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Error Bounds and Singularity Degree in Semidefinite Programming

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Pith's one-line read This paper proves that for a class of central paths in semidefinite programming, an upper bound on the maximum rank of the solution set turns the observable tail eigenvalues of the iterates into a computable lower bound on the…

desk verdict A useful, mostly solid paper: the forward-error lower bound is clean and new, but the Section 4 convergence bridge is asserted by citation and needs real proof before the hardness claims can be trusted. read the letter →

arxiv 1908.04357 v1 pith:KY7MZYSM submitted 2019-08-12 math.OC

Stefan Sremac , Hugo J. Woerdeman , Henry Wolkowicz This is my paper · ORCID
classification math.OC MSC 90C2290C25
keywords semidefiniteprogrammingforwarderrorbackwardsingularitydegreecentralpathmaximumrankfacialreductioneigenvalueratios
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

Semidefinite programming solvers can return a point with tiny violation of the optimality conditions but a large true distance to the solution set. This paper proves that along central paths converging to the relative interior of the solution set, any upper bound on the maximum rank of the solutions yields a computable lower bound on that true distance, formed from the tail eigenvalues of the iterate beyond the rank bound. That lower bound complements the classical upper bound, so the forward-backward error relation becomes two-sided for this class of paths. For the external central path defined by maximizing $\alpha\log\det X$ over the perturbed constraints $A(X)=b+\alpha A(B)$, the paper also shows that singularity degree (the number of facial-reduction steps needed to make the problem strictly feasible) greater than 1 forces at least one eigenvalue to vanish slower than $O(\alpha)$, making large singularity degree, in this path family, a sufficient condition for slow convergence.

What carries the argument

The key machinery is the maximum rank $r$ of the solution set $F$ (the largest rank attained by any solution), probed through two eigenvalue ratios: the adjacent-eigenvalue ratio $\lambda_i(X(\alpha))/\lambda_{i+1}(X(\alpha))$, whose first blow-up locates $r$, and the Q-convergence ratio $\lambda_i(X(\sigma^{k+1}))/\lambda_i(X(\sigma^k))$, whose limit inferior separates eigenvalues that vanish from those that do not. A rank upper bound becomes an error lower bound through the identities $\|X\|_F=\|\lambda(X)\|_2$ and $\langle X,Y\rangle\le \lambda(X)^T\lambda(Y)$: because every solution has rank at most $r$, the squared distance from $X(\alpha)$ to any solution must dominate the squared norm of the tail eigenvalues. The second mechanism is the external primal-dual central path with $Z(\alpha)=\alpha X(\alpha)^{-1}$, whose limit structure forces at least $\operatorname{sd}(F)$ different vanishing rates among the blocks of $Z(\alpha)$.

What would settle it

The sufficiency claim in Theorem 4.4 would be refuted by an explicit spectrahedron with singularity degree 2 whose external central path (4.1) has every vanishing eigenvalue of $X(\alpha)$ bounded by a constant times $\alpha$ as $\alpha\to 0$; following the path numerically for a small instance and measuring $\lambda_{r+1}(X(\alpha))/\alpha$ would settle it.

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Extended reading notes

Core claim

The paper's central claim is that the gap between backward and forward error in semidefinite programming can be certified from eigenvalue data collected along the path, without knowing the solution set. If a central path $\{X(\alpha):\alpha>0\}$ satisfies the assumptions that $X(\alpha)\succ 0$ and $X(\alpha)\to \bar X\in\operatorname{relint}(F)$, and if $r$ upper-bounds the maximum rank over $F$, then Theorem 3.10 gives $\epsilon_f(X(\alpha),F)\ge \|(\lambda_{r+1}(X(\alpha)),\dots,\lambda_n(X(\alpha)))^T\|_2$. The proof rests on the spectral norm identity and the trace bound $\langle X,Y\rangle\le \lambda(X)^T\lambda(Y)$. For the external central path defined by maximizing $\alpha\log\det X$ over $A(X)=b+\alpha A(B)$, the paper proves Theorem 4.4: if $\operatorname{sd}(F)>1$, some eigenvalue of $X(\alpha)$ converges to $0$ at a rate that is not $O(\alpha)$, and the dual path contains at least $\operatorname{sd}(F)$ distinct vanishing rates.

Load-bearing premise

The load-bearing premise is that the central path converges to a point in the relative interior of the solution set, the interior within the flat space that contains it, because only then does its rank equal the maximum rank over all solutions, and this condition cannot be verified from observable solver data.

Editorial extensions

If this is right

  • Practitioners can turn any rank upper bound, from Proposition 3.3 or Corollary 3.8, into a certified lower bound on the distance to the solution set without knowing the solution set.
  • For the external central path family, the classical upper bound and the new lower bound sandwich the forward error, so small backward error can coexist with large forward error exactly when singularity degree is large.
  • Maximum rank over $F$ can be estimated as the smallest index whose eigenvalue ratio blows up, and validated by comparing Q-convergence rates against thresholds $\sigma^{2^{-(d-1)}}$, yielding lower bounds on singularity degree.
  • Since $Z(\alpha)=\alpha X(\alpha)^{-1}$, the distinct vanishing rates of blocks in the dual path imply distinct Q-convergence rates in the primal path, explaining why interior-point solvers stall when the fastest block reaches machine precision.
  • For feasibility SDPs with bounded solution set, singularity degree greater than 1 rules out $O(\alpha)$ convergence of all vanishing primal eigenvalues, so slow convergence is forced along this path.

Reading between the lines

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

  • The eigenvalue-tail lower bound could be implemented as an online stopping criterion: declare a solution unreliable whenever $\|(\lambda_{r+1},\dots,\lambda_n)\|_2$ exceeds a user tolerance, even if the backward error looks solved; the paper demonstrates the gap numerically but does not propose this as an algorithm.
  • Because the relative-interior convergence assumption is not checkable from solver output, a natural stress test is to run multiple perturbed paths with different $B$ in (4.1) and see whether the rank and error bounds stabilize; instability would signal that the assumption is failing.
  • The paper's empirical observation that the number of distinct vanishing rates appears to upper-bound singularity degree suggests a tractable, rank-free route to singularity-degree estimation; proving or disproving this is left open by the authors.
  • Extending the sufficiency theorem beyond feasibility problems would require controlling the unknown optimal value $p^*$; the duality-gap proxy in Remark 3.9 is a step, and a plausible next direction is to prove analogous block-rate results for the primal-dual path of the shifted objective.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the gap between forward error (distance to the solution set) and backward error (residual in the optimality conditions) for semidefinite programs without Slater points. In Section 3 the authors prove a lower bound on forward error, complementing Sturm's upper bound, under the assumption that the path under consideration converges to the relative interior of the solution set. The lower bound is obtained by first bounding the maximum rank over the solution set via eigenvalue ratios along the path, and then using Fan's inequality. In Section 4 the authors analyze a family of external-type 'central' paths proposed by Potra and Sheng, claiming convergence to a point in the relative interior of the solution set, and use this to argue that large singularity degree is, for this path family, a sufficient condition for sublinear convergence of some vanishing eigenvalue. Numerical case studies in Section 5 illustrate the bounds and the rank/singularity-degree estimates on five spectrahedra.

Significance. If the results hold, the paper makes two useful contributions. First, Theorem 3.10 is cleanly proved and gives a computable lower bound on the unmeasurable forward error once an upper bound on the maximum rank is known, thereby complementing Sturm's one-sided upper bound; the proof via Fan's inequality and the rank bound is elementary and correct. Second, the eigenvalue-ratio criteria in Proposition 3.3 and Corollaries 3.7-3.8 provide a practical, though heuristic, way to estimate maximum rank and to lower-bound singularity degree, and the numerical section gives detailed, honest case studies showing the method at work. The main theoretical claim of Section 4 - that large singularity degree is sufficient for slow convergence of a natural central path - would be significant if the underlying convergence theorem (Theorem 4.2) is fully proved, because it would make Sturm's necessary condition two-sided for that path family. The paper is clear about which statements are rigorous and which are inferred from numerical evidence, which is a strength.

major comments (4)
  1. [Section 3.1, Lemma 3.4] The proof of Theorem 4.2 is not supplied. The text asserts that the result 'may be deduced' from Goldfarb-Scheinberg [11] and Milnor's lemma via Halicka et al. [12,13], but no deduction is shown. The path (4.1)-(4.2) is not the standard central path for a fixed SDP: the right-hand side itself is perturbed to b + alpha A(B), so the feasible set varies with alpha, and the hypotheses under which the cited convergence results transfer to this external path are not checked. This is load-bearing, because Corollary 4.3, Theorem 4.4, and Theorem 4.7 all depend on Theorem 4.2. The authors' own caveat that they are not aware of this exact result in the literature underscores the gap. A complete proof, or a reframing of Section 4 as conditional on an explicitly stated convergence assumption, is needed.
  2. [Section 4.2, Theorem 4.7] The proof of Lemma 3.4 is incomplete. In bounding S(alpha) by the maximum of the norms of the diagonal blocks Xi(alpha), it implicitly relies on control of the off-diagonal blocks of QX(alpha)Q^T, but Fact 2.3 only gives bounds on the diagonal blocks. The missing argument is that S(alpha) is positive definite, hence each off-diagonal block has norm at most the geometric mean of the corresponding diagonal block norms; since xi(i) is nondecreasing in i, this yields the claimed O(alpha^{xi(i)}) bound. Please include this argument explicitly.
  3. [Section 5, Corollary 3.8] The proof of Theorem 4.7 is not fully rigorous. The iterative construction of successive exposing vectors involves normalizing coefficient vectors and passing to limit points, and several key steps are compressed: the justification that (A*(y2))11 is positive semidefinite after normalization, the 'without loss of generality' block-diagonal assumption in (4.11), and the statement that 'by reasoning analogous' to the first step we may continue. In particular, the termination of the process and the claimed bound d in [sd(F), mbar] are only justified by an informal 'continue in this fashion'. Please provide a complete induction that tracks the rates of the coefficients at each reduction step and compares the number of steps with the facial reduction process defining sd(F).
  4. [Section 4.2, Theorem 4.4] The numerical estimates of maximum rank and singularity degree in Section 5 rely on choosing the threshold tau in Corollary 3.8 by visual inspection of the plots (Figures 5.1-5.5). Since the paper presents this as a method, a principled, reproducible rule for selecting tau, or a sensitivity analysis showing that the conclusions are stable over a range of tau, is needed. Without this, the applicability of the rank-bounding method to new instances is not fully demonstrated.
minor comments (4)
  1. [Section 3.1, Proposition 3.3] The heading 'Analaysis' should be 'Analysis'.
  2. [Section 3, Assumption 3.2] The phrase 'blows up' in Proposition 3.3 is informal; for a formal statement, use the liminf/limsup formulation as in Corollary 3.8, and clarify that in practice the ratio may blow up slowly.
  3. [Section 5, Table 5.1] Assumption 3.2(i) is asserted to hold for 'many of the well-known algorithms', but no specific justification is given. Since all of Section 3 depends on the limit lying in relint(F), a brief discussion of which families of central paths are known to satisfy this assumption (and which are not) would sharpen the scope of the results.
  4. [Section 1] The table would benefit from a note clarifying which entries are proven bounds versus heuristic estimates; in particular, the reported lower bounds on singularity degree for spec4 and spec5 are derived from the visual threshold choice, not from a theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rank bound is used as a hypothesis in a conditional eigenvalue inequality, and the Section 4 path convergence is imported from external, non-self citations; the caveats are rigor gaps, not circular reductions.

full rationale

The paper's main derivation chain is not circular. Theorem 3.10 is a direct consequence of Fan's eigenvalue inequality: given any upper bound r on the maximum rank over the solution set F, the tail eigenvalues of X(alpha) give a valid lower bound on the true forward error. The rank upper bound is not fitted to the quantity being predicted; Corollary 3.8 states a conditional equivalence involving a threshold tau, and Proposition 3.3 identifies the smallest index whose adjacent-eigenvalue ratio blows up. The numerical Section 5 estimates are checked against independently known values, and in spec3 the reported lower bound on singularity degree (4) falls short of the known value (5), which rules out calibration-by-construction. The Section 4 results rest on Theorem 4.2, whose proof is explicitly cited to Goldfarb-Scheinberg and to Milnor's lemma via Halicka, de Klerk, and Roos. Those are external, non-self citations supplying independent evidence; the authors' caveat 'we are not aware of this exact result in the literature' signals an incompleteness in the written deduction, but importing a known theorem is not circularity. The self-citations that appear (e.g., [5], [15], [24], [29]) are used for algorithmic directions, benchmark instances, or known lower bounds in case studies, not as the load-bearing justification of the theoretical claims. The visual choice of tau in Corollary 3.8 is a methodological weakness, but the theorem itself is a valid conditional statement, so no prediction is equivalent to its input by definition. Overall, no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper's input structure: Sturm's theorem, the relint-convergence assumption, standard spectral inequalities, and the Potra-Sheng path construction. The only hand-set quantities appear in the numerical section (σ = 0.6 and per-plot thresholds τ), which affect Table 5.1 but not the theorems. No new entities are postulated.

free parameters (2)
  • sigma (sampling ratio) = 0.6
    Q-convergence ratios are sampled at α = σ^k with σ = 0.6 (Section 5). The thresholds σ^{2^{-(d-1)}} and their numerical distinguishability from 1 depend on this choice; the authors note in Remark 3.9's vicinity that large singularity degree makes σ^{2^{-(sd-1)}} hard to distinguish from 1 numerically.
  • tau (per-instance threshold) = chosen by visual inspection; not reported numerically
    Corollary 3.8 requires a τ ∈ (0,1) 'numerically distinguishable from 1' to decide which eigenvalue-ratio curves converge to values below τ. In Section 5 this threshold is set by looking at the plots (e.g., spec2's reading is called 'speculative'), so the Table 5.1 bounds inherit this subjective choice. The theorems themselves are parameter-free.
assumptions (6)
  • standard math Sturm's error bounds (Facts 2.2 and 2.3): forward error is O(backward error^{2^{-sd(F)}}), and X(α) has an orthogonal block decomposition with diagonal blocks of size O(α^{ξ(i)}).
    Imported from Sturm [25] without proof; used as the backbone of Lemma 3.4 and of the thresholds in Corollaries 3.7-3.8.
  • domain assumption Assumption 3.2(i): the central path converges to a point in relint(F).
    Stated in Section 3; it is what links observed eigenvalue ratios to the maximum rank over F, since rank is constant on relint. It is not verifiable from solver data and is supplied for the Section 4 path by Theorem 4.2.
  • standard math Convergence and limit-point structure of the Potra-Sheng path (Theorem 4.2): unique limit with X̄ in relint(F) and Z̄ in relint of the dual cone.
    The paper deduces this from Goldfarb-Scheinberg [11] and Milnor's lemma via Halická et al. [12,13]; the deduction is asserted, not shown, and it underlies Corollary 4.3 and Theorems 4.4-4.7.
  • domain assumption Assumption 4.1: F bounded, nonempty, sd(F) ≥ 1, F ≠ {0}, A surjective; and b(α) = b + αA(B) for fixed B ≻ 0.
    Scope of the Section 4 hardness results; boundedness guarantees the determinant maximizer exists, and the perturbation b(α) defines the external path of Potra-Sheng.
  • standard math Spectral inequalities: Fan trace inequality ⟨X,Y⟩ ≤ λ(X)ᵀλ(Y) and eigenvalue interlacing for principal submatrices.
    Used in the proof of Theorem 3.10 (Fan) and in Lemma 3.4 and Corollary 4.8 (interlacing); standard linear algebra cited to [10,14].
  • standard math Rank-relative-interior fact: for a convex set in S^n_+, rank is constant on relint and equals the maximum over the set.
    Tacit in Proposition 3.3 and Lemma 3.4 when asserting that rank(X̄) equals the maximum rank over F; standard convex-analysis fact.

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Pith. "Pith review of Error Bounds and Singularity Degree in Semidefinite Programming." pith.science (2026). https://pith.science/paper/KY7MZYSM

@misc{pith2026190804357,
  author       = {Pith},
  title        = {Pith review of: Error Bounds and Singularity Degree in Semidefinite Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KY7MZYSM}},
  note         = {Machine review of arXiv:1908.04357}
}
read the original abstract

In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the unmeasurable `true error') and backward error (the measurable violation of optimality conditions). In his seminal work, Sturm provided an upper bound on forward error in terms of backward error and singularity degree. In this paper we provide a method to bound the maximum rank over all solutions and use this result to obtain a lower bound on forward error for a class of convergent sequences. This lower bound complements the upper bound of Sturm. The results of Sturm imply that semidefinite programs with slow convergence necessarily have large singularity degree. Here we show that large singularity degree is, in some sense, also a sufficient condition for slow convergence for a family of external-type `central' paths. Our results are supported by numerical observations.

Figures

Figures reproduced from arXiv: 1908.04357 by the authors.

Figure 5.1
Figure 5.1. The dashed lines coincide with the values [PITH_FULL_IMAGE:figures/full_fig_p019_5_1.png] view at source ↗
Figure 5
Figure 5. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 5
Figure 5. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗

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