Why it matters

LP underlies OR + planning + scheduling. Simplex is workhorse.

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The architecture

Vertices of feasible polytope.

Pivot: move to adjacent vertex improving objective.

Optimal when no improvement.

Simplex flowStart vertexfeasiblePivotimprove objectiveOptimalno improvementInterior point methods polynomial worst-case; simplex better for many practical LPs
Simplex.
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How it works end to end

Standard form: max c^T x s.t. Ax <= b, x >= 0.

Pivots via tableau.

Bland's rule prevents cycling.