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Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm.SimplexAlgorithm

Simplex Algorithm #

To obtain required vector in Linarith.SimplexAlgorithm.findPositiveVector we run the Simplex Algorithm. We use Bland's rule for pivoting, which guarantees that the algorithm terminates.

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inductive Linarith.SimplexAlgorithm.SimplexAlgorithmException :
Type

An exception in the SimplexAlgorithmM monad.

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    @[reducible, inline]
    abbrev Linarith.SimplexAlgorithm.SimplexAlgorithmM (matType : Nat → Nat → Type) [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] (α : Type) :
    Type

    The monad for the Simplex Algorithm.

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      def Linarith.SimplexAlgorithm.doPivotOperation {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] (exitIdx enterIdx : Nat) :
      Linarith.SimplexAlgorithm.SimplexAlgorithmM matType Unit

      Given indexes exitIdx and enterIdx of exiting and entering variables in the basic and free arrays, performs pivot operation, i.e. expresses one through the other and makes the free one basic and vice versa.

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        def Linarith.SimplexAlgorithm.checkSuccess {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] :
        Linarith.SimplexAlgorithm.SimplexAlgorithmM matType Bool

        Check if the solution is found: the objective function is positive and all basic variables are nonnegative.

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          def Linarith.SimplexAlgorithm.chooseEnteringVar {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] :
          Linarith.SimplexAlgorithm.SimplexAlgorithmM matType Nat

          Chooses an entering variable: among the variables with a positive coefficient in the objective function, the one with the smallest index (in the initial indexing).

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            def Linarith.SimplexAlgorithm.chooseExitingVar {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] (enterIdx : Nat) :
            Linarith.SimplexAlgorithm.SimplexAlgorithmM matType Nat

            Chooses an exiting variable: the variable imposing the strictest limit on the increase of the entering variable, breaking ties by choosing the variable with smallest index.

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              def Linarith.SimplexAlgorithm.choosePivots {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] :
              Linarith.SimplexAlgorithm.SimplexAlgorithmM matType (Nat × Nat)

              Chooses entering and exiting variables using (Bland's rule)[(https://en.wikipedia.org/wiki/Bland%27s_rule)] that guarantees that the Simplex Algorithm terminates.

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                def Linarith.SimplexAlgorithm.runSimplexAlgorithm {matType : Nat → Nat → Type} [Linarith.SimplexAlgorithm.UsableInSimplexAlgorithm matType] :
                Linarith.SimplexAlgorithm.SimplexAlgorithmM matType Unit

                Runs the Simplex Algorithm inside the SimplexAlgorithmM. It always terminates, finding solution if such exists.

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