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The feasibility alternative

Lax109476.FarkasAlternative · concepts/Lax109476/FarkasAlternative.lean · lax-109476

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    Natural Language Statement

    Theorem

    A real system Ax≤bAx\le b, x≥0x\ge0 is infeasible if and only if there is a vector y≥0y\ge0 with ATy≥0A^T y\ge0 and bTy<0b^T y<0.

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    3 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax109476.FarkasCertificate
    2
    3/-!
    4---
    5title: The feasibility alternative
    6type: theorem
    7---
    8A real system Ax≤bAx\le b, x≥0x\ge0 is infeasible if and only if there is a
    9vector y≥0y\ge0 with ATy≥0A^T y\ge0 and bTy<0b^T y<0.
    10
    11# Formalization notes
    12
    13This packages the existence and soundness directions as an equivalence.
    14Its proof assumes Farkas' lemma and the separately proved soundness of
    15infeasibility certificates.
    16-/
    17
    18namespace Lax109476.FarkasAlternative
    19
    20open Lax109476.LinearProgram Lax109476.FarkasCertificate
    21
    22/-- Infeasibility is equivalent to the existence of separating row multipliers. -/
    23axiom infeasible_iff_certificate :
    24 ∀ (m n : ℕ) (P : Program ℝ m n),
    25 ¬ IsFeasible P ↔ ∃ y : Fin m → ℝ, IsInfeasibilityCertificate P y
    26
    27end Lax109476.FarkasAlternative
    28
    Show Proof
    Formalization notes

    This packages the existence and soundness directions as an equivalence. Its proof assumes Farkas' lemma and the separately proved soundness of infeasibility certificates.

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