tests for local extrema in Lagrange multiplier method
Let be open in , and , be twice continuously differentiable functions. Assume that is a stationary point for on , and has full rank everywhere11Actually, only needs to have full rank, and the arguments presented here continue to hold in that case, although would not necessarily be a manifold then. on . Then we know that is the solution to the Lagrange multiplier system
for a Lagrange multiplier vector .
Our aim is to develop an analogue of the second derivative test for the stationary point .
The most straightforward way to proceed is to consider a coordinate chart for the manifold , and consider the Hessian of the function at . This Hessian is in fact just the Hessian form of expressed in the coordinates of the chart . But the whole point of using Lagrange multipliers is to avoid calculating coordinate charts directly, so we find an equivalent expression for in terms of without mentioning derivatives of .
To do this, we differentiate twice using the chain rule and product rule22Note that the “product” operation involved (second equality of (2)) is the operation of composition of two linear mappings. Think hard about this if you are not sure; it took me several tries to get this formula right, since multi-variable iterated derivatives have a complicated structure.. To reduce clutter, from now on we use the prime notation for derivatives rather than .
If we interpret as a bilinear mapping of vectors , then formula (2) really means
Naïvely, we might think that is simply restricted to the tangent space . This happens to be the first term in (4), but there is also an additional contribution by the second term involving ; intuitively, is the curvature of the surface (manifold) , “changing the geometry” of the graph of .
But the second term of (4) still involves . To eliminate it, we differentiate the equation twice.
or expressed as a quadratic form,
Thus, to understand the nature of the stationary point , we can study the modified Hessian:
For example, if this bilinear form is positive definite, then is a local minimum, and if it is negative definite, then is a local maximum, and so on. All the tests that apply to the usual Hessian in apply to the modified Hessian (8).
In coordinates of , the modified Hessian (8) takes the form
We emphasize that the vector can be restricted to lie in the tangent space , when studying the stationary point of restricted to .
In matrix form (9) can be written
where the columns of the matrix form a basis for .
|Title||tests for local extrema in Lagrange multiplier method|
|Date of creation||2013-03-22 15:28:52|
|Last modified on||2013-03-22 15:28:52|
|Last modified by||stevecheng (10074)|