Calculus
Lagrange Multipliers
Stand the objective up as a hill and the constraint becomes a fence winding across it. Walk the fence to its highest point. There the ground below is tangent to a contour line, which is exactly the statement that the two gradients line up.
Height f here—
Slope along the fence—
∠(∇f, ∇g)—
Multiplier λ—
Verdict—
Drag to orbit, scroll to zoom. The hill is f, the amber loop is f along the constraint circle, and the ball rolls to wherever the slope along the fence is zero.
What to observe
- The fence is the constraint circle lifted onto the hill, so its height is the objective. Optimizing under a constraint is just finding the highest point of that fence, and the flat spots are where walking along it stops changing your height.
- Watch the slope-along-the-fence readout as you drag. Where the contour on the ground crosses the circle the slope is nonzero, so you can still climb. At the top it passes through zero: the contour there is tangent to the circle.
- Tangent contours mean the two ground gradients point the same way, so∇f = λ∇g. The angle between them hits 0° exactly when the slope along the fence hits 0. That is the whole method in one picture.
- The multiplier λ is how steeply the peak height would rise if you let the circle grow. Switch objectives: x · y gives four flat spots, two peaks and two dips, and the fence has a top and a bottom to match.
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