2. (10 points) Consider the NLP with inequality...

2. (10 points) Consider the NLP with inequality constraints: Max f(x,y) = 10x - 2x^2 - x^3 + 8y - y^2 such that x + y ? 2 x, y ? 0. Use the KKT conditions to show that (x0, y0) = (1, 1) is NOT an optimal solution.

Transcript

00:01 Hi, in this problem we are given maximize f of xy equal to 10 x minus 2x squared minus x cube plus 8y minus y square such that x plus y is less than equal to 2 and x comma y greater than equal to 0.

00:19 Now we are required to show that x0 y0 equal to 1 1 is not an optimal solution.

00:25 So further, the lagrangian function for this can be written.

00:30 Written as l equal to 10x minus 2x squared minus x squared minus x cubed plus 8y minus y square minus 5 of x plus y minus 2.

00:49 Consider this as equation number 1.

00:52 Now the kkt conditions are a, 5 greater than equal to 0.

00:57 Consider this as equation number 2, b, del l, by del x equal to 0, which implies that 10 minus 4x minus 3 x squared minus 5 is equal to 0.

01:09 Consider this as equation number 3 and del l by del y equal to 0, which implies that 8 minus 2y minus 5 equal to 0.

01:19 Consider this as equation number 4.

01:22 See, 5 of x plus y minus 2 equal to 0.

01:26 Consider this as equation number 5 and d x plus y minus 2 less than equal to 0 consider this as equation number 6.

01:35 Now let us take case 1 where 5 is not equal to 0 so therefore this implies that from equation 3 and 4 we can write 10 minus 4x minus 3x square equal to 8 minus 2 y.

01:59 Further, this implies that 3x square plus 4x minus 2y is equal to 2.

02:08 Consider this as equation number 7.

02:11 Now further, from the equation 5, let us consider the case x plus y minus 2 equal to 0...

Question

2. (10 points) Consider the NLP with inequality constraints: Max f(x,y) = 10x - 2x^2 - x^3 + 8y - y^2 such that x + y ? 2 x, y ? 0. Use the KKT conditions to show that (x0, y0) = (1, 1) is NOT an optimal solution.

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