Maths 4.9Lesson outlineLesson 9 of 9 in Module 4
Constrained Optimisation and Lagrange Multipliers
Minimise a function subject to a constraint, and read the multiplier.
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In this lesson10 parts
- 01A minimum with a rule attached
- 02The tangency picture
- 03The Lagrangian
- 04What the multiplier measures
- 05Inequalities and KKT
- 06Checkpoint: constraints so far
- 07Ridge and lasso as constraints
- 08PCA as a constrained maximum
- 09Constrained optimization in Python
- 10Recap, challenge, and next: numbers on a computer
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- constraint
- a condition the answer must satisfy, such as
g(x, y) = c|x + y = 1 - Lagrange multiplier
λin∇f = λ∇gat a constrained optimum |λ = −2- Lagrangian
L = f − λ(g − c); its stationary point gives the constrained optimum | three equations forx,y,λ- active constraint
- an inequality that holds with equality at the optimum |
x + y ≤ 1at(1, 0) - complementary slackness
- either the constraint is tight or its multiplier is zero |
x + y ≤ 4: multiplier0 - KKT conditions
- the conditions for an optimum with inequality constraints | gradient condition, feasibility,
μ ≥ 0, slackness
Quiz
Your first pick on each question is the one that counts, and a right one earns a coin. Getting one wrong here is how the lesson sticks.