Question

a) Solve the following problem using graphical method (using the following graph): Minimize f(x,y) - 2x-y subject to the constraints x2+y's 20 y<x (1) (2) (In the space provided below the graph, please write down your solution clearly)

Answer 1

Plot Points (2) Graphical Method - Solving C Electrical Engineering question S Calculus I- Lagrange Multipli Objective Vocabulary Solve line x X X https://www.desmos.com/calculator/mhq4hsncnh Plot Points desmoS Create Account or Sign In (4,2) 15 10 10 15 (-3.732, -2.464) (-2,-4)5 (0.-4.472) desmos ENG 20:16 US 26-06-2019

It can be seen that the function

f(x,y) = 2x - y = K = constant is shown by parallel lines.

Putting the points we get

f(-4 , 2) = -10

f(-3.732 , -2.464) = -5

f(-2 , -4) = 0

Hence function returns minimum value = -10 when (x , y) = (-4 , 2)

EXTERIOR PENALTY FUNCTION is defined as follows

minimize fg(x, y, rg , rh) = f(x , y) + rg max(0 , h1 - c1) + rh max(0 , h2 - c2)

where h1 - c1 =0 and h2 - c2 =0 are the constrained equations

minimize fg(x, y, rg , rh) = 2 * x - y + rg max(0 , x² - y² - 20) + rh max(0 , y - x²)

minimize fg(x, y, rg , rh) = 2 * x - y + rg ( x² - y² - 20) ² + rh ( y - x²)² is the augmented cost function.

Now the partial derivative of the function fg taken individually with respect to x , y , rg and rh to get the final minimum value of the function f.