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CS-319 · computer science · mathematics · Optimization

Efficient Moment Relaxations for Polynomial Optimization

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A polynomial is a sum of squares (SOS) if it is $\sum_j q_j^2$ for polynomials $q_j$. Moment-SOS hierarchies impose positivity through increasingly large truncated moment matrices and corresponding degree-bounded SOS certificates.

A semialgebraic set is described by finitely many polynomial equalities and inequalities and finite Boolean combinations of these conditions. A homeomorphism is a continuous bijection with continuous inverse.

Statement recorded in the supplied source.
Can moment-relaxation methods solve polynomial optimization over closed basic semialgebraic sets efficiently, including computing the infimum and the points where it is attained? [S2]
Scope qualification.
The oracle/encoding model, approximation accuracy, boundedness, and meaning of efficient are not specified; infimum attainment is not assumed.
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Obtain efficient moment-based global optimization over the stated semialgebraic feasible sets, including minimizers.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002563 (OWR-12975-008). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Real Algebraic Geometry With A View Toward Systems Control and Free Positivity (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/17, pp. 977--1045 https://doi.org/10.4171/owr/2014/17.
  • S3 Mai--Lasserre--Magron--Wang, arXiv https://arxiv.org/abs/2012.08873. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002563 and source code OWR-12975-008 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-319-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-319

Efficient Moment Relaxations for Polynomial Optimization

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A polynomial is a sum of squares (SOS) if it is $\sum_j q_j^2$ for polynomials $q_j$. Moment-SOS hierarchies impose positivity through increasingly large truncated moment matrices and corresponding degree-bounded SOS certificates.

A semialgebraic set is described by finitely many polynomial equalities and inequalities and finite Boolean combinations of these conditions. A homeomorphism is a continuous bijection with continuous inverse.

Statement recorded in the supplied source.
Can moment-relaxation methods solve polynomial optimization over closed basic semialgebraic sets efficiently, including computing the infimum and the points where it is attained? [S2]
Scope qualification.
The oracle/encoding model, approximation accuracy, boundedness, and meaning of efficient are not specified; infimum attainment is not assumed.
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Obtain efficient moment-based global optimization over the stated semialgebraic feasible sets, including minimizers.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002563 (OWR-12975-008). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Real Algebraic Geometry With A View Toward Systems Control and Free Positivity $2015$, Oberwolfach Reports, DOI 10.4171/owr/2014/17, pp. 977--1045 https://doi.org/10.4171/owr/2014/17.
  • S3 Mai--Lasserre--Magron--Wang, arXiv https://arxiv.org/abs/2012.08873. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002563 and source code OWR-12975-008 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Exact infimum does not ensure a moment optimizer

The source explicitly separates computing the infimum from recovering minimizers if they exist, and discusses additional optimality equations. We test the bare unconstrained moment-relaxation route, not those strengthened methods. On the closed set $\mathbb R^2$ take \[ f(x,y)=(xy-1)^2+x^2. \] Its infimum is 0, approached at $(x,y)=(1/t,t)$ as $t\to\infty$, but no point attains it: $f=0$ would require both $x=0$ and $xy=1$.
Proposition: The order-two moment relaxation with $L(1)=1$ and positive semidefinite moment matrix indexed by $1,x,y,x^2,xy,y^2$ has infimum 0 for $L(f)$, but has no feasible optimizer attaining that value.
Proof: Positive semidefiniteness means $L(p^2)\ge0$ for every polynomial $p$ of degree at most two, hence $L(f)=L((xy-1)^2)+L(x^2)\ge0$. Evaluation at $(1/t,t)$ defines a feasible functional with rank-one moment matrix and objective $t^{-2}$, proving infimum 0. If a feasible functional attained zero, both nonnegative summands would vanish. For a positive semidefinite bilinear form $\langle p,q\rangle=L(pq)$, $\langle p,p\rangle=0$ implies $\langle p,q\rangle=0$ for every allowed $q$: positivity of $\langle p+uq,p+uq\rangle$ for every real $u$ forces its linear coefficient to vanish. Taking $p=x,q=y$ therefore gives $L(xy)=0$. Taking $p=xy-1,q=1$ gives $L(xy)-L(1)=0$, so $L(xy)=1$, a contradiction. Thus no optimizer exists. ∎

Verification and remaining gaps

At $t=1,2,10,100$, exact rational point-evaluation moments gave objective $t^{-2}$ and a positive semidefinite rank-one matrix (the outer product of the monomial vector). Its $y^4$ diagonal entry grows as $t^4$, making the escaping feasible sequence explicit. The failed inference is that equality between an SDP infimum and the polynomial infimum automatically produces an optimal moment vector or extractable minimizer. The source already acknowledges nonattainment, so this example does not contradict its qualified formulation or its optimality-ideal constructions. It gives no general exact-relaxation criterion and no full algorithm for arbitrary closed noncompact semialgebraic sets.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2014/17. Relevant passages read: 2014/17 pp. 1015--1016, infimum versus minimizer recovery, noncompact sets and optimality-equation relaxations.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch019/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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