Lagrange Multipliers in 50 Seconds: Where Gradients Line Up

Find the highest point, but you must stay on a path. How?

• Maximize f
• Stay on the path

Draw the hill as contour lines. Higher rings mean higher ground.

• f(x, y) is height
• Rings are equal height

Now add a rule. You may only walk along this curve.

• g(x, y) = 0
• Walk only on the curve

Slide along it. While the curve crosses contours, you can still climb.

• Crossing contours
• f keeps rising

At the best point, the curve just touches one contour. They are tangent.

• Curve touches one contour
• No more climbing

The gradient of f points uphill, perpendicular to its contour. The gradient of g is perpendicular to the curve.

• grad f ⟂ contour
• grad g ⟂ constraint

Tangent means the same perpendicular. So both gradients point along one line.

• Same tangent, same normal
• Parallel at the optimum

One equation captures it. Gradient of f equals lambda times gradient of g.

• One equation
• Lagrange condition

Lambda just scales one arrow into the other. Lesson thirty-seven puts numbers to it. Follow for more.

• λ scales the arrow
• Worked numbers next