ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 revised-simplex unbounded ray

Example

Minimize $- x_{1}$ subject to $x_{1} - x_{2} = 0$ and $x_{1}, x_{2} \geq 0$ . Use $B = (2)$ as the initial feasible basis.

Facts & Assumptions

Given: $A = (1, - 1)$ , $b = 0$ , $c = (- 1, 0)$ , and the basis $B = (2)$ .

[F1]

Primal revised-simplex pivot identities and certificates: Let $B$ be a primal-feasible basis, let $j$ be nonbasic with ${\overset{ˉ}{c}}_{j} < 0$ , and put $d = A_{B}^{- 1} a_{j}$ . If $d_{i} \leq 0$ for every basic row $i$ , then $r_{j} = 1$ , $r_{B} = - d$ , and $r_{k} = 0$ for the other nonbasic indices define a feasible ray with $A r = 0$ and $c^{T} r = {\overset{ˉ}{c}}_{j} < 0$ ; the LP is unbounded below.

Verification

1.1givenalgebra

The basis matrix is $A_{B} = - 1$ , so $x_{2} = (- 1)^{- 1} 0 = 0$ . The multiplier is $y = (- 1)^{- 1} c_{2} = 0$ , and the nonbasic reduced cost is ${\overset{ˉ}{c}}_{1} = - 1 - 1 \cdot 0 = - 1$ .

2.1givenF1step 1.1algebra∎

The entering direction coefficient is $d = A_{B}^{- 1} a_{1} = - 1$ . There is no row with $d_{i} > 0$ , so the ray has $r_{1} = 1$ and $r_{2} = - d = 1$ . Then $A r = 1 - 1 = 0$ , $r \geq 0$ , and $c^{T} r = - 1 < 0$ ; [F1] gives unboundedness below.

Depends on

Primal revised-simplex pivot identities and certificates

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Dimitris Bertsimas and John N. Tsitsiklis, Introduction to Linear Optimization, §§3.1–3.2 (standard reference, not scraped)
Michel Goemans, MIT 18.310 Lecture Notes: Linear Programming, §2.3 (standard reference, not scraped)