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 one-pivot primal and dual certificate trace

Example

First consider $\min x_{1} + 2 x_{2}$ subject to $x_{1} + x_{2} = 1$ and $x_{1}, x_{2} \geq 0$ . Start from basis $B = (2)$ , so $x = (0, 1)$ , $y = 2$ , and ${\overset{ˉ}{c}}_{1} = - 1$ . The primal ratio test pivots $x_{1}$ in at $θ = 1$ ; the new basis is $B = (1)$ with $x = (1, 0)$ , $y = 1$ , and ${\overset{ˉ}{c}}_{2} = 1$ .

For the same $A$ and $c$ but right-hand side $b = - 1$ , the basis $B = (1)$ remains dual feasible and has basic value $x_{1} = - 1$ . The dual-simplex row has no negative nonbasic coefficient, giving a Farkas certificate.

Facts & Assumptions

Given: The exact two-variable LP data in the Example and the basis definitions and pivot results below.

[F1]

Basis dictionaries and reduced costs for revised simplex: The basic vector $x (B)$ is defined by $x_{B} (B) = A_{B}^{- 1} b$ and $x_{j} (B) = 0$ for $j \in N$ ; the basis is primal feasible, and $x (B)$ is a basic feasible solution, exactly when $A_{B}^{- 1} b \geq 0$ .

[F2]

Primal revised-simplex pivot identities and certificates: At a primal-feasible basis, a nonbasic $j$ with ${\overset{ˉ}{c}}_{j} < 0$ and some positive entry of $d = A_{B}^{- 1} a_{j}$ admits a feasible minimum-ratio pivot with objective change $θ {\overset{ˉ}{c}}_{j}$ . If all nonbasic reduced costs are nonnegative at a primal-feasible basis, its basic vector is optimal.

[F3]

Dual-simplex invariant and Farkas infeasibility certificate: At a dual-feasible basis, for a row $i$ with $x_{B_{i}} < 0$ , if there is no nonbasic $j$ with $d_{i j} = (A_{B}^{- 1} a_{j})_{i} < 0$ , then $q = A_{B}^{- T} e_{i}$ satisfies $A^{T} q \geq 0$ and $b^{T} q < 0$ .

Verification

1.1F1givenalgebra

With $B = (2)$ , $A_{B} = 1$ , $x_{2} = 1$ , and $y = c_{2} = 2$ . Thus ${\overset{ˉ}{c}}_{1} = 1 - 2 = - 1$ , so $x_{1}$ is a valid entering variable.

1.2F1givenalgebra

For the changed right-hand side $b = - 1$ and basis $B = (1)$ , $x_{1} = - 1$ while $y = 1$ and ${\overset{ˉ}{c}}_{2} = 1 \geq 0$ , so the basis is dual feasible and primal infeasible. Its only nonbasic coefficient is $d_{12} = A_{B}^{- 1} a_{2} = 1 \geq 0$ .

2.1F1F2step 1.1algebra

The direction coefficient is $d = A_{B}^{- 1} a_{1} = 1$ and the ratio is $x_{2} / d = 1$ . The pivot yields $x_{1} = 1, x_{2} = 0$ and changes cost by $1 \cdot (- 1) = - 1$ , from $2$ to $1$ .

3.1F1F2step 2.1algebra

At the new basis $B = (1)$ , $y = c_{1} = 1$ and ${\overset{ˉ}{c}}_{2} = 2 - 1 = 1 \geq 0$ . The vector is feasible, so [F2] proves it is optimal; the primal and dual objective values are both $1$ .

4.1F3step 1.2algebra∎

There is no eligible negative coefficient. The certificate from [F3] is $q = 1$ , for which $A^{T} q = (1, 1) \geq 0$ and $b^{T} q = - 1 < 0$ . Thus the modified system has no nonnegative feasible vector.

Depends on

Used by

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Sources

Dimitris Bertsimas and John N. Tsitsiklis, Introduction to Linear Optimization, §§3.1–3.2 and 4.5 (standard reference, not scraped)
Michel Goemans, MIT 18.310 Lecture Notes: Linear Programming, §§2–3 (standard reference, not scraped)