help with $\nabla$ and Lagrangian in optimization / portfolio theory?

Question

So $\nabla$ as I know it from calculus means gradient.

We have

$\min \ \ \frac{1}{2}w^T\Sigma w$

$s.t. \ \ \ \ \ w^T1 = V_0, \ \ V_0 = 100$

where $w$ is weights in vector, $\Sigma$ is the covariance matrix and 1 is a vector with 1 values corresponding to $w$.

and Lagrangian

$L = \frac{1}{2}w^T\Sigma w + \lambda(V_0 - w^T1)$

How did we get the lagrangian? I it looked up in wiki (Lagrangian field theory) but don't really understand.

Also how do we get the first order condition:

$\nabla L = \Sigma w - \lambda1 = 0$ and

I know how to calculate this type of exercises because they are all the same but why do we do that we do?

Answer 1

The google search term you want is not Lagrangian but rather Lagrange multipliers (https://en.wikipedia.org/wiki/Lagrange_multiplier)