Franklin College
Erich Prisner,

Optimizing on surfaces with restraints:

Lagrange Multiplier

Let f be a function from R2 into R. Remember how to find global maximum points for this surface. What if we are restricted to a certain region? The boundary of the region should be given by one equation gi(x,y) = h.

How do we find the maximum f-value on this g-path? The idea is simple. Assume we are at point on this boundary path where the gradient vector is not perpendicular to the path. Then we could gain f-value by moving along. Therefore, at the highest f on the boundary, the gradients of f and g should point in the same direction, i.e.
(fx,fy) = l (gx,gy) .
In this way we get two equations with three unknowns, namely x, y, l. The third equation we need to solve the system is g(x,y)=h.