ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A two-variable semidefinite relaxation with a matching dual certificate

Example

The binary problem $\max {u_{1} u_{2} : u_{i} \in {- 1, 1}}$ lifts to $\max X_{12}$ with $X = u u^{T}$ , $X_{11} = X_{22} = 1$ , and $X ⪰ 0$ . Dropping $rank X = 1$ gives a two-by-two SDP relaxation.

Facts & Assumptions

Given: The SDP relaxation and the trace-inner-product primal-dual convention of Conic, LP, and convex QP primal-dual models.

[F1]

$X ⪰ 0$ iff $v^{T} X v \geq 0$ for every $v$ , and the positive-semidefinite cone is self-dual (The positive semidefinite cone is self-dual and its log determinant barrier has parameter n).

Verification

1.1F1givenalgebra

Write the minimization objective as $C ∙ X = - X_{12}$ with $C = (\begin{array}{cc} 0 & - 1 / 2 \\ - 1 / 2 & 0 \end{array})$ . For any feasible $X$ , [F1] applied to $e_{1} - e_{2}$ gives $2 - 2 X_{12} \geq 0$ , hence $- X_{12} \geq - 1$ . The feasible matrix $X_{*} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ attains $- 1$ .

2.1F1step 1.1algebra∎

The dual multiplier $y = (- 1 / 2, - 1 / 2)$ and slack $S = \frac{1}{2} (\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ satisfy $v^{T} S v = \frac{1}{2} (v_{1} - v_{2})^{2} \geq 0$ for every $v$ , hence $S ⪰ 0$ . Direct addition gives $diag (y) + S = C$ . Their dual value is $y_{1} + y_{2} = - 1$ , and $X_{*} ∙ S = 0$ . Thus the displayed primal and dual values match and certify the relaxation optimum.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (standard reference, not scraped)
Hamza Fawzi, Topics in Convex Optimisation, Lecture 16 (standard reference, not scraped)