Form the Lagrangian - objective minus lambda times constraint - and set all partial derivatives to zero.
The multiplier is not just bookkeeping. Lambda is the shadow price: the rate at which the optimum improves as the constraint is relaxed. In portfolio optimisation it tells you what an extra unit of risk budget is worth, which is often the answer the question actually wants.
Where it appears: mean-variance optimisation subject to a budget constraint is the canonical case, and it is the standard route to the efficient frontier.