Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers

Roland Hildebrand; Vitali Pirau

REVIEW 3 major objections 5 minor 8 references

Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that solving a convex relaxation of the barrier-parameter problem yields a valid self-concordant barrier whose parameter exceeds the optimum by at most a function $\tilde\nu(\nu_{\mathrm{rel}})$, and that near the lowest…

desk verdict Genuinely new convexification framework with a proven lower bound, but the headline upper bound is a numerical estimate rather than a theorem. read the letter →

arxiv 2507.01812 v2 pith:QS45DVCL submitted 2025-07-02 math.OC

Vitali Pirau , Roland Hildebrand This is my paper

classification math.OC MSC 90C2590C2690C5152A20

keywords self-concordantbarriersbarrierparameterconvexrelaxationconicprogramminginterior-pointmethodsnon-convexoptimizationhullpowercones

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Interior-point methods for conic programs run faster when the cone admits a self-concordant barrier with a small parameter $\nu$, but for most non-symmetric cones the optimal barrier is unknown and finding it is a non-convex, infinite-dimensional problem. This paper studies what is lost when that problem is replaced by its convex relaxation: it proves that the relaxation's optimal value $\nu_{\mathrm{rel}}$ brackets the true optimum $\nu_{\mathrm{opt}}$ from below, and that any solution of the relaxed constraints yields a genuine barrier with parameter no larger than a function $\tilde\nu(\nu_{\mathrm{rel}})$. The gap $\tilde\nu-\nu$ is second-order near the universal lower bound $\nu = 2$ — the paper gives the explicit asymptotic $\nu + O((\nu-2)^2)$ — and is numerically computed for $\nu \in [2.1, 5.1]$ by comparing a one-parameter family of non-convex bodies in $\mathbb{R}^3$ with their convex hulls. This makes the convex relaxation a viable route to near-optimal barriers for low-dimensional non-symmetric cones such as power cones.

What carries the argument

The central object is the non-convex body $P_\nu \subset \mathbb{R}^3$ of admissible derivative triples $(x_1,x_2,x_3) = (\tilde p, \tilde h, \tilde w)$, defined by the two-sided inequalities $\sqrt{x_2-x_1^2}/\sqrt{\nu-1} \le \pm x_1 + 1 \le \sqrt{\nu-1}\sqrt{x_2-x_1^2}$ and $|x_3 - 6x_2x_1 + 4x_1^3| \le 2(\nu-2)/\sqrt{\nu-1}\,(x_2-x_1^2)^{3/2}$. The affine coordinate change from the derivatives $(p,h,w)$ of a function on an interval to $(\tilde p,\tilde h,\tilde w)$, given in Lemma 6, makes local self-concordance conditions pointwise independent of the base point $x$. Theorem 1 then reduces admissibility of a function $f$ on a compact convex set to the requirement that every directional triple lie in $P_\nu$, and Corollary 2 shows that if $P_{\tilde\nu}$ contains $\operatorname{conv}(P_\nu)$, every convex combination of $\nu$-barriers is a $\tilde\nu$-barrier. The quantitative loss is thus captured entirely by the containment function $\tilde\nu(\nu) = \inf\{\tilde\nu : \operatorname{conv}(P_\nu) \subseteq P_{\tilde\nu}\}$, whose lower bound comes from a tangent-line argument on the projection of $\operatorname{conv}(P_\nu)$ and whose upper estimate comes from a numerical moment/SDP construction of the convex hull.

What would settle it

Compute the true value of $\tilde\nu(\nu)$ for one specific parameter, say $\nu = 3$, by an independent algebraic or interval-certified method (e.g., exact parametrization of the boundary of $\operatorname{conv}(P_3)$ and the containment test $\operatorname{conv}(P_3) \subseteq P_{\tilde\nu}$) and compare it with the tabulated value; a difference larger than the reported grid discrepancies would show the numerical upper bound is inaccurate. Alternatively, take two explicit $3$-barriers on a low-dimensional power cone, form their convex combination, and numerically test whether its barrier parameter exceeds the predicted $\tilde\nu(3)$; a violation would disprove the conjectured bound.

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Extended reading notes

Core claim

The paper establishes a two-sided bound that converts the non-convex barrier-parameter problem into a convex one without losing control of the objective: for any cone and any parameter $\nu \ge 2$, if the convexified derivative constraints are feasible, then there exists a self-concordant barrier with parameter at most $\tilde\nu(\nu)$, where $\tilde\nu$ is the least number such that the set $P_{\tilde\nu}$ contains the convex hull of the body $P_\nu$. Equivalently, in the authors' formulation, $\nu_{\mathrm{rel}} \le \nu_{\mathrm{opt}} \le \tilde\nu(\nu_{\mathrm{rel}})$. The quantitative content is the function $\tilde\nu$: it is bounded below by the explicit quadratic $\nu^2/8 + \nu/2 + 1/2$, has asymptotic gap $\tilde\nu(\nu) - \nu = O((\nu-2)^2)$ as $\nu \downarrow 2$, and is estimated numerically for $\nu \in [2.1, 5.1]$ by constructing $\operatorname{conv}(P_\nu)$ with semidefinite programming on a grid. The pointwise conditions use an affine change of variables in the first three derivatives of the barrier's restriction to each interval, which makes the feasibility set independent of the base point. The result means that solving the convex relaxation does not merely produce a lower bound; it produces a usable barrier with a known, modest parameter increase.

Load-bearing premise

The numerical computation of the convex hull of $P(\nu)$ and the pointwise maximum over a finite grid are assumed to give the true value of $\tilde\nu(\nu)$ at the stated accuracy; the paper explicitly flags the maximum as 'assumed to be a good estimate of the true maximum', so the upper bound $\nu_{\mathrm{opt}} \le \tilde\nu(\nu_{\mathrm{rel}})$ is certified only up to this numerical uncertainty, while the quadratic lower bound is rigorous.

Editorial extensions

If this is right

For any cone whose convexified problem can be solved, the barrier produced has parameter at most $\tilde\nu(\nu_{\mathrm{rel}})$, so the relaxation is not merely a heuristic lower bound but a constructive method for building usable barriers.
Near $\nu = 2$ the gap is second-order, so for low-dimensional cones where the optimal parameter is expected to be small, the convexified barrier is nearly optimal.
The rigorous lower bound $\tilde\nu(\nu) \ge \nu^2/8 + \nu/2 + 1/2$ holds for all $\nu \ge 2$ independently of the numerical table, giving a guaranteed worst-case degradation of the convex relaxation.
The numerical estimates of $\tilde\nu$ over $[2.1, 5.1]$ provide a practical performance guarantee for the relaxation in the range relevant to power cones and $p$-norm cones.
The reduction of barrier optimization to the containment $\operatorname{conv}(P_\nu) \subseteq P_{\tilde\nu}$ turns an infinite-dimensional functional problem into a finite-dimensional geometric one that can be attacked numerically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same convexification-loss paradigm would extend to higher-dimensional derivative spaces, but the explicit $\mathbb{R}^3$ geometry is specific to scalar functions on intervals; for cones of dimension at least 3 one would need tensor-valued analogues of $P_\nu$, and the quantitative results would not directly transfer.
The reported grid differences (0.5 between coarsest and finest, 0.09 between middle and finest) suggest that a certified upper bound on $\tilde\nu$ is within reach: an interval-arithmetic or exact algebraic version of the same containment test would turn the numerical table into a rigorous guarantee, complementing the already rigorous quadratic lower bound.
For a fixed finite-dimensional search space, such as polynomial or rational barriers, the convex relaxation can be solved as an SDP and the paper's bound predicts in advance how much parameter slack to expect; a direct self-concordance check then becomes a verification step rather than an open search.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read Pirau–Hildebrand. The genuinely new part is the convexification analysis: they reformulate the barrier-parameter minimization through the pointwise derivative body P(ν), and Corollary 2 gives a clean sufficient condition under which convex combinations of ν-barriers become ν̃-barriers, namely conv P(ν) ⊆ P(ν̃). The projection argument in Section 4 yields a rigorous lower bound ν̃ ≥ ν²/8 + ν/2 + 1/2, and that bound appears correct.

The problem is the upper bound. The headline estimate of ν̃ is computed on a finite (x1, x2) grid, using an SDP-based convex hull, and Section 4 says the maximum 'is assumed to be a good estimate of the true maximum.' Grid differences of 0.5 and 0.09 are shown, but they are not error bounds. No code, no data, and no explicit piecewise-algebraic expression for ν̃ are given, despite the abstract promising one. The conclusion's O((ν−2)²) is asserted, not proven. So the central quantitative claim is a numerical estimate with an honest caveat, not a certified theorem.

That said, the paper is not deceptive. The caveat is explicit, and the framework is sound as far as it goes. The rigorous lower bound alone is a real contribution. The reliance on Hildebrand [2] (a co-author) is legitimate—it is a published theorem, not a circular step—and no parameters are tuned to force the claimed bound.

The fix is straightforward: relabel the upper bound as a numerical conjecture or estimate, release the computational artifacts, and state the asymptotic claim conditionally. Or, if a certified upper bound is desired, derive one. Either way the paper becomes honest about what is proven versus what is conjectured.

This paper is for researchers in conic optimization, especially those working on non-symmetric cones (power, exponential, p-norm) where near-optimal barriers are scarce. It deserves a serious referee: the framework is important enough, and the lower-bound proof is solid, but the numerical claims need to be reframed as estimates. I would accept it with major revision.

Referee Report

3 major / 5 minor

Summary. The paper studies the convex relaxation of the problem of finding the lowest self-concordant barrier parameter for a cone. It reduces the infinite-dimensional barrier problem to pointwise constraints on an affine section, characterizes the feasible derivative triples by a non-convex body P_ν, and shows that if conv P_ν ⊆ P_ν̃ then convex combinations of ν-barriers are ν̃-barriers. The authors then compute numerically the least such ν̃ over a grid and report values for ν in [2.1,5.1]. The paper claims that for small ν_opt the relaxation is nearly tight and becomes asymptotically exact as ν_opt → 2, with degradation ν + O((ν−2)^2). A rigorous lower bound ν̃ ≥ (ν+2)^2/8 is derived from the projection of P(ν).

Significance. If the numerical estimate is correct, the paper provides the first quantitative bound on the loss incurred by convexifying the barrier parameter problem, and it offers a practical route to constructing suboptimal barriers from a convex relaxation. The theoretical reduction to the finite-dimensional body P_ν is clean, and the lower bound ν̃ ≥ (ν+2)^2/8 is derived rigorously. The main unsupported element is the certified upper bound on ν̃: the reported values rest on an unverified SDP/moment computation on a finite grid and on an explicit 'assumed' estimate. The paper also ships no code or data, so the central quantitative claim is not reproducible as written. The piecewise-algebraic formula promised in the introduction is not supplied.

major comments (3)

[Section 4 (numerical estimate of ν̃)] The paper's main quantitative claim, ν_opt ≤ ν̃(ν_rel), depends on the exact least ν̃ such that conv P(ν) ⊆ P(ν̃). The text states, however, that 'the maximum at each point is assumed to be a good estimate of the true maximum' and supports this only by comparing grids of steps s, 2s, and 4s. The reported absolute differences of 0.5 and 0.09 are not rigorous error bounds, and no code or data are provided to reproduce the SDP/moment computation. Consequently the upper bound is not certified; an underestimate of ν̃ by even a small amount would invalidate the claimed guarantee. This needs either certified numerical bounds, a complete specification of the computation, or an explicit restatement of the result as heuristic.
[Section 5 (Conclusion)] The final paragraph asserts that the degradation is ν + O((ν−2)^2). The rigorous part of the paper establishes only the lower bound ν̃ ≥ (ν+2)^2/8, i.e., ν̃ − ν ≥ (ν−2)^2/8. No proof of a matching upper bound is given; the numerical experiments can at most suggest it. Since the introduction and abstract advertise the asymptotic exactness as ν_opt → 2, this proof gap is load-bearing and must be closed or the claims weakened.
[Section 4 (moment/SDP construction)] The construction of conv P(ν) is described only as 'using the moment techniques presented in [3]' and 'semi-definite programming to describe the convex hull of rational curve segments.' As written this is not a verifiable numerical method: the exact SDP formulation, the tolerances, and the handling of the rational curve segments are unspecified. The statement that 'the true value... is equal to the limit of the calculated values on the grid as the step approaches zero' is a convergence claim, not an error bound. Please provide a reproducible specification or a certified computation.

minor comments (5)

[Section 4, Figures 3 and 4] The captions do not identify the axes or the meaning of the curves; please add labels and a legend.
[Introduction and Section 4] The introduction describes ν̃ as a 'piece-wise algebraic function', but the paper never supplies this function or proves that it is piecewise algebraic; please provide the explicit expression or state that only numerical values are computed.
[Section 2, Definition 2] The Lipschitz replacement of the self-concordance condition is justified only by a comment on practical usability; please spell out the closure/density relation between the C^3 barriers of Lemma 1 and the admissible functions of Definition 2, since the equality of optimal values is used implicitly.
[Lemma 4 proof] The existence of a uniform bound C for |p''_±| on a neighbourhood U for all initial points x0 ∈ U is asserted without argument; a short compactness argument would make the proof self-contained.
[Section 4, set inclusion step] The authors should state explicitly how the finite-grid computation of min/max x3 values yields the set inclusion conv P(ν) ⊆ P(ν̃); currently the text moves from per-point intervals to the global inclusion without a formal statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central reduction is a direct geometric inclusion plus a numerical estimate, not a fit to the target result.

full rationale

The paper's derivation chain is not circular. The characterization of self-concordant barriers in terms of admissible functions f on an affine section (Lemma 1) is cited from Hildebrand [2], a published prior theorem by a co-author. While this citation is load-bearing, it is a substantive external result with its own proof, not an assumption of the present conclusion, and the paper independently proves the converse sufficiency direction in Lemmas 4–6. The key convexification statement, Corollary 2, follows from the affine dependence of the derivative triple (p̃,h̃,w̃) on f and the geometric inclusion conv P_ν ⊆ P_ν̃; it does not presuppose the value of ν_opt. The lower bound ν̃ ≥ ν²/8 + ν/2 + 1/2 is derived by an explicit tangent-line calculation, and the numerical upper estimate in Section 4 is obtained by computing the minimal ν̃ from the geometry of P(ν) on a grid, with no parameter fitted to reproduce ν_opt. The statement that the grid maximum is 'assumed to be a good estimate' is a numerical accuracy caveat, not a circular reduction: an inaccurate estimate would weaken the certified guarantee, but it would not make the derivation equivalent to its inputs. The absence of a rigorous error bound for the finite-grid computation is a correctness/rigor concern, not a circularity concern. Therefore no specific circular step meeting the evidentiary standard can be quoted.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No physical free parameters are fitted, but the numerical nutilde values depend on computational discretization choices. The framework is built on Hildebrand's prior characterization [2] and an unproven closure equivalence. The upper bound also assumes that the grid and moment/SDP computations correctly capture the true convex hull of P(nu).

free parameters (2)

Grid step s for sampling P(nu) = approximately 0.012, with coarser grids 2s and 4s
The numerical estimate of the convex hull and of nutilde depends on this discretization. The paper reports only heuristic grid-degradation differences, not a certified convergence bound.
Range and density of nu samples = 101 uniformly distributed values on [2.1, 5.1]
The reported nutilde is the maximum over these sampled nu values and grid points, so behavior outside the sampled range is not demonstrated.

assumptions (4)

domain assumption Lemma 1 / [2, Theorem 2]: a self-concordant barrier on a cone corresponds exactly to a C^3 function f on an affine section satisfying the three stated conditions with gamma = (nu-2)/sqrt(nu-1).
Invoked at the start of Section 2; the entire reformulation into the body P(nu) depends on this external characterization from prior work by a co-author.
domain assumption Passing to the closure via the Lipschitz condition on f'' in Definition 2 does not change the practical barrier parameter problem.
Section 2 asserts this modification has no impact on practical usability because interior-point methods need only the first two derivatives, but it changes the feasible set compared to the original self-concordance condition.
ad hoc to paper The moment/SDP computation of conv P(nu) is exact for the rational curve segments involved.
Section 4 constructs the convex hull using techniques from [3] on a finite grid, but no exactness conditions or numerical tolerances are provided.
ad hoc to paper The finite-grid maximum of the minimal nutilde values is a good estimate of the true maximum over P(nu).
Section 4 states the maximum is 'assumed to be a good estimate of the true maximum'; grid-degradation differences of 0.09 and 0.5 are heuristic checks, not rigorous bounds.

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Cite this review

Pith. "Pith review of Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers." pith.science (2026). https://pith.science/paper/QS45DVCL

@misc{pith2026250701812,
  author       = {Pith},
  title        = {Pith review of: Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QS45DVCL}},
  note         = {Machine review of arXiv:2507.01812}
}

abstract

Self-concordant barriers are essential for interior-point algorithms in conic programming. To speed up the convergence it is of interest to find a barrier with the lowest possible parameter for a given cone. The barrier parameter is a non-convex function on the set of self-concordant barriers on a given cone, and finding an optimal barrier amounts to solving a non-convex infinite-dimensional optimization problem. In this work we study the degradation of the optimal value of the problem when the problem is convexified, and provide an estimate of the accuracy of the convex relaxation. The amount of degradation can be computed by comparing a 1-parameter family of non-convex bodies in $R^3$ with their convex hulls. Our study provides insight into the degree of non-convexity of the problem and opens up the possibility of constructing suboptimal barriers by solving the convex relaxation

Figures

Figures reproduced from arXiv: 2507.01812 by the authors.

**Figure 2.** Projections on the x1, x2 plane of the set P(ν) for ν = 5.1 and ˜ν = 6.30125. Using the moment techniques presented in [3], we have constructed the convex hull conv P(ν) numerically on the same grid, using semi-definite programming to describe the convex hull of rational curve segments. Subsequently, for each value of the parameter ν and for each point (x1, x2) in the grid, the minimum possible value of the param… view at source ↗

**Figure 3.** Value ˜ν calculated for 101 ν values uniformly distributed between 2.1 and 5.1 on different (x1, x2) grids [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Value ˜ν calculated for 101 ν values uniformly distributed between 2.1 and 5.1 on different (x1, x2) grids 5 Conclusion In this paper we have formulated necessary conditions for the existence of a barrier with given parameter ν and have constructed their convexificatio…

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Reference graph

Works this paper leans on

8 extracted references · 5 canonical work pages

[2]

Projectively self-concordant barriers

Roland Hildebrand. Projectively self-concordant barriers. Math. Oper. Res. , 47(3):2444--2463, 2022

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Moments, positive polynomials and their applications , volume 1

Jean Bernard Lasserre. Moments, positive polynomials and their applications , volume 1. World Scientific, 2009

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Characterization of the barrier parameter of homogeneous convex cones

Osman G\"uler. Characterization of the barrier parameter of homogeneous convex cones. Math. Program. , 81:55--76, 1998

work page 1998
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Self-scaled barriers and interior-point methods for convex programming

Yu E Nesterov and Michael J Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations research , 22(1):1--42, 1997

work page 1997
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Towards non-symmetric conic optimization

Yurii Nesterov. Towards non-symmetric conic optimization. Optimization methods and software , 27(4-5):893--917, 2012

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Interior-point Polynomial Algorithms in Convex Programming , volume 13 of SIAM Stud

Yurii Nesterov and Arkadii Nemirovskii. Interior-point Polynomial Algorithms in Convex Programming , volume 13 of SIAM Stud. Appl. Math. SIAM, Philadelphia, 1994

work page 1994
[7]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [2]

Projectively self-concordant barriers

Roland Hildebrand. Projectively self-concordant barriers. Math. Oper. Res. , 47(3):2444--2463, 2022

work page 2022

[2] [3]

Moments, positive polynomials and their applications , volume 1

Jean Bernard Lasserre. Moments, positive polynomials and their applications , volume 1. World Scientific, 2009

2009

[3] [1]

Characterization of the barrier parameter of homogeneous convex cones

Osman G\"uler. Characterization of the barrier parameter of homogeneous convex cones. Math. Program. , 81:55--76, 1998

work page 1998

[4] [4]

Self-scaled barriers and interior-point methods for convex programming

Yu E Nesterov and Michael J Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations research , 22(1):1--42, 1997

work page 1997

[5] [5]

Towards non-symmetric conic optimization

Yurii Nesterov. Towards non-symmetric conic optimization. Optimization methods and software , 27(4-5):893--917, 2012

work page 2012

[6] [6]

Interior-point Polynomial Algorithms in Convex Programming , volume 13 of SIAM Stud

Yurii Nesterov and Arkadii Nemirovskii. Interior-point Polynomial Algorithms in Convex Programming , volume 13 of SIAM Stud. Appl. Math. SIAM, Philadelphia, 1994

work page 1994

[7] [7]

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write newline

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