I am new to this. I am self learning to get ahead of my next years course and came across this question. I thought it would be a good question to look at due to it touching an many different aspects of optimization.
subject to $(x−1)^2 +(y−1)^2\leq 1$,
subject to $(x−1)^2 +(y+1)^2\leq 1$, where $(x, y)\in\Bbb R^2$.
(i) What are the set of feasible points for this problem? Using this, find the optimal point.
(ii) Write down the KKT conditions; are these conditions satisfied at the optimal point? Describe why.
(iii) Write the Lagrange dual problem, and find the optimal solution to the dual problem. Is this optimum solution attained? Does the strong duality theorem for convex programming apply to this scenario?
I have researched the basics of this course, and am trying to challenge myself with this question. Any help would be greatly appreciated.