Consider the following problem: Minimize f(x_1, x_2) = x_1^2 + x_2^2 subject to:
ID: 3142511 • Letter: C
Question
Consider the following problem: Minimize f(x_1, x_2) = x_1^2 + x_2^2 subject to: h(X): x_1^2 = x_2, g(X): X_1 + 2x_2 greaterthanorequalto 2 a) First solve it using graphical method. b) Next, suppose we wish to solve it using a penalty function approach. First define an augmented cost function, P(X, R), by introducing suitable penalty functions for violation of the constraints. Then minimize P(X, R) as an unconstrained optimization problem. Show how the solution varies as a function of R. Please explain how the solutuion depends on R and how R should be chosen to reach the true constrained minimum.Explanation / Answer
We are to minimise x12 + x2 2
such that x12 = x2 which means we are to minimise x2 + x22
Another constraint given is that x1+2x2 >= 2 or x22 + 2x2>= 2
For minimum value of x2 we are to consider that x22 + 2x2 = 2
let x2 = y
We have y2 + 2y -2=0;
y = -2 +- sqrt(4-(-8)) / 2 = -2 + sqrt(12) / 2 = -1 + sqrt 3 or -1 - sqrt(3)
If x2 = -1 + sqrt 3 then
function x12 + x22 = -1 + sqrt (3) + 1+3 - 2sqrt (3) = 3- sqrt(3)
if x2 = -1 - sqrt(3) then
function x12 + x22 = -1 - sqrt (3) + 1 + 3 +2sqrt(3) = 3+sqrt(3)
Of these two values the minimum is 3 - sqrt(3) that occurs at x2 = -1 + sqrt (3) and x1 = sqrt ( -1+sqrt(3))