# Lagrange multipliers and the Simplex Algorithm

I am trying to understand the Simplex Algorithm from a gradient perspective, and I am wondering if anyone knows of a method for determining the variables that should both enter and leave the basis of the Simplex Algorithm by using Lagrange multipliers?

That is, given an objective function and a set of linear constraints that form a convex polytope, how could one go about using the Lagrange multipliers at the current point (vertex of the polytope) to determine the direction (or constraint) to move toward during the next iteration?

• The Lagrange multipliers method is made to find the stationary points, it doesn't help to deal with the extremal points on the boundary. – Yves Daoust Jun 24 at 18:28
• Refer to Gilbert Strang $introduction\ to\ applied\ mathematics$. It gives some brilliant answers! – Book Book Book Jun 25 at 19:14