University⏱ 5 min

Lagrange Multipliers

Constrained optimization

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Lagrange Multipliers Explained

∇f∇g○ Contours f(x,y) = x²+y²– Constraint g = 0● Optimal point

Constraint

g(x,y) = x/a + y/b − 1 = 0

a1.50
b1.50

Solution

Optimal point(0.750, 0.750)
f(x*,y*)1.1250
λ2.2500

∇f = λ · ∇g

Gradients are parallel at the optimum

💡 Geometric intuition

The optimum occurs where a contour of f is tangent to the constraint g. At that point ∇f and ∇g are parallel – if they weren't, we could move along the constraint and improve f.