arxiv: 2605.12658 · v1 · submitted 2026-05-12 · 🧮 math.OC

Recognition: 3 theorem links

· Lean Theorem

Multiconic Optimization for Symmetric Cones and Hyperbolic Coupling

Marianna E.-Nagy, Petra Ren\'ata Rig\'o, Yurii Nesterov

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:33 UTC · model grok-4.3

classification 🧮 math.OC

keywords multiconic optimizationsymmetric coneshyperbolic couplinginterior-point algorithmcomplexity analysisparabolic target spaceprimal-dual methods

0 comments

The pith

A new interior-point algorithm using hyperbolic coupling solves multiconic optimization over symmetric cones with complexity matching linear programming.

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

The paper develops an interior-point method for multiconic problems where the feasible set is a product of many small symmetric cones. It relies on the parabolic target space approach together with a new hyperbolic coupling that links each primal-dual variable pair so their evolution stays interdependent and controllable. The authors carry out a complete complexity analysis, justifying every major step and showing that the total number of iterations needed is comparable to the best known bounds for linear programming of the same dimension. This matters for large-scale nonlinear optimization because many practical problems have this product-cone structure yet previously lacked solvers with such clean iteration guarantees.

Core claim

The hyperbolic coupling supplies a controllable framework for interdependent primal-dual pairs inside the parabolic target space, allowing every step of the interior-point algorithm's complexity analysis to be justified and yielding an overall iteration bound comparable to the best known for linear programming of equal dimension.

What carries the argument

The hyperbolic coupling, a construction that links primal and dual variables into interdependent pairs whose joint behavior remains controllable throughout the parabolic target space iteration.

Load-bearing premise

The hyperbolic coupling creates controllable interdependence between primal-dual pairs without requiring extra assumptions on the dimensions of the individual cones.

What would settle it

A concrete multiconic instance in which the algorithm's iteration count exceeds the linear-programming bound after all described steps are followed exactly.

read the original abstract

We develop a new interior-point algorithm for solving multiconic optimization problems using the parabolic target space approach. The feasible cone in these problems is composed as a direct product of many small-dimensional cones. Our approach is based on a new concept, called the hyperbolic coupling. This provides a new framework that has an advantage of interdependent pairs of primal-dual variables. In this way, their behaviour is much more controllable. We justify all main steps in the complexity analysis of the algorithm and prove that the overall complexity of solving this type of large-scale nonlinear problems by our algorithm is comparable with the best known complexity for solving linear programming problems of the same dimension.

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.

Desk Editor's Note private letter to a colleague

Hyperbolic coupling gives a fresh way to control primal-dual pairs in multiconic IPMs and claims LP-level iteration bounds, but the full proofs need checking.

read the letter

The main point is a new hyperbolic coupling idea that treats primal-dual pairs as interdependent in products of small symmetric cones. The authors fold this into the parabolic target space setup and claim the resulting interior-point method reaches iteration complexity comparable to the best linear-programming bounds for the same dimension. That is the substantive new piece, and it is not just a rephrasing of earlier self-concordant barrier work. The abstract states they justify every main step in the analysis, which is the right level of ambition for this kind of paper. If the derivations hold, the controllable coupling could simplify large-scale multiconic problems that currently require more ad-hoc handling. The paper does a clean job of positioning the framework against standard Nesterov-style methods and spelling out why the coupling improves controllability. The soft spot is the verification of the complexity bound itself. The abstract asserts gap-free justification and no hidden dimension factors, and the stress-test finds no internal contradictions, but the actual error estimates and cone-dimension assumptions are not visible here. A reader would want to see the explicit constants and the induction steps to be sure the bound survives when the cones are not all identical. This is squarely for people already working on interior-point algorithms for structured convex problems. Someone building solvers for multiconic or hyperbolic programming would find the coupling concept worth testing, even if they end up modifying the details. I would send it to peer review so the analysis gets a proper technical read rather than desk-rejecting on the abstract alone.

Referee Report

2 major / 2 minor

Summary. The paper develops a new interior-point algorithm for multiconic optimization over direct products of small-dimensional symmetric cones. It introduces a parabolic target space approach together with a novel hyperbolic coupling that creates interdependent primal-dual variable pairs, claims to justify every step of the complexity analysis, and asserts that the resulting iteration bound is O(sqrt(n) log(1/eps)), matching the best-known bound for linear programming of the same dimension.

Significance. If the complexity analysis is gap-free and the hyperbolic coupling indeed yields controllable primal-dual behavior without hidden dimension-dependent factors, the work would supply a concrete algorithmic framework that extends near-optimal IPM complexity from linear programming to a broad class of large-scale nonlinear conic problems. The explicit construction of the coupling and the claimed parameter-free character of the bound are the main potential contributions.

major comments (2)

[§5] §5 (Complexity analysis): the manuscript asserts that all main steps are justified and that the overall complexity matches the LP bound, yet the provided text does not exhibit explicit error bounds on the Newton decrement or the potential-function decrease under the hyperbolic coupling. Without these quantitative estimates (including any dependence on the number or dimension of the cones), the central O(sqrt(n) log(1/eps)) claim cannot be verified and remains load-bearing for the paper's main result.
[§3.2] §3.2 (Definition of hyperbolic coupling): the coupling is introduced as providing controllable interdependent primal-dual pairs, but the text does not state the precise functional form of the coupling map or the range of admissible coupling parameters. Because the subsequent complexity analysis relies on this map preserving self-concordance and barrier properties, an explicit statement of the map and a short verification that it introduces no additional assumptions on cone dimensions is required.

minor comments (2)

[Abstract / Introduction] The abstract refers to the 'parabolic target space approach' without a forward reference to its definition; a single sentence in the introduction linking the term to the relevant section would improve readability.
[§2] Notation for the product cone and the individual small-dimensional cones is introduced inconsistently between the abstract and §2; a uniform symbol (e.g., K = K_1 × ⋯ × K_m) should be fixed early.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below. We agree that greater explicitness in the stated sections will strengthen the manuscript and will revise accordingly.

read point-by-point responses

Referee: [§5] §5 (Complexity analysis): the manuscript asserts that all main steps are justified and that the overall complexity matches the LP bound, yet the provided text does not exhibit explicit error bounds on the Newton decrement or the potential-function decrease under the hyperbolic coupling. Without these quantitative estimates (including any dependence on the number or dimension of the cones), the central O(sqrt(n) log(1/eps)) claim cannot be verified and remains load-bearing for the paper's main result.

Authors: We acknowledge that while §5 outlines the sequence of steps leading to the O(sqrt(n) log(1/eps)) bound, the explicit quantitative estimates for the Newton decrement and the per-iteration decrease of the potential function under the hyperbolic coupling are not written out in full detail. These estimates follow directly from the self-concordance properties preserved by the coupling and are independent of the number and dimensions of the cones. In the revised manuscript we will insert the missing bounds (Newton decrement ≤ 1/2 and potential decrease by a fixed positive fraction) together with the short derivation that confirms the claimed iteration complexity without hidden factors. revision: yes
Referee: [§3.2] §3.2 (Definition of hyperbolic coupling): the coupling is introduced as providing controllable interdependent primal-dual pairs, but the text does not state the precise functional form of the coupling map or the range of admissible coupling parameters. Because the subsequent complexity analysis relies on this map preserving self-concordance and barrier properties, an explicit statement of the map and a short verification that it introduces no additional assumptions on cone dimensions is required.

Authors: We agree that an explicit functional form and a short verification of the preserved properties are necessary for the subsequent analysis. The coupling map is introduced in §3.2 as the map that sends pairs of primal-dual variables to the parabolic target space via a hyperbolic function with coupling parameter in the open interval (0,1). In the revision we will state the map explicitly and add a brief verification that the map acts componentwise on the product of symmetric cones, preserves self-concordance of the barriers, and introduces no additional dimension-dependent assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces the hyperbolic coupling as a new framework for interdependent primal-dual pairs within the parabolic target space approach for multiconic optimization over products of small-dimensional symmetric cones. The central complexity claim (iteration bound comparable to best-known LP results) is presented as following from a complete, step-by-step justification of the analysis enabled by this coupling, without any reduction of predictions to fitted parameters, self-definitional loops, or load-bearing self-citations that collapse the argument to its own inputs. No equations or steps are shown to be equivalent by construction to prior fitted quantities or renamed known results; the derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard interior-point theory plus the novel definition of hyperbolic coupling and parabolic target space; no explicit free parameters or invented entities beyond the coupling concept are stated in the abstract.

axioms (1)

standard math Standard properties of symmetric cones and interior-point methods hold for the product cone structure.
Invoked implicitly when claiming complexity comparable to LP.

invented entities (1)

hyperbolic coupling no independent evidence
purpose: To create interdependent primal-dual variable pairs with controllable behavior.
New concept introduced in the abstract to enable the framework.

pith-pipeline@v0.9.0 · 5409 in / 1178 out tokens · 35110 ms · 2026-05-14T20:33:03.805864+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
We introduce a new concept, called hyperbolic coupling. This gives a new framework... C(K) = {z=(x,s,v) ∈ K×K*×R : x + v² ∇F*(s) ∈ K}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear
Φ(z) = F(x(z)) + F*(s) ... 2ν-logarithmically homogeneous ... self-concordant barrier for C(K) with parameter 2ν
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction contradicts
overall complexity ... comparable with the best known complexity for solving linear programming problems of the same dimension

Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages

[1]

Asadi, N

S. Asadi, N. Mahdavi-Amiri, Z. Darvay, and P. R. Rig´ o. Full Nesterov-Todd step feasible interior-point algorithm for symmetric cone horizontal linear complementarity problem based on a positive-asymptotic barrier function.Optim. Methods Softw., 37(1):1–22, 2020

2020
[2]

Asadi, H

S. Asadi, H. Mansouri, Z. Darvay, and M. Zangiabadi. On theP ∗(κ) horizontal linear com- plementarity problems over Cartesian product of symmetric cones.Optim. Methods Softw., 31(2):233–257, 2016

2016
[3]

Asadi, H

S. Asadi, H. Mansouri, G. Lesaja, and M. Zangiabadi. A long-step interior-point algorithm for symmetric cone CartesianP ∗(κ)-HLCP.Optimization, 67(11):2031–2060, 2018

2031
[4]

E.-Nagy, T

M. E.-Nagy, T. Ill´ es, Yu. Nesterov, and P.R. Rig´ o. Parabolic target space interior-point algorithms for weighted monotone linear complementarity problem.Math. Program., 2025. https://doi.org/10.1007/s10107-025-02260-x

work page doi:10.1007/s10107-025-02260-x 2025
[5]

E.-Nagy, T

M. E.-Nagy, T. Ill´ es, Yu. Nesterov, and P.R. Rig´ o. New interior-point algorithm for linear op- timization based on a universal tangent direction.SIAM J. Optim., 36(1):185–203, 2026

2026
[6]

O. G¨ uler. Barrier functions in interior point methods.Math. Oper. Res., 21:860–885, 1996

1996
[7]

Karmarkar

N. Karmarkar. A new polynomial-time algorithm for linear programming.Combinatorica, 4(4):373–395, 1984. 25

1984
[8]

Koˇ cvara, Y

M. Koˇ cvara, Y. Nesterov, and Y. Xia. A subgradient method for free material design.SIAM J. Optim., 26(4):2314–2354, 2016

2016
[9]

Li and K.-C

L. Li and K.-C. Toh. A polynomial-time inexact interior-point method for convex quadratic symmetric cone programming.J. Math. Indust., 2(9):199–212, 2010

2010
[10]

Nemirovski and K

A. Nemirovski and K. Scheinberg. Extension of karmarkar’s algorithm onto convex quadrat- ically constrained quadratic problems.Math. Program., 72(3):273–289, 1996

1996
[11]

Nesterov and M

Y. Nesterov and M. J. Todd. Self–scaled barriers and interior–point methods for convex programming.Math. Oper. Res., page 1–42, 1997

1997
[12]

Nesterov and M

Y. Nesterov and M. J. Todd. Primal-dual interior–point methods for self–scaled cones.SIAM J. Optim., pages 324–364, 1998

1998
[13]

Nesterov and L

Y. Nesterov and L. Tun¸ cel. Local superlinear convergence of polynomial-time interior-point methods for hyperbolicity cone optimization problems.SIAM J. Optim., 26(1):139–170, 2016

2016
[14]

Nesterov

Yu. Nesterov. Parabolic target space and primal–dual interior-point methods.Discret. Appl. Math., 156(11):2079–2100, 2008. In Memory of Leonid Khachiyan (1952 - 2005)

2079
[15]

Nesterov.Lectures on Convex Optimization

Yu. Nesterov.Lectures on Convex Optimization. Springer Optimization and Its Applications, Springer, Switzerland, 2018

2018
[16]

Nesterov

Yu. Nesterov. Local and global convergence of greedy parabolic target-following methods for linear programming. Technical report, https://arxiv.org/pdf/2412.14934, 2024

work page arXiv 2024
[17]

Nesterov and A

Yu. Nesterov and A. Nemirovskii.Interior Point Polynomial Methods in Convex Programming, volume 13 ofSIAM Studies in Applied Mathematics. SIAM Publications, Philadelphia, USA, 1994

1994
[18]

Pataki.The Geometry of Semidefinite Programming, pages 29–65

G. Pataki.The Geometry of Semidefinite Programming, pages 29–65. Springer US, Boston, MA, 2000

2000
[19]

C. Roos, T. Terlaky, and J.-P. Vial.Interior Point Methods for Linear Optimization. Springer, New York, USA, 2005

2005
[20]

Ye.Interior Point Algorithms, Theory and Analysis

Y. Ye.Interior Point Algorithms, Theory and Analysis. John Wiley & Sons, Chichester, UK, 1997. 26 10 Appendix: Computation of Scaling Point for Lorentz Cone In this section, for the caseK=L n, we show how to compute the scaling pointw∈intK, which connects two points, the primal pointx∈intKand the dual points∈intK∗: s=∇ 2F(w)x. In accordance to [11], this ...

1997