Answer to Question #98012 in Operations Research for Dorothy

Question #98012
Use KKT conditions to find the optimal solution to this problem: maximize x1 - x2, such that x1^2 + x2^2 <= 1
1
Expert's answer
2019-11-11T10:35:32-0500

"x_1-x_2" -> max

"x_1^2+x_2^2<=1"

Lagrange:

"F(x_1,x_2, \\lambda)=x_1-x2-\\lambda*(x_1^2+x_2^2-1^2)"

we obtain the stationary conditions

"1-2\\lambda x_1=0\\\\1+2\\lambda x_2=0"

subtracting the first from the second we have:

"2\\lambda (x1+x2)=0=>x_1=-x_2"

substituting in the original equation we get

"x_1=0.707106781186548\\\\x_2=-0.707106781186548"

answer: "1.414213562373095"


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