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Strong duality

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

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    Natural Language Statement

    Theorem

    If a real primal linear program is feasible and bounded above, there are primal and dual feasible points x,yx,y with equal objective values:

    cTx=bTy.c^T x=b^T y.

    By weak duality, these points are optimal for their respective programs.

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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.LinearProgram
    2
    3/-!
    4---
    5title: Strong duality
    6type: theorem
    7---
    8If a real primal linear program is feasible and bounded above, there are
    9primal and dual feasible points x,yx,y with equal objective values:
    10cTx=bTy.c^T x=b^T y.
    11By weak duality, these points are optimal for their respective programs.
    12
    13# Formalization notes
    14
    15The statement includes attainment on both sides and requires no strict
    16feasibility or interior-point hypothesis. The proof network separates
    17primal attainment from the construction of optimal dual multipliers by
    18Farkas' lemma.
    19-/
    20
    21namespace Lax109476.StrongDuality
    22
    23open Lax109476.LinearProgram
    24
    25/-- A feasible bounded real program has matching primal and dual optimizers. -/
    26axiom exists_matching_optimizers :
    27 ∀ (m n : ℕ) (P : Program ℝ m n), IsFeasible P → IsBoundedAbove P →
    28 ∃ (x : Fin n → ℝ) (y : Fin m → ℝ),
    29 PrimalFeasible P x ∧ DualFeasible P y ∧ primalValue P x = dualValue P y
    30
    31end Lax109476.StrongDuality
    32
    Show Proof
    Formalization notes

    The statement includes attainment on both sides and requires no strict feasibility or interior-point hypothesis. The proof network separates primal attainment from the construction of optimal dual multipliers by Farkas' lemma.

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