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REVIEW 3 major objections 5 minor 32 references

On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences

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Pith's one-line read Moment-SOS hierarchy convergence rate tied to the Lojasiewicz exponent of the domain.

desk verdict Good idea, false dual transfer: the reduced SOS hierarchy has a constant duality gap, so Corollary 4.8 is false as stated. read the letter →

arxiv 2507.00572 v1 pith:AHZLU6YL submitted 2025-07-01 math.OC

Hoang Anh Tran , Toh Kim-Chuan This is my paper · ORCID
classification math.OC MSC 90C2290C2614P1044A60
keywords moment-SOShierarchypolynomialoptimizationLojasiewiczexponenttruncatedmomentsequencesHausdorffdistanceChristoffel-Darbouxkernelsum-of-squaressemi-algebraicset
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

This paper establishes that the convergence rate of the moment–sum-of-squares hierarchy for polynomial optimization over a compact basic semi-algebraic set is controlled by the Lojasiewicz exponent of the feasible set: the Hausdorff distance between $r$-level truncated pseudo-moment sequences and true truncated moment sequences is $O(1/r^L)$, giving the same rate for the hierarchy. This matters because previous $O(1/r^2)$ rates were known only for special sets such as balls, simplices, and hypercubes, and relied on explicit Christoffel–Darboux kernel formulas. The paper's geometric route works on general compact semi-algebraic sets after a lifting, and it automatically yields $O(1/r)$ for polytopes and sets satisfying the constraint qualification condition, $O(1/\sqrt{r})$ for domains satisfying Polyak–Lojasiewicz or strong convexity, and $O(1/r^2)$ for general polynomials on a sphere. It also introduces a reduced Schmüdgen-type certificate that keeps the same theoretical rates.

What carries the argument

The central object is the Hausdorff distance $d_k(R(X)_{2r})$ between the projection of the $r$-level pseudo-moment spectrahedron and the set $M_k(X)$ of $k$-truncated moment sequences of probability measures on $X$; Lemma 2.4 converts a bound on this distance into a bound on the optimal-value error of the hierarchy. The arguments are carried by three tools: the perturbed Christoffel–Darboux kernel on products of balls and simplices (Theorem 3.3), which supplies $O(1/r^2)$ approximation of nonnegative polynomials by elements of the preordering; the Lojasiewicz inequality, which bounds the distance to $X$ by a power $L$ of a defining polynomial; and a polynomial lifting $x \mapsto (x, g(x))$ that embeds a general compact semi-algebraic set into a ball–simplex product so the simple-set estimates apply. The reduced certificate $R(X)$ requires only $h_i^2 = 0$ and the inequalities $g_J \ge 0$, which is cheaper than the full Schmüdgen preordering yet still carries the same rate.

What would settle it

Compute the Hausdorff distance $d_k(R(X)_{2r})$ for a compact semi-algebraic set with known Lojasiewicz exponent $L$, such as $X = \{x : g(x) \ge 0\}$ with $g$ having a zero of order $L$ at a boundary point, for increasing $r$; if the distance decays strictly slower than $O(1/r^L)$, the theorem fails. A second targeted check is to search for a level $r$ where the Schmüdgen-type moment SDP (2.8) and its dual (2.9) have a nonzero duality gap, since strong duality is asserted but not proved for the preordering, and one gap would sever the claimed transfer from pseudo-moment sequences to the SOS hierarchy.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the error of the Schmüdgen-type moment hierarchy is a geometric quantity: the distance from the spectrahedron of $r$-level truncated pseudo-moment sequences to the set of true truncated moment sequences on $X$. Theorem 4.7 bounds this Hausdorff distance by $O(1/r^L)$ whenever the Archimedean condition holds, where $L$ is the Lojasiewicz exponent of the semi-algebraic description of $X$; Corollary 4.8 transfers that bound to the lower-bound hierarchy. The proof lifts $X$ into a product of a ball and a simplex using the map $x \mapsto (x, g_1(x), \dots, g_m(x))$, applies a perturbed Christoffel–Darboux kernel bound on that product, then projects back and uses the Lojasiewicz inequality to control how far points in the ambient product are from $X$. The paper also proves sharper rates in special cases and shows the same rate holds for a reduced certificate $R(X)$ instead of the full Schmüdgen preordering.

Load-bearing premise

The argument assumes that the moment-side semidefinite program and its sum-of-squares dual give exactly the same value at every level; if they ever differ, the claimed rate for the sum-of-squares hierarchy does not follow from the proven moment-side bound.

Editorial extensions

If this is right

  • If correct, for any compact basic semi-algebraic set with Lojasiewicz exponent $L$, both the lower-bound moment hierarchy and its SOS dual converge with error $O(1/r^L)$, matching the geometry of the feasible set.
  • For polytopes and sets satisfying the constraint qualification condition, the error becomes $O(1/r)$, a concrete improvement over general bounds.
  • For domains satisfying the Polyak–Lojasiewicz condition or defined by locally strongly convex polynomials, the error is $O(1/\sqrt{r})$.
  • For polynomial optimization over a sphere, the hierarchy converges at $O(1/r^2)$ even for non-homogeneous objectives, extending earlier results that only handled homogeneous polynomials.
  • The reduced certificate $R(X)$ can replace the Schmüdgen preordering without changing the asymptotic rate, with potential computational savings.

Reading between the lines

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

  • Editorial: because the proof works by lifting any compact semi-algebraic set into a ball–simplex product, the same method could in principle attach rates to hierarchies for sets that are images of simple sets under polynomial maps, as long as the violation functions remain controlled by a Lojasiewicz inequality.
  • Editorial: the result suggests a practical way to certify convergence rates for a given instance: estimate the Lojasiewicz exponent of the defining inequalities, then predict the decay of the hierarchy; this can be tested numerically on small random instances.
  • Editorial: if the rate is governed by $L$, then improving a domain's description (for instance, replacing a high-degree inequality by a lower-degree equivalent with exponent $1$) could improve the hierarchy's speed, a design principle not stated in the paper.
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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

3 major / 5 minor

Summary. This paper studies the convergence rate of Lasserre's moment-SOS hierarchy for polynomial optimization over a compact basic semi-algebraic set X defined by polynomial inequalities and equalities. The authors introduce a reduced Schmüdgen-type preordering R(X) in Section 2.1 and analyze the Hausdorff distance between the set of r-truncated pseudo-moment sequences satisfying the associated localizing constraints and the set Mk(X) of true k-truncated moment sequences. The main theorems (4.3, 4.7) bound this distance by O(1/r^L), where L is the Lojasiewicz exponent of X. Corollaries yield O(1/r) for polytopes and sets satisfying CQC, O(1/sqrt(r)) for sets satisfying the Polyak-Lojasiewicz or strong convexity conditions, and O(1/r^2) for products of simple sets and for the sphere. A separate result extends the CD-kernel rate O(1/r^2) to upper-bound hierarchies on products of unit balls and simplices. The paper is self-contained and contains detailed proofs of the CD-kernel estimates and of the Lojasiewicz exponent computations.

Significance. The geometric approach via Hausdorff distance is a genuine novelty: it separates the objective from the domain and ties the rate to a single geometric invariant, the Lojasiewicz exponent. The CD-kernel extension to products and the explicit rates for PL/strongly convex sets are useful contributions, and the appendices provide explicit constants. If the dual-side transfer were valid, the paper would substantially improve the state of the art for general compact semi-algebraic sets, going beyond the O(1/r^{1/10}) of [BMP25]. However, the paper's headline claim that the SOS hierarchy (the dual) converges at the same rate is not established and, for the reduced hierarchy as defined, is false. This is a load-bearing defect in the current version, though it is local to the sign restriction in the definition of R(X).

major comments (3)
  1. [§2.1–2.2, Eqs. (2.3), (2.8)–(2.9)] The dual problem (2.9) is not the dual of the moment problem (2.8). In (2.3), the multipliers τ_i are restricted to R_{\ge0}, but the constraint ℓ_y(h_i^2)=0 in (2.8) is an equality constraint whose Lagrange multiplier must be free. Consequently (2.9) is a strict subset of the true dual and a constant duality gap can appear. For example, take n=1, X={0} defined by h_1(x)=x and g_0(x)=1-x^2\ge0, and f(x)=x. For every r\ge1, the primal (2.8) forces y_2=0 and M_1(y)\succeq0, hence y_1=0 and mlb(f,R(X))_r=0. In the dual (2.9), any certificate x-c=\sigma_0+\tau x^2+\sigma_1(1-x^2) with \sigma_0,\sigma_1 SOS and \tau\ge0, evaluated at x=-1, gives -1-c=\sigma_0(-1)+\tau\ge0, so c\le-1, and c=-1 is attained by x+1=\tfrac12(x+1)^2+\tfrac12(1-x^2). Thus lb(f,R(X))_r=-1 for all r, while the right-hand side of Corollary 4.8 tends to 0. This makes Corollary 4.8 false as stated, and it also invalidates the claims about the dual hierarchy (2.9) in Theorem 4.14(2) and Corollary 4.18. The defect is local: allowing τ_i\in\mathbb{R} in (2.3) restores the true dual; in the example the supremum in (2.9) then equals 0 for all r, matching mlb. Since the primal feasible set M(R(X)_{2r}) is unchanged by this modification, the Hausdorff bound in Theorem 4.7 is not affected.
  2. [§2.3, Lemma 2.4 and the paragraph following it] Lemma 2.4 bounds only fmin-mlb(f,\cdot)_r, i.e., the error of the moment (primal) relaxation. The paragraph after the lemma asserts that the convergence rates of the pairs (2.4)-(2.5), (2.8)-(2.9), and (2.6)-(2.7) are the same as the rates of the Hausdorff distances. This requires strong duality between each primal and its stated dual, but strong duality is not proved for the Schmüdgen preordering pair (2.4)-(2.5) nor for the reduced pair (2.8)-(2.9). The citation [JH16] supports strong duality for the Putinar-type quadratic module under the Archimedean condition, not for the preordering or for R(X). Even for (2.4)-(2.5), no constraint qualification or Slater condition is verified, so the equality of fmin-lb and fmin-mlb remains an assumption. This gap is load-bearing because Corollary 4.8, Theorem 4.14(2), and Corollary 4.18 all state rates for the dual SOS hierarchies.
  3. [§4.4, Theorem 4.14 and Corollary 4.18] The statements that the 'reduced moment-SOS hierarchy (2.8), (2.9)' and the Schmüdgen-type hierarchy (2.4), (2.5) both converge at O(1/r) for polytopes and under CQC inherit both problems above. For the reduced hierarchy, the counterexample in the first comment shows the statement about (2.9) is not merely unproven but false when equalities are present. Since the definition of a polytope in this paper can include equality constraints (see (1.1)), the result as stated covers that case. The authors should restrict the dual-side claims to inequality-only sets, or fix R(X) and then prove strong duality for the corrected pair, before these corollaries can stand.
minor comments (5)
  1. [Lemma 4.6] The statement begins with '4. If If y ∈ Mk(φ(X))' containing a duplicated 'If'.
  2. [Remark 2.3] The remark writes the ball constraint as R−∥x∥^2≥0, while the rest of the paper uses R^2−∥x∥^2≥0; the two conventions are used interchangeably and should be harmonized.
  3. [Theorem 3.9] The statement says 'for any integer r > ...' and then bounds ub(f,T(X))_{mr}; since T(X)_{2mr} is the certificate cone used in the proof, it would be clearer to denote the level by 2mr or define ub with the degree parameter explicitly.
  4. [Section 4.2, Eq. (4.9)] In the displayed definition of φ(X), the inequality constraints p_0,...,p_{m+1} are listed, but the equality constraints include q_j(z)=u_j-g_j(x)=0; the notation q_j is not introduced before its use in Lemma 4.5, which may confuse readers.
  5. [Appendix A, Remark A.2] The remark asserts that Theorem A.1 can be proved by push-forward of the CD kernel, but the actual proof provided uses a general linear transformation and does not construct the CD kernel on A(X); the remark should either be removed or reconciled with the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hausdorff-distance rates are derived from the Lojasiewicz exponent and CD-kernel bounds, not from the target convergence rates.

full rationale

The paper's derivation chain is: for simple sets, Theorem 3.3 establishes the perturbed CD-kernel approximation and Corollary 3.4 derives the O(1/r^2) rate for the moment-SOS hierarchy without invoking the Hausdorff distance as an input; Theorem 3.5 then uses that independent rate to bound d_k(T(X)_{2r}). For general compact sets, Theorem 4.7 bounds d_k(R(X)_{2r}) by lifting to a product of simple sets and applying the Lojasiewicz inequality d_X(x) <= c max{|h_i(x)|}^L; the exponent L is an assumption on the domain, not a quantity fitted from the relaxation error or defined in terms of the target rate. Lemma 2.4 transfers the Hausdorff bound to fmin - mlb, and Corollary 4.8 transfers it to lb via the asserted strong duality between (2.8) and (2.9). The cited results [JH16], [Slo21], [FF21], and [BCR98] are external theorems used as black boxes; none is a self-citation, and none is invoked to rule out alternatives. The reduced certificate R(X) is introduced by definition and then estimated, not assumed to have the stated error. The only flagged concern is the omitted/possibly false strong-duality step for equality-constrained reduced hierarchies (Section 2.2 and Corollary 4.8): the dual (2.9) restricts tau_i >= 0 and may not be the true dual of (2.8), so the rate for lb(f,R(X))_r may fail, as in the skeptic's X={0}, f=x example. That is a correctness gap, not a circular reduction: the claimed rate is not identical to an input by construction, and no parameter is fit to make the prediction come out. Accordingly, no circular step is established, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper's central results rest on standard positivity certificates and the Lojasiewicz inequality. The only nonstandard input is the asserted strong duality for the preordering-based SDP pair, which is load-bearing for the dual SOS hierarchy claims and is not proven.

assumptions (5)
  • standard math Schmüdgen Positivstellensatz (Theorem 2.1)
    Used to guarantee convergence of Schmüdgen-type hierarchies and in Remark 2.3 to add a ball constraint without changing the convergence rate.
  • standard math Putinar Positivstellensatz (Theorem 2.2)
    Used for convergence of Putinar-type hierarchies under the Archimedean condition.
  • standard math Tchakaloff's theorem (cited as [Put97], [BT06])
    Used to represent moment sequences as convex combinations of point evaluations; invoked in Theorem 3.3, Theorem 3.5, Theorem 4.3, and Theorem 4.7.
  • standard math Lojasiewicz inequality for semi-algebraic functions (Lemma 4.2, cited as [BCR98])
    Core geometric input: relates distance to X to the maximum violation of the defining equations. Existence of the exponent L is guaranteed for semi-algebraic sets.
  • ad hoc to paper Strong duality for the Schmüdgen-type (preordering) moment/SOS SDP pair
    Asserted after Lemma 2.4 and used in Corollaries 4.8, 4.14, and 4.18 to transfer the primal moment hierarchy rate to the dual SOS hierarchy. Not proven; the cited [JH16] only covers the Putinar-type quadratic module under Archimedean condition.
invented entities (1)
  • Reduced preordering R(X)_2r independent evidence
    purpose: A weaker certificate than the Schmüdgen preordering, allowing the same convergence rate with fewer SDP constraints in the moment hierarchy.
    Well-defined mathematical construction introduced in Section 2.1. Its properties (inclusions and localizing constraints) are proven, so it has internal evidence, though it is not a physical entity.

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Pith. "Pith review of On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences." pith.science (2026). https://pith.science/paper/AHZLU6YL

@misc{pith2026250700572,
  author       = {Pith},
  title        = {Pith review of: On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHZLU6YL}},
  note         = {Machine review of arXiv:2507.00572}
}
abstract

The moment-SOS hierarchy is a widely applicable framework to address polynomial optimization problems over basic semi-algebraic sets based on positivity certificates of polynomial. Recent works show that the convergence rate of this hierarchy over certain simple sets, namely, the unit ball, hypercube, and standard simplex, is of the order $O(1/r^2)$, where r denotes the level of the moment-SOS hierarchy. This paper aims to provide a comprehensive understanding of the convergence rate of the moment-SOS hierarchy by estimating the Hausdorff distance between the set of truncated pseudo-moment sequences and the set of truncated moment sequences specified by Tchakaloff's theorem. Our results provide a connection between the convergence rate of the moment-SOS hierarchy and the Lojasiewicz exponent L of the domain under the compactness assumption, where we establish the convergence rate of $O(1/r^L)$. Consequently, we obtain the convergence rate of $O(1/r)$ for polytopes and sets satisfying the constraint qualification condition, $O(1/\sqrt{r})$ for domains that either satisfy the Polyak-Lojasiewicz condition or are defined by locally strongly convex polynomials. We also obtain the convergence rate of $O(1/r^2)$ for general polynomials over a sphere.

Figures

Figures reproduced from arXiv: 2507.00572 by the authors.

Figure 1
Figure 1. Flow chart for the results established in this paper. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Projection of a sequence y ∈ Mk(T (Y)2r) onto Mk(X ). Proof. We prove the theorem by evaluating the distance between an arbitrary pseudo-moment sequence y ∈ Mk(R(X )2r) and Mk(X ). First, we conduct two consecutive projections as follows: since Mk(X ) ⊂ Mk(Y), we first project y onto Mk(Y) and denote its projection by y. By Tchakaloff’s theorem, y can be written as a convex combination of some vk(xj ) ′ s with point… view at source ↗
Figure 3
Figure 3. illustrates our idea. After defining y and ye, we evaluate each term in (4.14). M2t(φ(X )) M2t(BR × ∆m K) M(R(φ(X ))2t) M(T (BR × ∆m K)2t) yb φ y ′ ye ′ ψk ◦ πk y=ψk(πk(yb φ)) y=ψk(πk(y ′ )) ye=ψk(πk(ye ′ )) Mk(X ) Mk(BR) Mk(R(X )2r) M(T (BR)2r) y ye y [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Proof of Theorem 4.12. Theorem 4.12. Let X ⊂ Ω o be either a strongly convex semi-algebraic set as in Assumption 1 or the violating function g satisfies the Polyak- Lojasiewicz condition, where Ω o is the interior of the convex set Ω. Then the error for the set of trun…

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