Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

The Effective Countable Generalized Moment Problem

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves explicit polynomial convergence rates for Moment-SoS relaxations of countable generalized moment problems, with rates that adapt to the underlying geometry and apply to optima, feasibility sets, and symmetric tensor…

desk verdict New general Moment-SoS rates for countable GMPs, with a real proof gap for negative objectives in the main theorem; the tensor application part is solid. read the letter →

arxiv 2501.09385 v4 pith:RMNFINGK submitted 2025-01-16 math.OC

Lucas Gamertsfelder (AROMATH) , Bernard Mourrain (AROMATH) This is my paper · ORCID
classification math.OC MSC 44A6090C2215A6914P10
keywords GeneralizedmomentproblemMoment-SoSrelaxationsconvergenceratesPositivstellensatzsymmetrictensordecompositionS-fullnesscountableconstraintsHausdorffdistance
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 explicit polynomial convergence rates for the Moment-Sum-of-Squares (Moment-SoS) hierarchy applied to the Generalized Moment Problem with countably many moment constraints on vectors of measures. Under three structural conditions — S-fullness, attainment of the dual optimum, and an Archimedean quadratic module — the gap between the true optimum and the $\ell$-th relaxation optimum decays no slower than $\kappa\ell^{-\theta}$, with an exponent $\theta$ determined by the geometry of the underlying semi-algebraic sets. The same rate bounds the Hausdorff distance between the truncated feasibility sets, and canonical relaxation optimizers converge weak$^*$ to the unique optimal measure when it exists. Applied to symmetric tensor decomposition on the unit ball, the results give explicit $\ell^{-2}$ error bounds.

What carries the argument

The argument hangs on a perturbed dual feasible point. Starting from a maximizer $v^*$ of the dual GMP, the paper forms $v_\varepsilon = v^* - \varepsilon w$ using the strictly positive vector $w$ supplied by S-fullness; then $q = f - h \cdot v_\varepsilon$ is a strictly positive polynomial, and an effective Positivstellensatz guarantees membership in the truncated quadratic module once $\ell \ge \max_i \gamma_i (q_{i,\max}/q_{i,\min})^{1/\theta_i}$. Since $v_\varepsilon$ is feasible for the $\ell$-th dual relaxation, weak duality and strong duality sandwich the gap as $d^* - d^*_\ell \le \varepsilon\, t \cdot w$; choosing $\varepsilon$ against the degree bound gives $\kappa\ell^{-\theta}$. For the feasibility-set statement, the machinery is a weighted moment norm embedding measures into a Banach space whose dual is the space of real-analytic functions with norm $\|f\|_A = \sum_\alpha \alpha! |f_\alpha|$, together with a separation argument that converts uniform value bounds into Hausdorff-distance bounds.

What would settle it

Take a concrete countable GMP on the unit ball that satisfies S-fullness and the Archimedean condition, write down its dual maximizer $v^*$, and compute the relaxation gap $d^* - d^*_\ell$ for increasing $\ell$; the theorem predicts $d^* - d^*_\ell \le \kappa\ell^{-2}$ with $\kappa$ given by (2.9), so a single instance whose gap is asymptotically larger than any constant times $\ell^{-2}$ would refute the main bound.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Moment-SoS relaxation sequence for a countable generalized moment problem and its dual are quantitatively close to the true problem: $0 \le p^* - p^*_\ell \le d^* - d^*_\ell \le \kappa\ell^{-\theta}$ for all sufficiently large $\ell$, and $\rho_H(F^{(k)}, L^{(k)}_\ell) \le \kappa'\ell^{-\theta}$ for fixed $k$. The constants $\kappa, \kappa'$ are explicit functions of the data — the strictly positive vector $w$ from S-fullness, the dual maximizer $v^*$, the cost function, and the geometry of the sets — and $\theta$ is the exponent available from an effective Positivstellensatz for the geometry at hand. A third theorem says that when the true optimum has a unique minimizing measure, every sequence of canonical relaxation optimizers converges to it in the weak$^*$ topology. The authors verify the hypotheses for positive and real symmetric tensor decomposition and derive the concrete bound $\theta = 2$ on the unit ball.

Load-bearing premise

The load-bearing premise is Assumption 1.9: the dual problem (1.3) must attain its supremum at some finite vector $v^*$, because the proof constructs the perturbed feasible point $v_\varepsilon = v^* - \varepsilon w$ from that maximizer; if no dual optimizer exists, the explicit $\kappa\ell^{-\theta}$ bound cannot even be stated.

Editorial extensions

If this is right

  • For every countable GMP satisfying Assumptions 1.3, 1.9, and 2.1, the objective gap $p^* - p^*_\ell$ (and $d^* - d^*_\ell$) is bounded by $\kappa\ell^{-\theta}$, upgrading qualitative convergence to a polynomial worst-case rate.
  • The same exponent $\theta$ controls the Hausdorff distance between the true truncated moment cone $F^{(k)}$ and the relaxed pseudo-moment cone $L^{(k)}_\ell$, so the geometry of feasible moment sequences converges at a polynomial rate, not just the optimal value.
  • If the GMP has a unique minimizer, every sequence of canonical relaxation optimizers converges weak$^*$ to it, so low-order moments of computed solutions are certified close to those of the true optimizer.
  • For symmetric tensor decomposition on the unit ball, the verified hypotheses give $\theta = 2$ with explicit constants, for both positive decompositions (Theorem 3.2) and real decompositions with a total-variation constraint (Theorem 3.7).
  • The rate is geometry-adaptive: changing the semi-algebraic sets changes $\theta$, because $\theta$ is inherited from the effective Positivstellensatz for that particular set.

Reading between the lines

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

  • Left implicit is that the same proof template should transfer to optimal control problems once controllability is used to verify Assumption 1.9; the missing piece is a practical way to bound $\|v^*\|_1$ and the support of $v^*$ from controllability data.
  • The Hausdorff-distance result suggests a stopping criterion that value bounds alone cannot provide: monitor the distance from the current pseudo-moment cone to the truncated moment cone to certify how close the relaxation is to true feasibility.
  • If the generic-uniqueness remarks for tensor decomposition are turned into a theorem, the weak$^*$-convergence result would turn the examples' finite-order exactness into a guarantee that the nuclear-norm heuristic recovers the minimal decomposition.
  • The paper's examples show the bounds can be pessimistic; an inference is that the true worst-case exponent may be higher for structured data, and identifying the sharp exponent is a natural next problem.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The paper's central claim is that the Moment-SoS relaxation sequence for a countable generalized moment problem and its dual are quantitatively close to the true problem: $0 \le p^* - p^*_\ell \le d^* - d^*_\ell \le \kappa\ell^{-\theta}$ for all sufficiently large $\ell$, and $\rho_H(F^{(k)}, L^{(k)}_\ell) \le \kappa'\ell^{-\theta}$ for fixed $k$. The constants $\kappa, \kappa'$ are explicit funct

Signed reviews

No signed human review yet.

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. The paper develops convergence-rate theory for Moment-Sum-of-Squares relaxations of countable Generalized Moment Problems with vector measures. Under S-fullness, dual optimum attainment, and Archimedean assumptions, it claims polynomial rates for the convergence of optimal values (Theorem 2.4 / Corollary 2.5), for the Hausdorff distance between truncated feasibility sets (Theorem 2.11), and for the weak-* convergence of canonical optimizers (Theorem 2.13). The framework is then applied to symmetric tensor decomposition, with separate treatments of positive and signed decompositions and explicit O(ell^{-2}) bounds in Theorems 3.2 and 3.7.

Significance. If the main results hold, the paper provides a unified quantitative convergence theory for a broad class of GMPs, with rates adapting to the geometry of the underlying semialgebraic sets through effective Positivstellensatz exponents. The tensor decomposition application is a genuine strength: the authors verify the structural assumptions for both positive and signed decompositions, provide explicit rates, and their numerical examples illustrate finite convergence in practice. The proofs are analytic rather than machine-checked, but the constants are expressed in terms of identifiable objects (the dual optimizer v*, the S-fullness witness w, and imported effective Positivstellensatz bounds) rather than fitted parameters, and the statements are falsifiable through the predicted polynomial rates.

major comments (3)
  1. [Theorem 2.11, proof] The proof's choice of epsilon is invalid for objectives with f_i,max <= 0. In (2.12), the inequality q_i,max <= 2 f_i,max + ||v*||_1 h*_{i,max} requires choosing epsilon <= f_i,max/(h_i . w)_max, and no positive epsilon exists when f_i,max is zero or negative. Assumptions 1.3, 1.9, and 2.1 do not restrict f to nonnegative functions, so the theorem is not proven for the stated class of objectives. The constant kappa in (2.9) can even be negative, for example with a single constraint h=1, t=1, and f=-10, where the formula gives kappa = -10 gamma^theta. The gap is repairable by replacing f_i,max with max(f_i,max,0) or by balancing epsilon against ell, but the resulting constant differs from (2.9); as written, Theorem A and Corollary 2.5 are not established for arbitrary continuous or polynomial objectives. The tensor applications use nonnegative Psi and are not affected.
  2. [Lemma 2.10 / Theorem 2.11] The proof of Theorem 2.11 applies Theorem 2.4 to every f in R_k with ||f||_A = 1 in order to use the forward direction of Lemma 2.10. However, Theorem 2.4 requires Assumption 1.9 for the GMP with that specific objective, and the assumptions of Theorem 2.11 do not imply dual attainment for all such f. S-fullness gives strong duality but not attainment; indeed Assumption 1.9 is introduced precisely because attainment is not automatic. Thus the value-gap bound needed for Lemma 2.10 is not established for all f, and the Hausdorff bound does not follow as written. A repair would be to impose a uniform dual-attainment condition on the unit ball of R_k or to prove the value-gap bound by a direct primal argument.
  3. [Lemma 2.10 / Theorem 2.11] Lemma 2.10 explicitly assumes that p*_ell is attained for every f in R_k, and Theorem 2.11 invokes this lemma without proving attainment. The truncated feasible set is not automatically compact: the restriction of the S-fullness witness w to J_ell need not belong to (K_ell)^*, because an extension of u_ell to K has arbitrary nonnegative components outside J_ell, and elements of L_ell are pseudo-moment functionals that need not be representable by measures. Consequently, the compactness argument for F in Lemma 1.4 does not transfer to the truncated feasible set. The authors should either prove boundedness and attainment for the truncated problems or modify Lemma 2.10 to work with epsilon-optimal points instead of exact optimizers.
minor comments (5)
  1. [Throughout] The unresolved macro '/suppress L' appears in Theorem 2.3 and Theorem 2.11, for instance as 'theta = 1/(2.5n/suppress L)' and 'gamma = gamma'(n,g) deg(p)^{-3.5n/suppress L}'. This should be replaced by the intended Lojasiewicz exponent so that the claimed explicit constants are actually readable.
  2. [Theorem 3.2] The constant kappa in Theorem 3.2 appears to omit the factor t . w = F_0 and the power gamma^2 that would follow from Theorem 2.4 when theta = 2; please reconcile the displayed bound with (2.9).
  3. [Example 3.9] The narrative says that the relaxation (3.5)_ell is solved, but the displayed optimization problem is the augmented formulation (3.15); the text should identify the actual program being solved.
  4. [Remark 3.4 and Remark 3.8] The appeal to [41, Theorem 2.2.11] for generic uniqueness of truncated minimizers is terse; a sentence explaining why the singular normal set has measure zero and how this yields uniqueness of the truncated minimizer would improve readability.
  5. [Abstract and header] The running title 'THE EFFECTIVE COUNT ABLE GENERALIZED MOMENT PROBLEM' contains a spacing typo; 'countable' should appear as one word.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence-rate theorems follow from explicitly stated assumptions and independently proven effective Positivstellensatz bounds.

full rationale

The paper's main derivation chain (Theorem 2.4, Corollary 2.5, Theorem 2.11, Theorem 2.13) is conditional: under Assumptions 1.3, 1.9 and 2.1 it bounds d* - d*_l by κ l^{-θ} using the effective Positivstellensatz results stated as Theorem 2.3. The constants κ and θ depend explicitly on the assumed dual optimizer v*, the S-fullness witness w, the geometric exponents γ_i and θ_i, and the problem data f, h, t, K. These are inputs of the theorem, not quantities being predicted; no parameter is fitted to the optimal value or to the feasibility sets whose convergence is asserted. The self-citations to Baldi–Mourrain's effective Putinar theorem provide a published, independently proven rate engine, and other cases of Theorem 2.3 cite Slot and Laurent–Slot, so the rate engine is not a private or unverified assumption. In the tensor applications, Propositions 3.1 and 3.6 verify S-fullness and dual attainment rather than assuming the conclusion of the convergence theorem. The numerical examples explicitly state that the proved rates are pessimistic, separating the proven bound from observed finite convergence. A possible technical gap in the ε-selection step for negative f_i,max would be a correctness or soundness issue, not a circularity, and does not affect this verdict.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The main theorems rest on three structural assumptions, S-fullness, dual attainment, and the Archimedean condition, plus imported effective Positivstellensatz bounds. No new particles or forces are introduced, and no parameters are fitted to observed data. The total-variation bound L in the real tensor application is the only hand-chosen numeric input that enters the final rate.

free parameters (1)
  • L (total-variation bound in real tensor decomposition GMP) = not fitted; chosen with L > sum_i |omega_i|
    Introduced in (3.11) to restore S-fullness for signed decompositions. It appears linearly in the rate constant kappa of Theorem 3.7 and is a modeling choice rather than a fitted constant.
assumptions (7)
  • domain assumption Assumption 1.3 (S-fullness): there exists w in K* such that h.w is strictly positive on every S_i.
    Makes the feasible set compact, gives strong duality, and provides the perturbation direction w used throughout Theorem 2.4.
  • domain assumption Assumption 1.9: the dual GMP attains its supremum at some v*.
    The explicit rates and constants in Theorem 2.4 are built from v*; if no dual optimizer exists, the perturbed dual construction and the stated bounds do not apply.
  • domain assumption Assumption 2.1: each quadratic module Q(g_i) is Archimedean.
    Needed for Putinar's Positivstellensatz and for identifying the relaxation cones with cones of positive measures.
  • standard math Effective Putinar Positivstellensatz bounds of Theorem 2.3, imported from references [4,7,28,43].
    The rates use exponents theta_i and constants gamma_i, ell_i,0 without reproving them. In the general case the constants depend on a Lojasiewicz exponent that is not explicitly computed.
  • domain assumption Theorem 2.13 assumes the GMP has a unique optimal measure.
    Needed to upgrade subsequential weak* convergence to convergence of the full sequence of canonical optimizers.
  • domain assumption For tensor decomposition, the existence of the decomposition (3.3) or (3.8) with support points on the unit ball and, for the positive case, a basis of R/I_xi in degree at most d.
    Guarantees feasibility and is used in Proposition 3.1 to prove dual attainment and S-fullness.
  • domain assumption For real tensor decomposition, L is chosen with L > sum of |omega_i|.
    Restores S-fullness when the decomposition has signed weights and appears in the rate constant of Theorem 3.7.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Effective Countable Generalized Moment Problem." pith.science (2026). https://pith.science/paper/RMNFINGK

@misc{pith2026250109385,
  author       = {Pith},
  title        = {Pith review of: The Effective Countable Generalized Moment Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMNFINGK}},
  note         = {Machine review of arXiv:2501.09385}
}
abstract

We establish new convergence rates for the Moment-Sum-of-Squares (Moment-SoS) relaxations for the Generalized Moment Problem (GMP) with countable moment constraints on vectors of measures, under dual optimum attainment, $S$-fullness and Archimedean conditions. These bounds, which adapt to the geometry of the underlying semi-algebraic set, apply to both the convergence of optima, and to the convergence in Hausdorff distance between the relaxation feasibility set and the GMP feasibility set. We show that under the previous conditions, the sequence of optimizers of the relaxations converge to the optimizer of the GMP for the weak$^*$ topology, provided this optimal measure is unique. This research provides quantitative geometry-adaptive rates for GMPs cast as linear programs on measures. It complements earlier analyses of specific GMP instances (e.g., polynomial optimization) as well as recent methodological frameworks that have been applied to volume computation and optimal control. We apply the convergence rate analysis to symmetric tensor decomposition problems, providing new effective error bounds for the convergence of the Moment-SoS hierarchies for tensor decomposition.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Efficient Tensor Decomposition via Moment Matrix Extension

    math.AG 2025-06 conditional novelty 6.0 of 10

    Generic order-4 symmetric tensors of rank up to 2n+1 are efficiently decomposable via moment matrix extension, with a conjectured extension to O(n^2) rank.

  2. Convergence rates for polynomial optimization on set products

    math.OC 2025-05 conditional novelty 6.0 of 10

    An O(1/t^2) convergence rate is established for Schmüdgen-type hierarchies on products of spheres, with a recipe for general set products and an application to quantum Wasserstein distances.

Reference graph

Works this paper leans on

51 extracted references · 46 canonical work pages · cited by 2 Pith papers

  1. [39]

    Schlosser, M

    C. Schlosser, M. Tacchi, and A. Lazarev , Convergence rates for the moment-SoS hierarchy, preprint arXiv:2402.00436, (2024)

    arXiv 2024

  2. [1]

    N. I. Akhiezer , The classical moment problem and some related questions in a nalysis, SIAM, 2020

    work page 2020

  3. [2]

    E. J. Anderson , A review of duality theory for linear programming over topol ogical vector spaces, Journal of mathematical analysis and applications, 97 (19 83), pp. 380–392

    work page

  4. [3]

    R. B. Ash , Measure, integration, and functional analysis , Academic Press, 2014

    work page 2014

  5. [4]

    Baldi and B

    L. Baldi and B. Mourrain , On the effective Putinar’s Positivstellensatz and moment ap prox- imation, Mathematical Programming, 200 (2023), pp. 71–103, https: //doi.org/10.1007/ s10107-022-01877-6

    work page 2023

  6. [5]

    Baldi and B

    L. Baldi and B. Mourrain , Exact Moment Representation in Polynomial Optimization , Journal of Symbolic Computation, (2024), p. 102403, https: //doi.org/10.1016/j.jsc.2024. 102403

    work page doi:10.1016/j.jsc.2024 2024

  7. [6]

    Baldi, B

    L. Baldi, B. Mourrain, and A. Parusi ´nski, On /suppress Lojasiewicz inequalities and the effective Putinar’s positivstellensatz , Journal of Algebra, 662 (2025), p. 741–767, https://doi.o rg/ 10.1016/j.jalgebra.2024.08.022

    work page doi:10.1016/j.jalgebra.2024.08.022 2025

  8. [7]

    Baldi and L

    L. Baldi and L. Slot , Degree bounds for Putinar’s Positivstellensatz on the Hype rcube, SIAM Journal on Applied Algebra and Geometry, 8 (2024), p. 1–25, h ttps://doi.org/10.1137/ 23m1555430

    work page 2024

Show all 51 references
  1. [8]

    Bourbaki , Topological vector spaces: Chapters 1–5 , Springer Science & Business Media, 2013

    N. Bourbaki , Topological vector spaces: Chapters 1–5 , Springer Science & Business Media, 2013

    2013

  2. [9]

    Boyd and L

    S. Boyd and L. V andenberghe , Convex optimization , Cambridge university press, 2004

    2004

  3. [10]

    Brachat, P

    J. Brachat, P. Comon, B. Mourrain, and E. Tsigaridas , Symmetric tensor decomposition, Linear Algebra and its Applications, 433 (2010), pp. 1851–1 872

    2010

  4. [11]

    M. C. Brambilla and G. Ottaviani , On the Alexander–Hirschowitz theorem, Journal of Pure and Applied Algebra, 212 (2008), pp. 1229–1251

    2008

  5. [12]

    R. E. Curto and L. A. Fialkow , Solution of the Truncated Complex Moment Problem for Flat Data, vol. 568 of Memoirs of the American Mathematical Society, A merican Mathematical Society, 1996

    1996

  6. [13]

    R. E. Curto and L. A. Fialkow , Truncated k-moment problems in several variables , Journal of Operator Theory, 54 (2005), pp. 189–226

    2005

  7. [14]

    F ang and H

    K. F ang and H. F awzi, The sum-of-squares hierarchy on the sphere and application s in quan- tum information theory , Mathematical Programming, 190 (2020), p. 331–360, https: //doi. org/10.1007/s10107-020-01537-7

    2020 doi

  8. [15]

    F azel, Matrix rank minimization with applications , PhD thesis, PhD thesis, Stanford Uni- versity, 2002

    M. F azel, Matrix rank minimization with applications , PhD thesis, PhD thesis, Stanford Uni- versity, 2002

    2002

  9. [16]

    Gaitsgory and S

    V. Gaitsgory and S. Rossomakhine , Linear programming approach to deterministic long run average problems of optimal control, SIAM journal on control and optimization, 44 (2006), 24 L. GAMERTSFELDER AND B. MOURRAIN pp. 2006–2037

    2006

  10. [17]

    Gaitsgory, S

    V. Gaitsgory, S. Rossomakhine, and N. Thatcher , Approximate solution of the HJB in- equality related to the infinite horizon optimal control pro blem with discounting, Dynamics of Continuous, Discrete and Impulsive Systems Series B: App lications and Algorithms, 19 (2012), pp. 65–92

    2012

  11. [18]

    Huang, J

    L. Huang, J. Nie, and Y.-X. Yuan , Finite convergence of moment-sos relaxations with nonreal radical ideals, SIAM Journal on Optimization, 34 (2024), pp. 3399–3428, ht tps://doi.org/ 10.1137/23M1605430

    2024 doi

  12. [19]

    E. d. Klerk and M. Laurent , A survey of semidefinite programming approaches to the gener- alized problem of moments and their error analysis, in W orld W omen in Mathematics 2018: Proceedings of the First W orld Meeting for W omen in Mathemat ics (WM) 2, Springer, 2019, pp. 17–56

    2018

  13. [20]

    T. G. Kolda and B. W. Bader , Tensor decompositions and applications , SIAM review, 51 (2009), pp. 455–500

    2009

  14. [21]

    Korda, D

    M. Korda, D. Henrion, and C. N. Jones , Convergence rates of moment-sum-of-squares hierarchies for optimal control problems , Systems & Control Letters, 100 (2017), pp. 1–5

    2017

  15. [22]

    H. J. Landau , Moments in mathematics , vol. 37, American Mathematical Soc., 1987

    1987

  16. [23]

    Lasserre , The K-moment problem for continuous linear functionals , Transactions of the American Mathematical Society, 365 (2013), pp

    J. Lasserre , The K-moment problem for continuous linear functionals , Transactions of the American Mathematical Society, 365 (2013), pp. 2489–2504

    2013

  17. [24]

    J. B. Lasserre , Global optimization with polynomials and the problem of mom ents, SIAM Journal on optimization, 11 (2001), pp. 796–817

    2001

  18. [25]

    J. B. Lasserre , A semidefinite programming approach to the generalized problem of moments , Mathematical Programming, 112 (2008), pp. 65–92

    2008

  19. [26]

    J. B. Lasserre , The moment-SOS hierarchy , in Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018, W orld Scientific, 2018 , pp. 3773–3794

    2018

  20. [27]

    Laurent and B

    M. Laurent and B. Mourrain , A generalized flat extension theorem for moment matrices , Archiv der Mathematik, 93 (2009), p. 87–98, https://doi.or g/10.1007/s00013-009-0007-6

    2009 doi

  21. [28]

    Laurent and L

    M. Laurent and L. Slot , An effective version of Schm¨ udgen’s Positivstellensatz for the hy- percube, Optimization Letters, 17 (2023), pp. 515–530

    2023

  22. [29]

    Mourrain and A

    B. Mourrain and A. Oneto , On minimal decompositions of low rank symmetric tensors , Linear Algebra and its Applications, 607 (2020), pp. 347–37 7

    2020

  23. [30]

    Nie , The A-truncated k-moment problem , Foundations of Computational Mathematics, 14 (2014), p

    J. Nie , The A-truncated k-moment problem , Foundations of Computational Mathematics, 14 (2014), p. 1243–1276, https://doi.org/10.1007/s10208-0 14-9225-9

    2014 doi

  24. [31]

    Nie , Optimality conditions and finite convergence of Lasserre’s hierarchy, Mathematical Programming, 146 (2014), pp

    J. Nie , Optimality conditions and finite convergence of Lasserre’s hierarchy, Mathematical Programming, 146 (2014), pp. 97–121

    2014

  25. [32]

    Nie and M

    J. Nie and M. Schweighofer , On the complexity of Putinar’s Positivstellensatz , Journal of Complexity, 23 (2007), pp. 135–150, https://doi.org/10.1 016/j.jco.2006.07.002

    2007

  26. [33]

    Pucci , The Veronese variety and catalecticant matrices , Journal of Algebra, 202 (1998), pp

    M. Pucci , The Veronese variety and catalecticant matrices , Journal of Algebra, 202 (1998), pp. 72–95

    1998

  27. [34]

    Putinar , Positive polynomials on compact semi-algebraic sets , Indiana University Mathe- matics Journal, 42 (1993), pp

    M. Putinar , Positive polynomials on compact semi-algebraic sets , Indiana University Mathe- matics Journal, 42 (1993), pp. 969–984

    1993

  28. [35]

    Recht, M

    B. Recht, M. F azel, and P. A. Parrilo , Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization , SIAM review, 52 (2010), pp. 471–501

    2010

  29. [36]

    Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics, Prince- ton University Press, 1997

    R. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics, Prince- ton University Press, 1997

    1997

  30. [37]

    Rudin , Real and Complex Analysis , Mathematics series, McGraw-Hill, 1987

    W. Rudin , Real and Complex Analysis , Mathematics series, McGraw-Hill, 1987

    1987

  31. [38]

    Rudin, Functional Analysis, International series in pure and applied mathematics, McG raw- Hill, 1991

    W. Rudin, Functional Analysis, International series in pure and applied mathematics, McG raw- Hill, 1991

    1991

  32. [40]

    Schm ¨udgen, The Moment Problem , Springer International Publishing, 2017, https://doi

    K. Schm ¨udgen, The Moment Problem , Springer International Publishing, 2017, https://doi. org/10.1007/978-3-319-64546-9

    2017 doi

  33. [41]

    Schneider , Convex bodies: the Brunn–Minkowski theory , vol

    R. Schneider , Convex bodies: the Brunn–Minkowski theory , vol. 151, Cambridge university press, 2013

    2013

  34. [42]

    Schweighofer , On the complexity of Schm¨ udgen’s Positivstellensatz, Journal of Complex- ity, 20 (2004), pp

    M. Schweighofer , On the complexity of Schm¨ udgen’s Positivstellensatz, Journal of Complex- ity, 20 (2004), pp. 529–543, https://doi.org/10.1016/j.j co.2004.01.005

    2004 doi

  35. [43]

    Slot , Sum-of-squares hierarchies for polynomial optimization a nd the Christoffel–Darboux Kernel, SIAM Journal on Optimization, 32 (2022), p

    L. Slot , Sum-of-squares hierarchies for polynomial optimization a nd the Christoffel–Darboux Kernel, SIAM Journal on Optimization, 32 (2022), p. 2612–2635, htt ps://doi.org/10.1137/ 21m1458338

    2022

  36. [44]

    Tacchi, J

    M. Tacchi, J. B. Lasserre, and D. Henrion , Stokes, Gibbs, and volume computation of semi-algebraic sets, Discrete & Computational Geometry, 69 (2023), pp. 260–283 . THE EFFECTIVE COUNTABLE GENERALIZED MOMENT PROBLEM 25 Appendix A. Linear programming proofs. Proof of Lemma 1.4...

    2023

  37. [45]

    sup t · v → − inf c · −v,

  38. [46]

    Specifically, we obtain −d∗ = inf t · v s.t

    b → −f . Specifically, we obtain −d∗ = inf t · v s.t. h · v + f ∈ C +, v ∈ K ∗. Since K is closed, K ∗∗ = K (see e.g. [36, p. 121], [9, p. 53]) and by the Riesz representation theorem ( C+)∗ = M+. It is then readily verified with reference to the above substitutions and (1.6) th...

  39. [47]

    Let f ∈ (D♭ i )∗

    Real-analyticity of ϕ(f ). Let f ∈ (D♭ i )∗. Then |f (dα)| ≤ ‖ f ‖∗ i ‖dα‖i = ‖f ‖∗ i sup β∈Nni |dα(xβ)|/β! = ‖f ‖∗ i /α!. The absolute value of the α-th term in the series of ϕ(f ) is |f (dα)xα| = |f (dα)||xα| ≤ (‖f ‖∗ i /α!)|xα| and the α! in the denominator ensures converge...

  40. [48]

    Let f, g ∈ (D♭ i )∗ and a, b ∈ R

    Linearity. Let f, g ∈ (D♭ i )∗ and a, b ∈ R. The fact ϕ(af + bg) = aϕ(f ) + bϕ(g) follows readily from the definition of ϕ

  41. [49]

    Let ϕ(f ) = ϕ(g)

    Injectivity. Let ϕ(f ) = ϕ(g). By linearity, this is equivalent to showing that if ϕ(F ) = 0 for F = f − g, then F = 0. The condition ϕ(F ) = 0 means that ∑ α∈Nni F (dα)xα = 0. This implies that all coefficients are zero: F (dα) = 0 for all α ∈ Nni. From the definition of the A-n...

  42. [50]

    Let h =∑ α∈Nni hαxα i ∈ A

    Surjectivity. Let h =∑ α∈Nni hαxα i ∈ A. Take fh(λ) def = ∑ α∈Nni hαλ(xα i ). We verify that fh is well-defined and continuous. Note that the above series converg es absolutely for any λ ∈ D ♭ i : |fh(λ)| ≤ ∑ α∈Nni |hα||λ(xα i )| ≤ ∑ α∈Nni |hα|(‖λ‖iα!) = ‖λ‖i ( ∑ α∈Nni α!|hα| )...

  43. [51]

    We show: THE EFFECTIVE COUNTABLE GENERALIZED MOMENT PROBLEM 27 (a) ‖f ‖∗ i ≤∑ α∈Nni α!|f (dα)|, (b) ‖f ‖∗ i ≥∑ α∈Nni α!|f (dα)|, and (c) ‖ϕ(f )‖A =∑ α∈Nni α!|f (dα)|

    Norm preservation. We show: THE EFFECTIVE COUNTABLE GENERALIZED MOMENT PROBLEM 27 (a) ‖f ‖∗ i ≤∑ α∈Nni α!|f (dα)|, (b) ‖f ‖∗ i ≥∑ α∈Nni α!|f (dα)|, and (c) ‖ϕ(f )‖A =∑ α∈Nni α!|f (dα)|. (a) Let f ∈ (D♭ i )∗ and λ ∈ D ♭ i with ‖λ‖i ≤ 1. Then, |⟨f, λ⟩| = ⏐ ⏐ ⏐ ⏐ ⏐ ∑ α∈Nni f (dα)...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.