A tool designed for constrained optimization problems uses a method attributed to Joseph-Louis Lagrange. This method involves constructing a new function, often denoted as L, which incorporates the original objective function and the constraints, each multiplied by a Lagrange multiplier. This constructed function allows for the identification of stationary points that represent potential solutions to the optimization problem, satisfying both the objective and constraints. For instance, maximizing area within a fixed perimeter would utilize this approach.
This mathematical approach offers a powerful framework for solving a wide array of problems across diverse fields such as physics, economics, and engineering. It provides a systematic way to handle constraints, eliminating the need for complex substitutions or graphical methods. Its development in the 18th century marked a significant advancement in optimization theory, providing a more elegant and efficient solution compared to previous techniques. This methodology remains fundamental to modern optimization and control theory.