Lab 3: A Lagrange Multiplier by Hand

DATA 442/642 · Week 3 · PDF

One equality constraint, stationary points, and a contour picture.

Work in a Jupyter notebook. Number the exercises. Do notes 3.1 through 3.3 first. You will need numpy and matplotlib.

The original assignment PDF is lab-3-assignment.pdf.

When you are done: File → Download as → HTML, then upload the HTML on Canvas.

1. One equality, by hand (3 points)

1. Write the Lagrangian $ℒ(\mathit{\theta },\lambda )=J(\mathit{\theta })+\lambda g(\mathit{\theta })$.
2. Set ${\nabla }_{\mathit{\theta }}ℒ=\mathbf{0}$ and $g=0$. Solve for ${\theta }_1$, ${\theta }_2$, and $\lambda$. Show the algebra in the notebook (markdown or comments), then check the numbers in code.
3. Say what $\mathit{\theta }$ and $\lambda$ mean for a model whose parameters must satisfy a computational budget written as $g=0$.

2. Picture (2 points)

Contour plot of $J$ in the $({\theta }_1,{\theta }_2)$ plane. Draw the line $g=0$. Mark the constrained minimizer. The unconstrained min is the origin; your star should not be there.