Lagrange Multipliers
Optimise subject to a constraint by setting the gradient of the objective parallel to the gradient of the constraint, which is the moment the level curves touch.
What a learner can do afterwards
- Set up the multiplier equations for a constrained problem
- Explain the tangency condition in terms of level curves
- Compare the multiplier method with substituting the constraint directly
1 · Read
Optimising with a constraint means staying on a curve while pushing the objective as far as possible. The best feasible point is where a level curve of f just touches the constraint instead of crossing it. Touching curves share a tangent, so grad f = lambda grad g, solved together with the constraint itself.
A rectangle with perimeter 20 obeys x + y = 10. The multiplier equations give x = y, so the winner is a 5 by 5 square with area 25. Likewise, x + y on x squared + y squared = 2 peaks at x = y = 1 with value 2.
Lambda is not a spare part; it is the exchange rate. It says how much the optimum improves per unit of relaxed constraint. When substitution is easy both routes agree, but on a circle isolating a variable creates square roots that are painful to differentiate.
The method finds suspects; the comparison convicts. Evaluate f at every candidate, then keep the largest or smallest, and check boundary behaviour plus any nonsmooth points. For x squared + y squared with x + y = 4, symmetry picks (2, 2) and the minimum 8.
Set the gradients parallel, add the constraint, solve, and compare candidates before declaring a winner.
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