Practical Strength of Lift-and-Project Relaxations
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Definitions and mathematical statement
Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.A lift-and-project relaxation strengthens a linear relaxation by introducing products of constraints or variables and then projecting back to the original coordinates. Sherali-Adams and Lovasz-Schrijver are particular hierarchies for this construction.
Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. This is an empirical comparison direction. No instance distribution, benchmark protocol, or quantified improvement is stated.Source-record limitation: An empirical comparison on MIPLIB is requested without a mathematical target or evaluation protocol.
Statement recorded in the supplied source.
Determine the practical strength of the Sherali–Adams and Lovász–Schrijver lift-and-project relaxations for mixed-integer programs, including their performance on MIPLIB instances relative to the basic lift-and-project closure. [S2]Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.Short English statement
Compare the practical strength of the stated integer-programming relaxations.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000350 (OWR-1111-005). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 Combinatorial Optimization (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/50, pp. 2833--2932 https://doi.org/10.4171/owr/2005/50.
- Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30000350 and code OWR-1111-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.Statement licence: CC-BY-SA-4.0 · Source file
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gpt 6 astra ultra attempt on CS-644
Practical Strength of Lift-and-Project Relaxations
Definitions and mathematical statement
Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.A lift-and-project relaxation strengthens a linear relaxation by introducing products of constraints or variables and then projecting back to the original coordinates. Sherali-Adams and Lovasz-Schrijver are particular hierarchies for this construction.
Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. This is an empirical comparison direction. No instance distribution, benchmark protocol, or quantified improvement is stated.Source-record limitation: An empirical comparison on MIPLIB is requested without a mathematical target or evaluation protocol.
Statement recorded in the supplied source.
Determine the practical strength of the Sherali–Adams and Lovász–Schrijver lift-and-project relaxations for mixed-integer programs, including their performance on MIPLIB instances relative to the basic lift-and-project closure. [S2]Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.Short English statement
Compare the practical strength of the stated integer-programming relaxations.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000350 (OWR-1111-005). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 Combinatorial Optimization $2006$, Oberwolfach Reports, DOI 10.4171/owr/2005/50, pp. 2833--2932 https://doi.org/10.4171/owr/2005/50.
- Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30000350 and code OWR-1111-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or 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.An exact first-level comparison on the triangle
The report's empirical question compares lift-and-project relaxations on mixed-integer instances. Its benchmark figures are historical source observations, not measurements performed here. We test the stronger-relaxation route on one fully specified $0$--$1$ instance: maximize $x_1+x_2+x_3$ over the triangle stable-set relaxation \[ P=\{0\le x_i\le1:x_i+x_j\le1\quad(i\ne j)\}. \] The original relaxation has optimum $3/2$ at $(1/2,1/2,1/2)$, because summing the three edge inequalities gives that upper bound. The integer optimum is 1.Proposition: Multiplying all defining inequalities by each $x_i$ and $1-x_i$, and consistently linearizing products with $x_i^2=x_i$, projects exactly to the integer hull $\{x\ge0:\sum_i x_i\le1\}$. The single-coordinate disjunction $x_1=0$ or $x_1=1$ also yields that hull for this instance.
Proof: Write $y_{ij}=y_{ji}$ for $x_ix_j$. The products of nonnegativity constraints imply $y_{ij}\ge0$. Multiplying $1-x_i-x_j\ge0$ by $x_i$ and linearizing gives $-y_{ij}\ge0$, hence all three distinct-pair moments vanish. Multiplying $1-x_1-x_2\ge0$ by $1-x_3$ now gives \[
1-x_1-x_2-x_3+y_{13}+y_{23}\ge0,
\] so $\sum_i x_i\le1$. Conversely every point of this simplex is a convex combination of $0,e_1,e_2,e_3$; their actual products satisfy all multiplied inequalities. Since the lifted constraints are linear, the same convex combination is a feasible lift. Thus projection equals the simplex. For the disjunction, the branch $x_1=0$ is the two-dimensional simplex $x_2,x_3\ge0,x_2+x_3\le1$, with vertices $0,e_2,e_3$. The branch $x_1=1$ forces $x_2=x_3=0$. The convex hull of the two branches is therefore the same simplex. ∎
Verification and remaining gaps
The feasible binary vectors and objective values were enumerated; the moment cancellations and branch hulls were checked with exact coefficients. This instance closes the entire gap under both procedures and therefore supplies no evidence of a strict practical advantage of one over the other. The attempted inference that more lifted variables necessarily imply a strictly better projected bound fails here. Runtime, numerical behavior, valid inequalities on general instances and the requested empirical comparison remain untested. This known small example is not a new benchmark result.References and source attribution
Primary workshop report: https://doi.org/10.4171/owr/2005/50. Relevant passages read: 2005/50 pp. 2886--2888, basic and strengthened lift-and-project procedures and historical benchmark comparisons.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 OpenAIgpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch 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 batch016/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.