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A method to solve constrained optimization problems
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). It is named after the mathematician Joseph-Louis Lagrange. 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.
The method can be summarized as follows: in order to find the maximum or minimum of a function subjected to the equality constraint , form the Lagrangian function
and find the stationary points of considered as a function of and the Lagrange multiplier . The negative sign in front of is arbitrary; a positive sign works equally well. 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 .
The following is known as the Lagrange multiplier theorem.
Let be the objective function, be the constraints function, both belonging to (that is, having continuous first derivatives). Let be an optimal solution to the following optimization problem such that rank :
(Here denotes the matrix of partial derivatives, .)
Then there exist unique Lagrange multipliers such that .
The Lagrange multiplier theorem states that at any local maxima (or minima) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then 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. 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 0 in every feasible direction.
Figure 1: The red curve shows the constraint g(x, y) = c. The blue curves are contours of f(x, y). The point where the red constraint tangentially touches a blue contour is the maximum of f(x, y) along the constraint, since d1 > d2.
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 , in which case the constraint is written , as in Figure 1.) We assume that both and have continuous first partial derivatives. We introduce a new variable () called a Lagrange multiplier (or Lagrange undetermined multiplier) and study the Lagrange function (or Lagrangian or Lagrangian expression) 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 (stationary points are those points where the first partial derivatives of are zero). 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, 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. Viewed in this way, it is an exact analogue to testing if the derivative of an unconstrained function is 0, that is, we are verifying that the directional derivative is 0 in any relevant (viable) direction.
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.
Suppose we walk along the contour line with g = c. We are interested in finding points where f almost does not change as we walk, since these points might be maxima.
There are two ways this could happen:
We could touch a contour line of f, since by definition f does not change as we walk along its contour lines. This would mean that the tangents to the contour lines of f and g are parallel here.
We have reached a "level" part of f, meaning that f does not change in any direction.
To check the first possibility (we touch a contour line of f), notice that since the gradient of a function is perpendicular to the contour lines, the tangents to the contour lines of f and g are parallel if and only if the gradients of f and g are parallel. Thus we want points (x, y) where g(x, y) = c and
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. (In some conventions is preceded by a minus sign).
Notice that this method also solves the second possibility, that f is level: if f 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
Note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers. Note that implies . To summarize
The method generalizes readily to functions on variables
which amounts to solving n + 1 equations in n + 1 unknowns.
The constrained extrema of f are critical points of the Lagrangian , but they are not necessarily local extrema of (see Example 2 below).
The fact that solutions of the Lagrangian are not necessarily extrema also poses difficulties for numerical optimization. This can be addressed by computing the magnitude of the gradient, as the zeros of the magnitude are necessarily local minima, as illustrated in the numerical optimization example.
Figure 2: A paraboloid constrained along two intersecting lines.
Figure 3: Contour map of Figure 2.
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 (see Figure 3); 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 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 (constrained) extrema. We therefore seek such that any allowable direction of movement away from is perpendicular to (otherwise we could increase by moving along that allowable direction). 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
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 (which depends on a choice of Riemannian metric) can be replaced with the exterior derivative.
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 0 is a regular value.
Let and be the exterior derivatives. Stationarity for the restriction at means Equivalently, the kernel contains In other words, and are proportional vectors. 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
Let and be as in the above section regarding the case of a single constraint. Rather than the function described there, now consider a smooth function with component functions for which is a regular value. Let be the submanifold of defined by
is a stationary point of if and only if contains . For convenience let and where denotes the tangent map or Jacobian The subspace has dimension smaller than that of , namely and belongs to if and only if belongs to the image of Computationally speaking, the condition is that belongs to the row space of the matrix of or equivalently the column space of the matrix of (the transpose). If denotes the exterior product of the columns of the matrix of the stationary condition for at becomes
Once again, in this formulation it is not necessary to explicitly find the Lagrange multipliers, the numbers such that
Interpretation of the Lagrange multipliers
Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, by parametrising the constraint's contour line, that is, if the Lagrangian expression is
So, ?k is the rate of change of the quantity being optimized as a function of the constraint parameter.
As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action, the time integral of the difference between kinetic and potential energy. Thus, the force on a particle due to a scalar potential, F = -?V, can be interpreted as a Lagrange multiplier determining the change in action (transfer of potential to kinetic energy) following a variation in the particle's constrained trajectory.
In control theory this is formulated instead as costate equations.
Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: if we denote values at the optimum with an asterisk, then it can be shown that
For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context ?k* is the marginal cost of the constraint, and is referred to as the shadow price.
Sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian matrix of second derivatives of the Lagrangian expression.
Illustration of the constrained optimization problem 1a
Suppose we wish to maximize subject to the constraint . The feasible set is the unit circle, and the level sets of f are diagonal lines (with slope -1), so we can see graphically that the maximum occurs at , and that the minimum occurs at .
For the method of Lagrange multipliers, the constraint is
Now we can calculate the gradient:
Notice that the last equation is the original constraint.
The first two equations yield
By substituting into the last equation we have:
which implies that the stationary points of are
Evaluating the objective function f at these points yields
Thus the constrained maximum is and the constrained minimum is .
Illustration of the constrained optimization problem 1b
Now we modify the objective function of Example 1a so that we minimize instead of again along the circle . Now the level sets of f are still lines of slope -1, and the points on the circle tangent to these level sets are again and . These tangency points are maxima of f.
On the other hand, the minima occur on the level set for f = 0 (since by its construction f cannot take negative values), at and , where the level curves of f are not tangent to the constraint. The condition that correctly identifies all four points as extrema; the minima are characterized in particular by
Illustration of the constrained optimization problem
In this example we will deal with some more strenuous calculations, but it is still a single constraint problem.
Suppose we want to find the maximum values of
with the condition that the - and -coordinates lie on the circle around the origin with radius . That is, subject to the constraint
As there is just a single constraint, we will use only one multiplier, say .
The constraint is identically zero on the circle of radius . See that any multiple of may be added to leaving unchanged in the region of interest (on the circle where our original constraint is satisfied).
Apply the ordinary Lagrange multiplier method by letting:
Now we can calculate the gradient:
Notice that (iii) is just the original constraint. (i) implies or. If then by (iii) and consequently from (ii). If , substituting this into (ii) we get . Now substituting this into (iii) and solving for gives . Thus there are six critical points of :
Evaluating the objective at these points, we find that
Note that while is a critical point of , it is not a local extremum of We have
Given any neighbourhood of , we can choose a small positive and a small of either sign to get values both greater and less than . This can also be seen from the fact that the Hessian matrix of evaluated at this point (or indeed at any of the critical points) is an indefinite matrix. Each of the critical points of is a saddle point of .
For this to be a probability distribution the sum of the probabilities at each point must equal 1, so our constraint is:
We use Lagrange multipliers to find the point of maximum entropy, , across all discrete probability distributions on . We require that:
which gives a system of n equations, , such that:
Carrying out the differentiation of these n equations, we get
This shows that all are equal (because they depend on ? only). By using the constraint
Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on n points.
Example 4: Numerical optimization
Lagrange multipliers cause the critical points to occur at saddle points.
The magnitude of the gradient can be used to force the critical points to occur at local minima.
The critical points of Lagrangians occur at saddle points, rather than at local maxima (or minima). Unfortunately, many numerical optimization techniques, such as hill climbing, gradient descent, some of the quasi-Newton methods, among others, are designed to find local maxima (or minima) and not saddle points. For this reason, one must either modify the formulation to ensure that it's a minimization problem (for example, by extremizing the square of the gradient of the Lagrangian as below), or else use an optimization technique that finds stationary points (such as Newton's method without an extremum seeking line search) and not necessarily extrema.
As a simple example, consider the problem of finding the value of x that minimizes , constrained such that . (This problem is somewhat pathological because there are only two values that satisfy this constraint, but it is useful for illustration purposes because the corresponding unconstrained function can be visualized in three dimensions.)
Using Lagrange multipliers, this problem can be converted into an unconstrained optimization problem:
The two critical points occur at saddle points where x = 1 and x = -1.
In order to solve this problem with a numerical optimization technique, we must first transform this problem such that the critical points occur at local minima. This is done by computing the magnitude of the gradient of the unconstrained optimization problem.
First, we compute the partial derivative of the unconstrained problem with respect to each variable:
If the target function is not easily differentiable, the differential with respect to each variable can be approximated as
where is a small value.
Next, we compute the magnitude of the gradient, which is the square root of the sum of the squares of the partial derivatives:
(Since magnitude is always non-negative, optimizing over the squared-magnitude is equivalent to optimizing over the magnitude. Thus, the ''square root" may be omitted from these equations with no expected difference in the results of optimization.)
The critical points of h occur at x = 1 and x = -1, just as in . Unlike the critical points in , however, the critical points in h occur at local minima, so numerical optimization techniques can be used to find them.
The Lagrange multiplier method has several generalizations. In nonlinear programming there are several multiplier rules, e.g. the Carathéodory-John Multiplier Rule and the Convex Multiplier Rule, for inequality constraints.
Methods based on Lagrange multipliers have applications in power systems, e.g. in distributed-energy-resources (DER) placement and load shedding.
^Lasdon, Leon S. (2002). Optimization Theory for Large Systems (Reprint of the 1970 Macmillan ed.). Mineola, New York: Dover. ISBN0-486-41999-1. MR1888251.
^Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: Springer-Verlag. pp. 136-193 (and Bibliographical comments on pp. 334-335). ISBN3-540-56852-2. MR1295240.
^Gautam, Mukesh; Bhusal, Narayan; Benidris, Mohammed (2020). "A Sensitivity-based Approach to Adaptive Under-Frequency Load Shedding". 2020 IEEE Texas Power and Energy Conference (TPEC). IEEE. pp. 1-5. doi:10.1109/TPEC48276.2020.9042569.
Geometric Representation of Method of Lagrange Multipliers Provides compelling insight in 2 dimensions that at a minimizing point, the direction of steepest descent must be perpendicular to the tangent of the constraint curve at that point. [Needs InternetExplorer/Firefox/Safari] Mathematica demonstration by Shashi Sathyanarayana