Infinitesimal calculus MOC

Lagrangian function

In its essence, the Lagrangian function Λ is a way to package all the information required to optimise a function under a constraint into a single function which can be optimised in the normal (unconstrained) fashion. For a function 𝑓(⃗𝐯) under a constraint 𝑔(⃗𝐯) =𝑘, the corresponding Lagrangian function is given by

Λ(⃗𝐯,𝜆)=𝑓(⃗𝐯)−𝜆(𝑔(⃗𝐯)−𝑘)

Using this function, optimising 𝑓 is a matter of solving ⃗∇Λ(⃗𝐜,𝜆) =⃗𝟎. This has the same effect as solving the system of equations used for a Lagrange multiplier.

In order to determine the nature of these saddle points of the Lagrangian, one must use the Second derivative test for multivariable functions by taking the Hessian matrix of the Lagrangian:

1. If det(𝐉2Λ) >0 it is a local maximum
2. If det(𝐉2Λ) <0 it is a local minimum
3. If det(𝐉2Λ) =0 it is neither a local maximum nor a local minimum.

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