Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. Once stationary points have been identified from the first-order necessary conditions, the definiteness of the bordered Hessian matrix determines whether those points are maxima, minima, or saddle points. In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. Once stationary points have been identified from the first-order necessary conditions, the definiteness of the bordered Hessian matrix determines whether those points are maxima, minima, or saddle points. The Lagrange multiplier theorem roughly states that at any stationary point of the function that also satisfies the equality constraints, the gradient of the function at that point can be expressed as a linear combination of the gradients of the constraints at that point, with the Lagrange multipliers acting as coefficients. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The great advantage of this method is that it allows the optimization to be solved without explicit parameterization in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. The method can be summarized as follows: in order to find the stationary points of a function f ( x ) {displaystyle f(x)} subject to the equality constraints g i ( x ) = 0 {displaystyle g_{i}(x)=0} , i = 1 , 2 , … , m {displaystyle i=1,2,ldots ,m} , form the Lagrangian function and find the stationary points of L {displaystyle {mathcal {L}}} considered as a function of x {displaystyle x} and the Lagrange multiplier λ {displaystyle lambda } . For the case of only one constraint and only two choice variables (as exemplified in Figure 1), consider the optimization problem (Sometimes an additive constant is shown separately rather than being included in g, in which case the constraint is written g(x, y) = c, as in Figure 1.) We assume that both f and g have continuous first partial derivatives. We introduce a new variable (λ) called a Lagrange multiplier (or Lagrange undetermined multiplier) and study the Lagrange function (or Lagrangian or Lagrangian expression) defined by where the λ term may be either added or subtracted. If f(x0, y0) is a maximum of f(x, y) for the original constrained problem, then there exists λ0 such that (x0, y0, λ0) is a stationary point for the Lagrange function (stationary points are those points where the first partial derivatives of L {displaystyle {mathcal {L}}} are zero). Also, it must be assumed that ∇ g ≠ 0. {displaystyle abla g eq 0.} However, not all stationary points yield a solution of the original problem, as the method of Lagrange multipliers yields only a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist, but if a particular candidate solution satisfies the sufficient conditions, it is only guaranteed that that solution is the best one locally – that is, it is better than any permissible nearby points. The global optimum can be found by comparing the values of the original objective function at the points satisfying the necessary and locally sufficient conditions. The method of Lagrange multipliers relies on the intuition that at a maximum, f(x, y) cannot be increasing in the direction of any such neighboring point that also has g = 0. If it were, we could walk along g = 0 to get higher, meaning that the starting point wasn't actually the maximum. We can visualize contours of f given by f(x, y) = d for various values of d, and the contour of g given by g(x, y) = c.