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 equation constraints. It is named after the mathematician Joseph-Louis Lagrange.

Summary and rationale

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. 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 or Lagrangian. In the general case, the Lagrangian is defined as
for functions ; the notation denotes an inner product. The value is called the Lagrange multiplier.
In simple cases, where the inner product is defined as the dot product, the Lagrangian is
The method can be summarized as follows: in order to find the maximum or minimum of a function subject to the equality constraint, find the stationary points of considered as a function of and the Lagrange multiplier. This means that all partial derivatives should be zero, including the partial derivative with respect to.
or equivalently
The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function, which can be identified among the stationary points from the definiteness of the bordered Hessian matrix.
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. Further, the method of Lagrange multipliers is generalized by the Karush–Kuhn–Tucker conditions, which can also take into account inequality constraints of the form for a given constant.

Statement

The following is known as the Lagrange multiplier theorem.
Let be the objective function and let be the constraints function, both belonging to . Let be an optimal solution to the following optimization problem such that, for the matrix of partial derivatives, :
Then there exists a unique Lagrange multiplier such that
The Lagrange multiplier theorem states that at any local maximum of the function evaluated under the equality constraints, if constraint qualification applies, then the gradient of the function can be expressed as a linear combination of the gradients of the constraints, with the Lagrange multipliers acting as coefficients. This is equivalent to saying that any direction perpendicular to all gradients of the constraints is also perpendicular to the gradient of the function. Or still, saying that the directional derivative of the function is in every feasible direction.

Single constraint

For the case of only one constraint and only two choice variables, consider the optimization problem
We assume that both and have continuous first partial derivatives. We introduce a new variable called a Lagrange multiplier and study the Lagrange function defined by
where the term may be either added or subtracted. If is a maximum of for the original constrained problem and then there exists such that is a stationary point for the Lagrange function. The assumption is called constraint qualification. 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, cannot be increasing in the direction of any such neighboring point that also has. If it were, we could walk along to get higher, meaning that the starting point wasn't actually the maximum. Viewed in this way, it is an exact analogue to testing if the derivative of an unconstrained function is, that is, we are verifying that the directional derivative is 0 in any relevant direction.
We can visualize contours of given by for various values of, and the contour of given by.
Suppose we walk along the contour line with We are interested in finding points where almost does not change as we walk, since these points might be maxima.
There are two ways this could happen:
  1. We could touch a contour line of, since by definition does not change as we walk along its contour lines. This would mean that the tangents to the contour lines of and are parallel here.
  2. We have reached a "level" part of, meaning that does not change in any direction.
To check the first possibility, notice that since the gradient of a function is perpendicular to the contour lines, the tangents to the contour lines of and are parallel if and only if the gradients of and are parallel. Thus we want points where and
for some where
are the respective gradients. The constant is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier..
Notice that this method also solves the second possibility, that is level: if is level, then its gradient is zero, and setting is a solution regardless of.
To incorporate these conditions into one equation, we introduce an auxiliary function
and solve
Note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers.
Note that implies as the partial derivative of with respect to is
To summarize
The method generalizes readily to functions on variables
which amounts to solving equations in unknowns.
The constrained extrema of are critical points of the Lagrangian, but they are not necessarily local extrema of .
One may reformulate the Lagrangian as a Hamiltonian, in which case the solutions are local minima for the Hamiltonian. This is done in optimal control theory, in the form of Pontryagin's maximum principle.
The fact that solutions of the method of Lagrange multipliers are not necessarily extrema of the Lagrangian, also poses difficulties for numerical optimization. This can be addressed by minimizing the magnitude of the gradient of the Lagrangian, as these minima are the same as the zeros of the magnitude, as illustrated in [|Example 5: Numerical optimization].

Multiple constraints

The method of Lagrange multipliers can be extended to solve problems with multiple constraints using a similar argument. Consider a paraboloid subject to two line constraints that intersect at a single point. As the only feasible solution, this point is obviously a constrained extremum. However, the level set of is clearly not parallel to either constraint at the intersection point ; instead, it is a linear combination of the two constraints' gradients. In the case of multiple constraints, that will be what we seek in general: The method of Lagrange seeks points not at which the gradient of is a multiple of any single constraint's gradient necessarily, but in which it is a linear combination of all the constraints' gradients.
Concretely, suppose we have constraints and are walking along the set of points satisfying Every point on the contour of a given constraint function has a space of allowable directions: the space of vectors perpendicular to The set of directions that are allowed by all constraints is thus the space of directions perpendicular to all of the constraints' gradients. Denote this space of allowable moves by and denote the span of the constraints' gradients by Then the space of vectors perpendicular to every element of
We are still interested in finding points where does not change as we walk, since these points might be extrema. We therefore seek such that any allowable direction of movement away from is perpendicular to . In other words, Thus there are scalars such that
These scalars are the Lagrange multipliers. We now have of them, one for every constraint.
As before, we introduce an auxiliary function
and solve
which amounts to solving equations in unknowns.
The constraint qualification assumption when there are multiple constraints is that the constraint gradients at the relevant point are linearly independent.

Modern formulation via differentiable manifolds

The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a differentiable manifold In what follows, it is not necessary that be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient can be replaced with the exterior derivative

Single constraint

Let be a smooth manifold of dimension Suppose that we wish to find the stationary points of a smooth function when restricted to the submanifold defined by where is a smooth function for which is a regular value.
Let and be the exterior derivatives of and. Stationarity for the restriction at means Equivalently, the kernel contains In other words, and are proportional 1-forms. For this it is necessary and sufficient that the following system of equations holds:
where denotes the exterior product. The stationary points are the solutions of the above system of equations plus the constraint Note that the equations are not independent, since the left-hand side of the equation belongs to the subvariety of consisting of decomposable elements.
In this formulation, it is not necessary to explicitly find the Lagrange multiplier, a number such that