Constrained Extrema

The extrema that arise in practical problems usually come with constraints: maximise profit subject to a budget, minimise the surface area subject to a volume. Under such conditions the extrema on the constraint surface cannot be found directly with the stationary point condition of the previous chapter — because the extremum point need not be a stationary point of the objective function.

The Lagrange multiplier method turns a constrained extremum into an unconstrained one: by introducing an undetermined multiplier, the constraint function is folded into the objective function, so that at the optimum the gradients of the two are parallel. The value of the multiplier itself has a clear meaning — it measures how much the objective function improves when the constraint is relaxed by one unit.

This chapter also explains why the method requires the constraints to be written as equalities, and how to handle the case where the constraint is an inequality.