Why it matters

Simplex underlies most LP solvers. Understanding shapes optimization intuition.

Advertisement

The architecture

Start at feasible vertex.

Move to adjacent vertex with better objective.

Repeat until optimum.

Simplex traversalFeasible vertexcorner of polytopeAdjacent vertexbetter objectiveOptimumno better neighborInterior-point methods: alternative with polynomial worst case
Simplex mechanics.
Advertisement

How it works end to end

Pivoting: exchange basic + non-basic variables.

Bland's rule / smallest index: avoid cycling.

Big-M / two-phase: handle no initial feasible point.

Duality: primal-dual pair with related solutions.