constrain optimization KKT condition -- 5
Budget: $10 – $30 USD
Consider the following nonlinear optimisation problem.
Objective function:
f(x) =x1^3+x2^3+x3^3
Constraints: (x1 + x2 + x3)^2 = 48
x1^2+x2^2+x3^3= 24
(a) Write all the KKT equations/inequalities and use them to identify all candidate optimisers.
(b) Use the second-order sufficient conditions to determine if any candidates are strict local maximisers or minimisers.
Note: there is a high degree of symmetry in this problem; use it to your advantage. There are 12 candidates, but finding 4 and invoking symmetry will let you find all of them. Cases of variables equal to each other or not will be useful.
Objective function:
f(x) =x1^3+x2^3+x3^3
Constraints: (x1 + x2 + x3)^2 = 48
x1^2+x2^2+x3^3= 24
(a) Write all the KKT equations/inequalities and use them to identify all candidate optimisers.
(b) Use the second-order sufficient conditions to determine if any candidates are strict local maximisers or minimisers.
Note: there is a high degree of symmetry in this problem; use it to your advantage. There are 12 candidates, but finding 4 and invoking symmetry will let you find all of them. Cases of variables equal to each other or not will be useful.