Lagrange Multipliers

To maximize $f(x)$ subject to $g(x) = 0$, the Lagrangian is

At an optimum, $\nabla \mathcal{L} = 0$, which translates to $\nabla f = \lambda \nabla g$. The constraint and objective gradients must be parallel — there's no remaining freedom to improve $f$ while staying on the constraint.

For multiple constraints, add a multiplier per constraint. For inequality constraints, the KKT (Karush-Kuhn-Tucker) conditions extend Lagrange and underpin most modern optimization.

In quant finance, Lagrangians solve the textbook minimum-variance portfolio: minimize $w^\top \Sigma w$ subject to $\sum w_i = 1$. The first-order conditions give a clean closed-form solution proportional to $\Sigma^{-1} \mathbf{1}$.