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arxiv: 2604.15072 · v1 · submitted 2026-04-16 · 🧮 math.OC

Duality attainment and strict feasibility of the generalized moment problem and its relaxations

Sami Halaseh , Victor Magron , Mateusz Skomra This is my paper

Pith reviewed 2026-05-10 10:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords generalized moment problemsum-of-squares hierarchiesduality attainmentstrict feasibilitysemialgebraic setsquadratic modulesmoment relaxationsGröbner bases
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The pith

Under a relative interior assumption on the feasible moments, both the dual of the generalized moment problem and all its sum-of-squares relaxations attain their optima.

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

The generalized moment problem seeks a measure supported on a compact semialgebraic set X that optimizes a linear functional while satisfying finitely many polynomial moment equalities. The authors focus on two structural cases for X: sets with nonempty interior, or sets defined as the common zero locus of polynomials that form a Gröbner basis for a real radical ideal. Under the additional hypothesis that the feasible moment sequences have nonempty relative interior inside their affine subspace, they prove that the infinite-dimensional dual attains its value and that every finite-dimensional sum-of-squares strengthening also attains. This supplies conditions under which strong duality holds exactly and the moment-SOS hierarchy meets the true optimum without duality gaps.

Core claim

We prove that if the relative interior of the moment cone intersects the interior of the feasible set in a certain way, then the dual GMP attains its value, and each SOS strengthening attains as well. We give two proofs for the SOS case: one via closedness of quadratic modules and one constructing a strictly feasible measure using exponential densities from Csiszár.

What carries the argument

The relative interior assumption on the feasible set of moment sequences, which forces closedness of the quadratic module and existence of a strictly feasible representing measure.

If this is right

  • Strong duality holds between the primal GMP and its dual under the stated conditions.
  • Each sum-of-squares relaxation admits an optimal solution that can be recovered by solving a finite semidefinite program.
  • The results apply directly when X is a product of spheres, covering GMP instances that arise in tensor optimization.
  • The same attainment statements hold for GMP formulations appearing in quantum information theory under the relative-interior hypothesis.

Where Pith is reading between the lines

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

  • The attainment guarantees may allow one to certify exact optimality of a candidate measure without solving the full hierarchy to convergence.
  • Similar interior-point arguments could be tested on non-compact support sets or on problems with countably many moment constraints.
  • When the relative-interior condition is violated, small generic perturbations of the right-hand sides may restore attainment while changing the optimal value by an arbitrarily small amount.

Load-bearing premise

The feasible set of moment sequences has nonempty relative interior inside the affine subspace cut out by the moment constraints.

What would settle it

A concrete semialgebraic set X and a finite list of polynomial moment constraints satisfying the relative-interior hypothesis for which either the infinite-dimensional dual or one finite SOS relaxation fails to attain an optimal solution.

read the original abstract

The generalized moment problem (GMP) is an infinite dimensional linear problem over the cone of finite nonnegative Borel measures. When a GMP instance involves finitely many polynomial moment constraints, moment/sum-of-squares hierarchies provide a sequence of bounds converging to the optimal value. We consider GMP instances with measures supported over a compact basic semialgebraic set $X$. We study the case when $X$ has nonempty interior, and the case when $X$ is the vanishing set of prescribed polynomials forming a Gr\"obner basis of the ideal they generate, which we assume is real radical. Under a relative interior assumption, we show attainment of the infinite dimensional dual problem, and attainment of each associated finite dimensional sum-of-squares strengthening. For the latter we present two disjoint proofs. The first is obtained by adapting results regarding the closedness of quadratic modules, and the second builds on Csisz\'ar's work on exponential density constructions to find a strictly feasible measure. Finally, we discuss the special case where $X$ is the product of spheres, and applications of our results to GMP instances arising from tensor optimization and quantum information theory.

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

0 major / 4 minor

Summary. The paper considers the generalized moment problem (GMP) over compact basic semialgebraic sets X, with finitely many polynomial moment constraints. Under a relative-interior assumption on the feasible moment set, together with structural hypotheses on X (nonempty interior, or X the real-radical vanishing set of a Gröbner basis), it proves attainment of the infinite-dimensional dual and attainment of every finite-dimensional sum-of-squares relaxation. Two independent proofs are supplied for the SOS attainments: one adapting closedness results for quadratic modules, the other constructing strictly feasible measures via Csiszár-type exponential densities. Special cases (products of spheres) and applications to tensor optimization and quantum information are discussed.

Significance. If the stated results hold, the paper supplies explicit, checkable conditions guaranteeing both dual attainment in the infinite-dimensional GMP and attainment at every level of the moment-SOS hierarchy. The provision of two disjoint proofs—one algebraic and one analytic—adds robustness. The applications to tensor and quantum-information GMP instances illustrate concrete utility. These guarantees strengthen the theoretical foundation for using moment-SOS hierarchies in polynomial optimization, control, and quantum information.

minor comments (4)
  1. [§2] The relative-interior hypothesis is load-bearing for both proofs; a short paragraph immediately after its definition, illustrating the condition on a simple example (e.g., the unit ball or a product of spheres), would help readers verify that the hypothesis is satisfied in typical applications.
  2. [§4] In the quadratic-module closedness argument, the precise invocation of the relevant theorem from real algebraic geometry (including any additional compactness or archimedeanness hypotheses) should be stated explicitly rather than left as “adapting results regarding the closedness of quadratic modules.”
  3. [§5] The exponential-density construction would benefit from a brief remark on how the Csiszár-type measure is shown to lie in the relative interior of the moment cone under the paper’s structural assumptions on X.
  4. [throughout] A few typographical inconsistencies appear in the notation for the quadratic module and its truncated versions; uniform use of subscripts or superscripts throughout would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recommending minor revision. The report correctly identifies the main results on dual attainment and SOS relaxation attainment under the stated relative-interior and structural assumptions on the support set X. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves attainment of the infinite-dimensional dual and finite-dimensional SOS relaxations for the generalized moment problem under an explicit relative-interior assumption on the feasible moment set, together with structural hypotheses on the support set X (nonempty interior or real-radical ideal generated by a Gröbner basis). The two proofs for the SOS case—one via closedness of quadratic modules and one via Csiszár-type exponential densities—are presented as disjoint and rely on external results from real algebraic geometry and functional analysis. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the arguments are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard domain assumptions from real algebraic geometry and convex analysis; no free parameters, new entities, or ad-hoc axioms are introduced beyond the problem setup.

axioms (3)
  • domain assumption X is a compact basic semialgebraic set
    Stated as the support of the measures in the GMP instances under study.
  • domain assumption The ideal is real radical when X is a variety
    Assumed for the vanishing-set case to ensure the quadratic module behaves well.
  • domain assumption The feasible moment set has nonempty relative interior
    Central hypothesis used to obtain attainment of both the dual and the relaxations.

pith-pipeline@v0.9.0 · 5501 in / 1494 out tokens · 73738 ms · 2026-05-10T10:44:46.914419+00:00 · methodology

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