The simplex method solves a linear program by exploiting the fact that an optimum can be sought among vertices of the feasible polyhedron.

After putting the problem in a suitable algebraic form, a basis selects variables that determine one basic feasible solution. A simplex pivot exchanges one basic and one nonbasic variable, moving to an adjacent vertex.

The entering variable is chosen because it can improve the objective. The leaving variable is chosen so the move remains feasible.

The algorithm repeats improving pivots until no adjacent feasible pivot can improve the objective, at which point optimality can be certified for the linear program.

Degeneracy can produce pivots with no objective improvement, and special cases can reveal infeasibility or an unbounded objective.

Simplex is therefore not a generic local-search method: its steps are tied to the combinatorial geometry of linear constraints.