Yesterday I learned the beautiful method of Lagrange Multiplier, to find the minima or maxima of multivariate functions. eg

If I ask you what’s the minimum value of a function, f(x,y) = xy, subject to the constraint 5x-y = 4, the answer would be -4/5. The method is this. Ensure that first-partials of function f(x,y) exist. That is *f _{x}* = (∂

*f*)/(∂

*x*) is not zero, and

*f*= (∂

_{y}*f*)/(∂

*y*) is not zero. Else method won’t work.

For the given example the f(x,y) = xy, because *f _{x}* =

*y*and

*f*=

_{y}*x*, our method will work.

According to the method, we have 3 equations:

(1) *f _{x}* +

*λg*= 0

_{x}(2) *f _{y}* +

*λg*= 0

_{y}(3) 5*x* − *y* = 4

We already know what are *f _{x}* and

*f*, right? We already evaluated them in this example;

_{y}*f*=

_{x}*y*and

*f*=

_{y}*x*. But what’s

*λ*? Its called Lagrange “undetermined” multiplier. It need not be determined from above 3 equations, for determining extrema. (minima and maxima) …

Note that 3 unknowns (x, y, *λ*) and 3 equations (1, 2, 3) are there. Also *g* is the constraint function, from the constraint equation: 5x – y = 4, when g(x,y) = 0. So let’s evaluate everything to find the minima.

As we have already seen, *f _{x}* = (∂

*f*)/(∂

*x*), so since f = xy,

*f*=

_{x}*y*. Similarly

*f*=

_{y}*x*. (very simple derivatives)

Now *g _{x}* = (∂

*g*)/(∂

*x*) = (∂(5

*x*−

*y*− 4))/(∂

*x*) = 5 .

Similarly *g _{y}* = (∂

*g*)/(∂

*y*) = (∂(5

*x*−

*y*− 4))/(∂

*y*) = − 1.

So 1, 2, 3 becomes:

(4) *y* + 5*λ* = 0

(5) *x* − *λ* = 0

(6) 5*x* − *y* = 4

We have 3 equations, 4, 5 and 6, and 3 unknowns, x, y, *λ*. Lets solve for x, y.

From 4 and 5,

(7) *y* + 5*x* = 0

and 5*x* − *y* = 4

So, 2*y* = − 4 or *y* = − 2. So *x* = (4 + *y*) ⁄ 5 = 2 ⁄ 5.

So *f*(*x*, *y*)* _{minima}* =

*x**

*y*= − 2*2 ⁄ 5 = − 4 ⁄ 5.

QED.

PS: The method can easily be extended to more variables, with x, y, z we will have 4 unknowns x, y, z, *λ* and 4 equations, involving them. For more than one constraint equation, we simply add over the constraint part in the above equations, that is, there will be summation sign before *g _{x}*,

*g*; etc; ∑

_{y}*λg*etc.

_{x}Document generated by eLyXer 1.2.5 (2013-03-10) on 2015-05-29T11:22:52.887000

Categories: calculus of variation, Mathematics

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