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Unbounded linear programs have improving recession directions

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

proven

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    Natural Language Statement

    Theorem

    If the feasible objective values are unbounded above, there is a nonnegative direction dd satisfying Ad≤0Ad\le0 and cTd>0c^Td>0. Thus unboundedness has a geometric witness, in addition to the soundness of such witnesses.

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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.RecessionDirection
    2import Lax109476.FarkasLemma
    3import Lax109476.WeakDuality
    4
    5/-!
    6---
    7title: Unbounded linear programs have improving recession directions
    8type: theorem
    9---
    10If the feasible objective values are unbounded above, there is a nonnegative
    11direction dd satisfying Ad≤0Ad\le0 and cTd>0c^Td>0. Thus unboundedness has a
    12geometric witness, in addition to the soundness of such witnesses.
    13
    14# Formalization notes
    15
    16The coefficients and direction are real. This theorem does not give rational
    17encoding bounds or an algorithm for recovering the direction. Empty dimensions
    18are included.
    19-/
    20
    21namespace Lax109476.RecessionExistence
    22
    23open Lax109476.LinearProgram Lax109476.RecessionDirection
    24
    25/-- Unbounded feasible objective values force an improving recession direction. -/
    26axiom exists_improving_direction :
    27 ∀ (m n : ℕ) (P : Program ℝ m n), ¬ IsBoundedAbove P →
    28 ∃ d : Fin n → ℝ, IsImprovingDirection P d
    29
    30end Lax109476.RecessionExistence
    31
    Show Proof
    Formalization notes

    The coefficients and direction are real. This theorem does not give rational encoding bounds or an algorithm for recovering the direction. Empty dimensions are included.

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