Maximization or minimization of a function f(x) subject to conditions gi(x) 0, i = 1, 2, …, m (which is often refer to as optimization), is frequently studied through the Lagrange-multiplier method. Lagrangian function (often "Lagrangian") is composed from both objective function and restricting conditions into one function
The sign "+" holds when gi(x) 0, and sign "-" when gi(x) 0. New variables ui are called Lagrangian multipliers.
Lagrangian function enables one obtain an optimization problem without constraints, and the optimum is find by equating the first partial derivatives to zero and using ideas of traditional mathematical analysis. Thus Lagrangian function associated to nonlinear programming problem utilises the problem of finding the solution.