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). 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 formula_1 subjected to the equality constraint formula_2, form the Lagrangian function and find the of formula_4 considered as a function of formula_5 and the Lagrange multiplier formula_6. 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 formula_7 for a given constant formula_8. Statement. The following is known as the Lagrange multiplier theorem. Let formula_9 be the objective function, formula_10 be the constraints function, both belonging to formula_11 (that is, having continuous first derivatives). Let formula_12 be an optimal solution to the following  optimization problem such that rank formula_13: Then there exists a unique Lagrange multiplier formula_18 such that formula_19. 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. Single constraint. 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 formula_22, in which case the constraint is written formula_23, as in Figure 1.) We assume that both formula_24 and formula_25 have continuous first partial derivatives. We introduce a new variable (formula_26) called a Lagrange multiplier (or Lagrange undetermined multiplier) and study the Lagrange function (or Lagrangian or Lagrangian expression) defined  by where the formula_26 term may be either added or subtracted. If formula_29 is a maximum of formula_30 for the original constrained problem and formula_31, then there exists formula_32 such that (formula_33) is a stationary point for the Lagrange function (stationary points are those points where the first partial derivatives of formula_4 are zero). The assumption formula_35 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 0, that is, we are verifying that the directional derivative is 0 in any relevant (viable) 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: To check the first possibility (we touch a contour  line of ), 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 formula_26 where are the respective gradients. The constant formula_26 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 formula_26 is preceded by a minus sign). Notice that this method also solves the second possibility, that  is level: if is level, then its gradient is zero, and setting formula_41 is a solution regardless of formula_42. 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 formula_45 implies formula_46, as the partial derivative of formula_4 with respect to formula_6 is formula_49, which clearly is zero if and only if formula_50. To summarize The method generalizes readily to functions on formula_52 variables which amounts to solving equations in unknowns. The constrained extrema of are  critical points of the Lagrangian formula_4, but they are not necessarily "local extrema" of formula_4 (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 . 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 formula_56 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 formula_56 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 formula_58 constraints and are walking along the set of points satisfying formula_59. Every point formula_60 on the contour of a given constraint function formula_61 has a space of allowable directions: the space of vectors perpendicular to formula_62. 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 formula_63 and denote the span of the constraints ' gradients by formula_64. Then formula_65, the space of vectors perpendicular to every element of formula_64. We are still interested in finding points where formula_56 does not change as we walk, since these points might be (constrained) extrema. We therefore seek formula_60 such that any allowable direction of movement away from formula_60 is perpendicular to formula_70 (otherwise we could increase formula_56 by moving along that allowable direction). In other words, formula_72. Thus there are scalars formula_73 such that These scalars are the Lagrange multipliers. We now have formula_58 of them, one for every constraint. As before, we introduce an auxiliary function  and solve which amounts to solving formula_78 equations in formula_78 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 formula_58. In what follows, it is not necessary that formula_58 be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient formula_82 (which depends on a choice of Riemannian metric) can be replaced with the  exterior derivative formula_83. Single constraint. Let formula_58 be a smooth manifold of dimension formula_85. Suppose that we wish to find the stationary points formula_5 of a smooth function formula_87 when restricted to the submanifold formula_88 defined by formula_89 where formula_90 is a smooth function for which 0 is a regular value. Let formula_91 and formula_92 be the exterior derivatives. Stationarity for the restriction formula_93 at formula_94 means formula_95 Equivalently, the kernel formula_96 contains formula_97 In other words, formula_98 and formula_99 are proportional 1-forms. For this it is necessary and sufficient that the following system of formula_100 equations holds: where  formula_102 denotes the exterior product. The stationary points formula_5 are the solutions of the above system of equations plus the constraint formula_104 Note that the formula_105 equations are not independent, since the left-hand side of the equation belongs to the subvariety of formula_106 consisting of decomposable elements. In this formulation, it is not necessary to explicitly find the Lagrange multiplier, a number formula_6 such that formula_108 Multiple constraints. Let formula_58 and formula_56 be as in the above section regarding the case of a single constraint. Rather than the function formula_22 described there, now consider a smooth function formula_112 with  component functions formula_113 for which formula_114 is a regular value. Let formula_88 be the submanifold of formula_58 defined by formula_117 formula_5 is a stationary point of formula_93 if and only if formula_96 contains formula_121. For convenience let formula_122 and formula_123 where formula_124 denotes the tangent map or Jacobian formula_125 The subspace formula_126 has dimension smaller than that of formula_127, namely formula_128 and formula_129 formula_126 belongs to formula_127 if and only if formula_132 belongs to the image of formula_133 Computationally speaking, the condition is that formula_134 belongs to the row space of the matrix of formula_135 or equivalently the column space  of the matrix of formula_136 (the transpose). If formula_137 denotes the exterior product of the columns of the matrix of formula_138 the stationary condition for formula_93 at formula_5 becomes Once again, in this formulation it is not necessary to explicitly find the Lagrange multipliers, the numbers formula_142 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 then So, 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, , 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 is the marginal cost of the constraint, and is referred to as the shadow price. Sufficient conditions. 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. Examples. Example 1. Example 1a. Suppose we wish to maximize formula_147 subject to the constraint formula_148. The feasible set is the unit circle, and the level sets of are diagonal lines (with slope −1), so we can see graphically that the maximum occurs at formula_149, and that the minimum occurs at formula_150. For the method of Lagrange multipliers, the constraint is hence a  function that is equivalent to formula_153 when formula_154 is set to formula_155. 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 formula_4 are Evaluating the objective function at these points yields Thus the constrained maximum is formula_164 and the constrained minimum is formula_165. Example 1b. Now we modify the objective function of Example 1a so that we minimize formula_166 instead of formula_167 again along the circle formula_168. Now the level sets  of formula_56 are still lines of slope −1, and the points on the circle tangent to these level sets are again formula_170 and formula_171. These tangency points are maxima of formula_56. On the other hand, the minima occur on the level set for formula_56 = 0 (since by its construction formula_56 cannot take negative values), at formula_175 and formula_176, where the level curves of formula_56 are not tangent to the constraint. The condition that formula_178 correctly identifies all four points as extrema; the minima are characterized in particular by formula_179 Example 2. This example deals with more strenuous calculations , but it is still a single constraint problem. Suppose one wants to find the maximum values of with the condition that the formula_5- and formula_182-coordinates lie on the circle around the origin with radius formula_183. That is, subject to the constraint As there is just a single constraint, there is a single multiplier, say formula_6. The constraint formula_154 is identically zero on the circle of radius formula_183. Any multiple of formula_154 may be added to formula_154 leaving formula_154 unchanged in the region of interest (on the circle where our original constraint is satisfied). Applying the ordinary Lagrange multiplier method  yields from which the gradient can be calculated: And therefore: (iii) is just the original constraint. (i) implies formula_194 "or" formula_195. If formula_194 then formula_197 by (iii) and consequently formula_198 from (ii). If formula_195, substituting this into (ii) yields formula_200. Substituting this into (iii) and solving for formula_182 gives formula_202. Thus there are six critical points of formula_4: Evaluating the objective at these points, one finds that Therefore, the objective function attains the global maximum (subject to the constraints) at formula_206 and the global minimum at formula_207 The point formula_208 is a local minimum of formula_56 and formula_210 is a  local maximum of formula_56, as may be determined by consideration of the Hessian matrix of formula_212. Note that while formula_213 is a critical point of formula_4, it is not a local extremum of formula_215 We have Given any neighbourhood of formula_213, one can choose a small positive formula_218 and a small formula_219 of either sign to get formula_4 values both greater and less than formula_221. This can also be seen from the Hessian matrix of formula_4 evaluated at this point (or indeed at any of the critical points) which is an indefinite matrix. Each of the critical points of formula_4  is a saddle point of formula_4. Example 3: Entropy. Suppose we wish to find the discrete probability distribution on the points formula_225 with maximal information entropy. This is the same as saying that we wish to find the least structured probability distribution on the points formula_226. In other words, we wish to maximize the Shannon entropy equation: For this to be a probability distribution the sum of the probabilities formula_228 at each point formula_229 must equal 1, so our constraint is: We use Lagrange multipliers to find the point of maximum entropy, formula_231, across all discrete probability distributions formula_232 on  formula_233. We require that: which gives a system of equations, formula_235, such that: Carrying out the differentiation of these equations, we get This shows that all formula_238 are equal (because they depend on only). By using the constraint we find Hence, the uniform distribution is the distribution with the greatest entropy, among distributions on points. Example 4: Numerical optimization. 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 that minimizes formula_241, constrained such that formula_242. (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 and . 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 formula_218 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: The critical points of occur at and , just as in formula_4. Unlike the critical points in formula_4, however, the critical points in occur at local minima, so numerical optimization techniques can be used to find them. Applications. Control theory. 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 programming. 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. Power systems. Methods based on Lagrange multipliers have applications in power systems, e.g. in distributed-energy-resources (DER) placement and load shedding. External links. Exposition For additional text and interactive applets