Constraint Minimization using Lagrange Multipliers
AIM: For this problem, you will minimize a simple non-linear function by using Lagrange Multipliers.
Minimize the following function, 5-(x-2)^2 -2(y-1)^2 subject to the following constraint x + 4y = 3.
SOLUTION:
What is Lagrange Multipliers?
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.
According to Lagrange theory:
For single constraint, the auxiliary function is `L(x,y,lamda)=f(x,y)-lamda.g(x,y)`
From the given question, `f(x,y)=5-(x-2)^2-2(y-1)^2` and `g(x,y)=x+4y-3`
Therefore,
`L(x,y,lamda)=5-(x-2)^2-2(y-1)^2 - lamda(x+4y-3)`
`L(x,y,lamda)=5-(x-2)^2-2(y-1)^2 - lamdax-4lamday+3lamda`
To successfully minimize the given function, we have to partially derivate L with respect to x, y and `lamda`
`(delL)/(delx) =0`
`-2x+4+lamda=0` ................equation 1
`(delL)/(dely)=0`
`-y+lamda+1=0` ...............equation 2
`(delL)/(dellamda)=0`
`x+4y=3` ...............equation 3
Solve the equation 1, 2 and 3,
equation 1 - equation 2, we get
`-2x+4+lamda+y-lamda-1=0`
`y=2x-3` ......substitute the value of y in equation 3, we get
`x+4(2x-3)-3=0`
`x+8x-12-3=0`
`9x-15=0`
`x=5/3` ......substitute the value of x in y, we get
`y=2(5/3)-3`
`y=1/3` .......substitute the value of y in equation 2, we get
`-1/3+lamda+1=0`
`lamda =-2/3`
Therefore for `x=5/3`, `y=1/3` and `lamda=-2/3` we can minimize the function `5-(x-2)^2-2(y-1)^2` subject to the constraint `x+4y=3`.