It is a strategy for finding the local maxima and minima of a function subject to equality constraints.or this to be a probability distribution the sum of the probabilities at each point must equal 1. The method of Lagrange multipliers is a method for finding extrema of a function of several variables restricted to a given subset.
and. $\lambda g_{y}(x,y)=0$, where $\lambda $ is the Lagrange multiplier. Thus. $f_{x}(x,y)-\lambda g_{x}(x,y)=0$, $f_{y}(x,y)-\lambda g_{y}(x,y)=0$. for some $(x,y,\lambda)=(x^{\ast },y^{\ast },\lambda ^{\ast })$. Now we set $F(x,y,\lambda)=f(x,y)-\lambda g(x,y)$. The lowest value of the objective function, subject to the given constraint, sits at $(x,y,f(x,y))$, and the Lagrange multiplier does not have an immediate physical interpretation. (The multipliers have meaning when appearing in certain contexts, more info on that here .). 1 where λ is called the Lagrangian multiplier. 2  In effect, this method imposes a penalty upon any proposed solution that is proportional to the extent to which the constraint is violated. 3  By choosing the constant of proportionality large enough the solution can be forced into compliance with the constraint. An Example of the Use ofthe Lagrangian Multiplier Method to Solve a Constrained Maximization Problem. Let Q=output, L=labor input and K=capital input where Q = L2/3K1/3. The cost of resources used is C=wL+rK, where w is the wage rate and r is the rental rate for capital. The method of Lagrange multipliers relies on the intuition that f(x, y) cannot be increasing at a maximum in the direction of any neighboring point where g = 0. If it were, we could walk along g = 0 to get higher, meaning that the starting point wasn't actually the maximum. 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. But the constraint function is always equal to c, so dg 0 /dc = 1. Thus, df 0 /dc = λ 0. That is, the Lagrange multiplier is the rate of change of the optimal value with respect to changes in the constraint. This is a powerful result, but be careful when using it! In particular, you have to make sure that your constraint function is written in just the right way. As a final note, I'll say a few words about what the Lagrange multiplier means, in ways inspired by both physics and economics. In our mostly geometrical discussion so far, λ was just an artificial variable that lets us compare the directions of the gradients without worrying about their magnitudes. Lagrange Multipliers. The method of Lagrange multipliers is a method for finding extrema of a function of several variables restricted to a given subset. Let us begin with an example. In this section we apply the method of Lagrange multipliers instead. The key to understanding this technique is the following figure, which plots the constraint function and the level curves of f(x,y). 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 partial derivatives of are zero). 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. In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints. 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. ECONOMIC APPLICATIONS OF LAGRANGE MULTIPLIERSMaximization of a function with a constraint is common in economic situations. The ﬁrst section consid-ers the problem in consumer theory of maximization of the utility function with a ﬁxed amount of wealth tospend on the commodities. The equations givenby the Lagrange multiplier method are∂U∂x i = λp i for 1 ≤ i ≤ nw 0 = p 1 x 1 +···+p n x n .A solution satisﬁesλ =1p i ∂U∂x i for all i.Thus, the ratio of the marginal utility to price is the same for each commodity.
http://math.stackexchange.com/questions/674/how-do-lagrange-multipliers-work-to-find-the-lowest-value-of-a-function-subject http://www.sjsu.edu/faculty/watkins/lagrange.htm https://en.wikipedia.org/wiki/Lagrange_multiplier http://www.slimy.com/%7Esteuard/teaching/tutorials/Lagrange.html http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lagrang/lagrang.html https://en.wikipedia.org/wiki/Lagrange_multiplier https://en.wikipedia.org/wiki/Lagrange_multiplier http://www.math.northwestern.edu/~clark/285/2006-07/handouts/lagrange-econ.pdf
what is lagrangian multiplier