Lagrange Multipliers

Optimize with a constraint by matching the gradient of the objective to the constraint's gradient.

What is it?

At a constrained optimum of $f$ subject to $g = c$ , the gradients are parallel: $\nabla f = λ \nabla g$ .

Minimum-variance portfolios and resource allocation questions are constrained optimizations.

Think of it when you maximize or minimize a smooth function subject to an equality constraint.

Think of this method when…

Set $\partial_{x} (f - λ g) = 0$ for every variable and solve with the constraint.

\nabla f (x) = λ \nabla g (x), g (x) = c

Two uncorrelated assets have variances 1 and 4. Find the fully invested weights that minimize portfolio variance.

1
Minimize $w_{1}^{2} + 4 w_{2}^{2}$ subject to $w_{1} + w_{2} = 1$ .
2
Conditions: $2 w_{1} = λ$ and $8 w_{2} = λ$ , so $w_{1} = 4 w_{2}$ , giving $w_{1} = 4/5$ , $w_{2} = 1/5$ .
3
Minimum variance.
$(\frac{4}{5})^{2} + 4 (\frac{1}{5})^{2} = \frac{16}{25} + \frac{4}{25} = 0.8$
Answer: $w = (4/5, 1/5)$ , variance $0.8$

Minimize $x^{2} + y^{2}$ with no constraint. Do you need a multiplier?

20 QuantSolve questions are tagged with this method.