Consider the following NLP problem :
Write the Khun-Tucker conditions for the problem
Verify if (x,y,z)=(-3,2,5) is a solution for Khun Tucker conditions
Show that optimum x*=(1,5/2,0) is a KKT point?
Solution
first write a function :
L = 2*x^2 + y^2 + z^2 +x*y +2*x*z + x+ y +10 + (Lambda1)(21 - x- 2*y - 4*z) + (lambda2)(4 - x-3*y -2*z)
differentiate L w.r.t. x , y, z and equate them to zero ;
lambda1 and lambda 2 > 0 and   (21 - x- 2*y - 4*z) , (4 - x-3*y -2*z) are <=0 and (Lambda1)(21 - x- 2*y - 4*z) = 0 and (lambda2)(4 - x-3*y -2*z) = 0 ;
by differentiating we get , 4*x + y +2*z +1 +(lambda1)(-1) + (lambda2)(-1) = 0 ;
2*y + x + 1 +(lambda1)(-2)+(lambda2)(-3) = 0 ;
2*z + 2*x + (lambda1)(-4) + (lambda2)(-2) = 0 ;
above conditions are kuhn tucker conditions
substitite x = -3, y=2, z=5 above;
get lambda1 and lambda2 values and cross check their sign
lambda1 = 1 and lambda2 = 0 ;
but lambda1 and lambda2 >0 ; and  (Lambda1)(21 - x- 2*y - 4*z) = 0 and (lambda2)(4 - x-3*y -2*z) = 0 does hold true , hence K-T conditions satisfied hence K-T conditions are satisfied
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Framing an Appropriate Research Question 6b9b26d93da94caf993c038d9efcdedb.pdf
Consider the following NLP problem - Write the Khun-Tucker conditions.docx
1. Consider the following NLP problem :
Write the Khun-Tucker conditions for the problem
Verify if (x,y,z)=(-3,2,5) is a solution for Khun Tucker conditions
Show that optimum x*=(1,5/2,0) is a KKT point?
Solution
first write a function :
L = 2*x^2 + y^2 + z^2 +x*y +2*x*z + x+ y +10 + (Lambda1)(21 - x- 2*y - 4*z) + (lambda2)(4 -
x-3*y -2*z)
differentiate L w.r.t. x , y, z and equate them to zero ;
lambda1 and lambda 2 > 0 and   (21 - x- 2*y - 4*z) , (4 - x-3*y -2*z) are <=0 and
(Lambda1)(21 - x- 2*y - 4*z) = 0 and (lambda2)(4 - x-3*y -2*z) = 0 ;
by differentiating we get , 4*x + y +2*z +1 +(lambda1)(-1) + (lambda2)(-1) = 0 ;
2*y + x + 1 +(lambda1)(-2)+(lambda2)(-3) = 0 ;
2*z + 2*x + (lambda1)(-4) + (lambda2)(-2) = 0 ;
above conditions are kuhn tucker conditions
substitite x = -3, y=2, z=5 above;
get lambda1 and lambda2 values and cross check their sign
lambda1 = 1 and lambda2 = 0 ;
but lambda1 and lambda2 >0 ; and  (Lambda1)(21 - x- 2*y - 4*z) = 0 and (lambda2)(4 - x-
3*y -2*z) = 0 does hold true , hence K-T conditions satisfied hence K-T conditions are satisfied
