tests for local extrema in Lagrange multiplier method

Let U be open in n, and f:U, g:Um be twice continuously differentiable functions. Assume that pU is a stationary point for f on M=g-1({0}), and Dg has full rank everywhere11Actually, only Dg(p) needs to have full rank, and the argumentsMathworldPlanetmath presented here continue to hold in that case, although M would not necessarily be a manifold then. on M. Then we know that p is the solution to the Lagrange multiplierMathworldPlanetmath system

Df(p)=λDg(p), (1)

for a Lagrange multiplier vector λ=(λ1,,λm).

Our aim is to develop an analogue of the second derivative testMathworldPlanetmath for the stationary point p.

The most straightforward way to proceed is to consider a coordinate chart α:VM for the manifold M, and consider the HessianMathworldPlanetmath of the function fα:Vn at 0=α-1(p). This Hessian is in fact just the Hessian form of f:M expressed in the coordinatesPlanetmathPlanetmath of the chart α. But the whole point of using Lagrange multipliers is to avoid calculating coordinate charts directly, so we find an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath expression for D2(fα)(0) in terms of D2f(p) without mentioning derivativesPlanetmathPlanetmath of α.

To do this, we differentiate fα twice using the chain ruleMathworldPlanetmath and product ruleMathworldPlanetmath22Note that the “productPlanetmathPlanetmathoperationMathworldPlanetmath 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 formulaMathworldPlanetmathPlanetmath right, since multi-variable iterated derivatives have a complicated structureMathworldPlanetmath.. To reduce clutter, from now on we use the prime notation for derivatives rather than D.

(fα)′′(0)=((fα)α)(0)=((f′′α)αα+(fα)α′′)(0)=f′′(α(0))α(0)α(0)+f(α(0))α′′(0)=f′′(p)α(0)α(0)+f(p)α′′(0). (2)

If we interpret (fα)′′(0) as a bilinear mapping of vectors u,vn-m, then formula (2) really means

(fα)′′(0)(u,v)=f′′(p)(α(0)u,α(0)v)+f(p)(α′′(0)(u,v)). (3)

To obtain the quadratic formMathworldPlanetmath, we set v=u; also we abbreviate the vector α(0)u by h, which belongs to the tangent space TpM of M at p. So,

(fα)′′(0)u2=f′′(p)h2+f(p)(α′′(0)u2). (4)

Naïvely, we might think that (fα)′′(0) is simply f′′(p) restricted to the tangent space TpM. This happens to be the first term in (4), but there is also an additional contribution by the second term involving α′′(0); intuitively, α′′(0) is the curvature of the surface (manifold) M, “changing the geometry” of the graph of f.

But the second term of (4) still involves α. To eliminate it, we differentiate the equation gα=0 twice.

0=(gα)′′(0)=g′′(p)α(0)α(0)+g(p)α′′(0). (5)

(It is derived the same way as (2) but with f replaced by g.) Now we can substitute (5) and (1) in (2) to eliminate the term f(p)α′′(0):

(fα)′′(0)=f′′(p)α(0)α(0)+λg(p)α′′(0)=f′′(p)α(0)α(0)-λg′′(p)α(0)α(0), (6)

or expressed as a quadratic form,

(fα)′′(0)u2=f′′(p)h2-λg′′(p)h2. (7)

Thus, to understand the nature of the stationary point p, we can study the modified Hessian:

f′′(p)-λg′′(p), restricted to TpM (8)

For example, if this bilinear formPlanetmathPlanetmath is positive definitePlanetmathPlanetmath, then p is a local minimumMathworldPlanetmath, and if it is negative definite, then p is a local maximum, and so on. All the tests that apply to the usual Hessian in n apply to the modified Hessian (8).

In coordinates of n, the modified Hessian (8) takes the form

i=1nj=1n(2fxixj|p-k=1mλk2gkxixj|p)hihj. (9)

We emphasize that the vector h can be restricted to lie in the tangent space TpM, when studying the stationary point p of f restricted to M.

In matrix form (9) can be written

B=[2fxixj-k=1mλk2gkxixj]ij. (10)

But again, the test vector h need only lie on TpM, so if we want to apply positivePlanetmathPlanetmath/negative definiteness tests for matrices, they should instead be applied to the projected or reduced Hessian:

ZtBZ (11)

where the columns of the n×(n-m) matrix Z form a basis for TpM=kerg(p)n.

Title tests for local extrema in Lagrange multiplier method
Canonical name TestsForLocalExtremaInLagrangeMultiplierMethod
Date of creation 2013-03-22 15:28:52
Last modified on 2013-03-22 15:28:52
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 6
Author stevecheng (10074)
Entry type Result
Classification msc 26B12
Classification msc 49-00
Classification msc 49K35
Related topic HessianForm
Related topic RelationsBetweenHessianMatrixAndLocalExtrema
Defines projected Hessian
Defines reduced Hessian