In mathematical optimization, the method of Lagrange multipliers (or method of Lagrange's undetermined multipliers, named after Joseph-Louis Lagrange[1]) 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 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:[2]

  1. isolate any possible singular point of the solution set of the constraining equations,
  2. find all the stationary points of the Lagrange function,
  3. establish which of those stationary points and singular points are global maxima (or minima, in case of minimization problems) of the objective function.

Contents

Single constraintEdit

 
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

maximize f(x, y)
subject to g(x, y) = 0.

(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   are zero). Also, it must be assumed that   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.[3][4][5][6][7] 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.

Suppose we walk along the contour line with g = c. We are interested in finding points where f does not change as we walk, since these points might be maxima.

There are two ways this could happen:

  1. We could be following a contour line of f, since by definition f does not change as we walk along its contour lines. This would mean that the contour lines of f and g are parallel here.
  2. We have reached a "level" part of f, meaning that f does not change in any direction.

To check the first possibility (we are following a contour line of f), notice that since the gradient of a function is perpendicular to the contour lines, 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) = 0 and

 

for some λ

where

 

are the respective gradients. The constant λ 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 λ 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 λ = 0 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 g(x, y) = 0. To summarize

 

The method generalizes readily to functions on   variables

 

which amounts to solving   equations in   unknowns.

The constrained extrema of f are critical points of the Lagrangian  , but they are not necessarily local extrema of   (see Example 2 below).

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 minimum principle.

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.

Multiple constraintsEdit

 
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 λ1, λ2, ..., λM 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 method of Lagrange multipliers is generalized by the Karush–Kuhn–Tucker conditions, which can also take into account inequality constraints of the form h(x) ≤ c.


Modern formulation via differentiable manifoldsEdit

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  .[8] 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  .

Single constraintEdit

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 of  .

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  

Multiple constraintsEdit

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 multipliersEdit

Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, if the Lagrangian expression is

 

then

 

So, λk 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 λ* is the marginal cost of the constraint, and is referred to as the shadow price.

Sufficient conditionsEdit

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.[9]

ExamplesEdit

Example 1Edit

 
Illustration of the constrained optimization problem

Example 1aEdit

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

 

hence

 

Now we can calculate the gradient:

 

and therefore:

 

Notice that the last equation is the original constraint.

The first two equations yield

 

By substituting into the last equation we have:

 

so

 

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  .

Example 1bEdit

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  

Example 2Edit

 
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 x and y coordinates lie on the circle around the origin with radius 3, that is, subject to the constraint

 

As there is just a single constraint, we will use only one multiplier, say λ.

The constraint g(x, y) is identically zero on the circle of radius 3. See that any multiple of g(x, y) may be added to f(xy) leaving f(xy) unchanged in the region of interest (on the circle where our original constraint is satisfied).

Apply the ordinary Langrange multiplier method. Let:

 

Now we can calculate the gradient:

 

And therefore:

 

Notice that (iii) is just the original constraint. (i) implies x = 0 or λ = −y. If x = 0 then   by (iii) and consequently λ = 0 from (ii). If λ = −y, substituting in (ii) we get x2 = 2y2. Substituting this in (iii) and solving for y gives y = ±1. Thus there are six critical points of  :

 

Evaluating the objective at these points, we find that

 

Therefore, the objective function attains the global maximum (subject to the constraints) at   and the global minimum at   The point   is a local minimum of f and   is a local maximum of f, as may be determined by consideration of the Hessian matrix of  .

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  

Example 3: EntropyEdit

Suppose we wish to find the discrete probability distribution on the points   with maximal information entropy. This is the same as saying that we wish to find the least structured probability distribution on the points  . In other words, we wish to maximize the Shannon entropy equation:

 

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 λ only). By using the constraint

 

we find

 

Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on n points.

Example 4: Numerical optimizationEdit

 
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).[10] 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.

ApplicationsEdit

Control theoryEdit

In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in Pontryagin's minimum principle.

Nonlinear programmingEdit

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.[11]

See alsoEdit

ReferencesEdit

  1. ^ Mécanique Analytique sect. IV, 2 vols. Paris, 1811 https://archive.org/details/mcaniqueanalyt01lagr
  2. ^ Bramanti, Marco (2004). Matematica: Calcolo Infinitesimale e Algebra Lineare (2nd ed.). Bologna: Zanichelli. pp. 462–468. ISBN 978-88-08-07547-5.
  3. ^ Bertsekas, Dimitri P. (1999). Nonlinear Programming (Second ed.). Cambridge, MA.: Athena Scientific. ISBN 1-886529-00-0.
  4. ^ Vapnyarskii, I.B. (2001) [1994], "Lagrange multipliers", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
  5. ^ Lasdon, Leon S. (2002). Optimization Theory for Large Systems (Reprint of the 1970 Macmillan ed.). Mineola, New York: Dover. ISBN 0-486-41999-1. MR 1888251.
  6. ^ 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.&nbsp, 334–335). ISBN 3-540-56852-2. MR 1295240.
  7. ^ Lemaréchal, Claude (2001). "Lagrangian relaxation". In Jünger, Michael; Naddef, Denis (eds.). Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000. Lecture Notes in Computer Science. 2241. Berlin: Springer-Verlag. pp. 112–156. doi:10.1007/3-540-45586-8_4. ISBN 3-540-42877-1. MR 1900016.
  8. ^ Lafontaine, Jacques (2015). An Introduction to Differential Manifolds. Springer. p. 70. ISBN 9783319207353.
  9. ^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). McGraw-Hill. p. 386. ISBN 0-07-010813-7.
  10. ^ Heath, Michael T. (2005). Scientific Computing: An Introductory Survey. McGraw-Hill. p. 203. ISBN 978-0-07-124489-3.
  11. ^ Pourciau, Bruce H. (1980). "Modern Multiplier Rules". The American Mathematical Monthly. 87 (6): 433–452. doi:10.2307/2320250. JSTOR 2320250.

Further readingEdit

  • Beavis, Brian; Dobbs, Ian M. (1990). "Static Optimization". Optimization and Stability Theory for Economic Analysis. New York: Cambridge University Press. pp. 32–72. ISBN 0-521-33605-8.
  • Beveridge, Gordon S. G.; Schechter, Robert S. (1970). "Lagrangian Multipliers". Optimization: Theory and Practice. New York: McGraw-Hill. pp. 244–259. ISBN 0-07-005128-3.
  • Carter, Michael (2001). "Equality Constraints". Foundations of Mathematical Economics. Cambridge: MIT Press. pp. 516–549. ISBN 0-262-53192-5.
  • Hestenes, Magnus R. (1966). "Minima of functions subject to equality constraints". Calculus of Variations and Optimal Control Theory. New York: Wiley. pp. 29–34.
  • Wylie, C. Ray; Barrett, Louis C. (1995). "The Extrema of Integrals under Constraint". Advanced Engineering Mathematics (Sixth ed.). New York: McGraw-Hill. pp. 1096–1103. ISBN 0-07-072206-4.

External linksEdit

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