Optimisation of function of n variables

Second degree (quadratic) Taylor approximation

del/nabla = upside down delta

del/nabla = gradient

f(x) = f(a) +  delf(a)*(x-a) + 1/2*(x-a)^T*f"(a)*(x-a)

When finding stationary points, the second bit disappears like before

f(x) = f(a) + 1/2*(x-a)^T*f"(a)*(x-a)

Classifying the stationary point requires a Hessian, a matrix of second derivatives

Find the stationary points from first derviatives

Put points into Hessian. Get the eigenvalues. If all negative = maximum. If all positive = minimum.

You need to look at the e-values.

If f"(a) > 0, positive semidefinite. Surface lies above tangent plane. Convex. a is global minimum.

If f"(a) < 0, negative semidefinite. Surface lies below tangent plane. Concave. a is global maximum.

Principal minors have to be used to classify maximum or minimum. ++++... and -+-+... is the only combos you care about. +++ is min and alternate is negative. All other is indefinite.

put second derivatives into matrix

substitute points into matrix

find det of matrix

if zero then undefinable

if negative then saddlepoint

if positive then stationary point

principal minor to find nature of stationary point

positive min negative max