as the standard primal-dual IPM.

    Authors: We accept that an explicit argument is desirable. In the revised manuscript we will add a short paragraph (or appendix remark) in §4 showing that the Newton system arising from the alternative complementarity condition is asymptotically equivalent, near a strict complementarity solution, to the standard primal-dual Newton system. Consequently the local quadratic convergence rate of the underlying interior-point method is retained. Solution accuracy is likewise unchanged because the KKT conditions satisfied at termination are identical to those of the classical formulation. The ablation experiments already indicate that final objective values and constraint violations match to machine precision; the added analysis will place this observation on a theoretical footing. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit algebraic change to complementarity yields bounded spectra by construction of the new representation.

full rationale

The paper's central contribution is an alternative complementarity representation that is introduced directly in the method derivation rather than fitted to data or justified solely by self-citation. The claim that linear systems remain spectrally bounded follows from the algebraic substitution itself (as described in the abstract and method sections), with empirical verification via ablation rather than any reduction of the target property to the paper's own fitted quantities or prior self-cited uniqueness results. No load-bearing step equates a prediction or bound to an input by definition, and the derivation remains self-contained against external IPM benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard convex optimization assumptions and the implicit function theorem for differentiation; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond the choice of the alternative complementarity form itself.

axioms (1)
  • domain assumption The problem is convex and satisfies standard constraint qualifications so that the KKT system is well-defined.
    Invoked implicitly when applying the interior-point method and the implicit function theorem for gradients.

pith-pipeline@v0.9.0 · 5728 in / 1300 out tokens · 29800 ms · 2026-05-20T09:56:27.963592+00:00 · methodology

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Reference graph

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