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LinearProgramming

LinearProgramming(coefficientsOfLinearObjectiveFunction, constraintList, constraintRelationList)

: the LinearProgramming function provides an implementation of George Dantzig's simplex algorithm for solving linear optimization problems with linear equality and inequality constraints.

See:

>>> LinearProgramming[{-2, 1, -5}, {{1, 2, 0},{3, 2, 0},{0,1,0},{0,0,1}}, {{6,-1},{12,-1},{0,1},{1,0}}]
{4.0,0.0,1.0}

solves the linear problem:

Minimize -2x + y - 5

with the constraints:
  x  + 2y <=  6
  3x + 2y <= 12
        y >= 0