Question

Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.

Answer 1

Let g(x, y, z) = x + y + z and h(x, y, z) = x^2 + y^2 + z^2, so the problem is to find the maximum of f(x, y, z) subtect to the constraints g(x, y, z) = 1 and h(x, y, z) = 1 we have nabla f = lambda nabla g + mu nabla h < = > (e^x, 1, 1) = lambda(1, 1, 1) + mu (alpha x, alpha y, alpha z) which given e^x = lambda + 2x mu - (1) 1 = lambda + 2y mu - (2) 1 = lambda + 2z mu - (3) x + y + z = 1 - (4) x^2 + y^2 + z^2 = 1 - (5) from (2) and (3) we have 2 mu (y - z) = 0, so either mu = 0 or y = z, so if mu = 0, then by (2) lambda = 1 and by (1) x = 0, so, by (4) & (5) , we have y + 3 = 1 & y^2 + 3^2 = 1 put z = 1 - y in y^2 + z^2 = 1 = > y^2 + (1 - y) ^2 = 1 = > y^2 + 1 + y^2 - 2y = 1 = > 2 y^2 - alpha y = 0 = > 2y (y - 1) = 0 so either y = 0 or y = 1 if y = 0 then z = 1 if y = 1 then z = 0 so, we have (x, y, z) = (0, 0, 1) (0, 1, 0) and if y = 3 then by (4) to (5) we have x + 2y = 12 x^2 + 2 y^2 = 1, put x = 1 - 2 y in x^2 + 2 y^2 = 1 then, (1 - 2 y) ^2 + 2 Y^2 = 1 = > 1 + 4y^2 - 44y + 2 y^2 = 1 = > 6y^2 - 4y = 0 = > 2y(3y - 2) = 0 = > y = 0 or y = 2/3 if y = 0, then x = 1 and then z = 0 if y = 2/3 then x = 1 - 2 times 2/3 = 1 - 4/3 = - 1/3 and then - 2/3 + 2/3 + 3 = 1 = > z = 1 - 1/3 = 2/3 so, we have (x, y, z) = (1, 0, 0) , (- 1/3, 2/3, 2/3)