Optimization problems are related to minimizing a function $($usually termed loss, cost or error function$)$ or

maximizing a function $($such as the likelihood$)$ with respect to some variable x$.$ The Karush$-$Kuhn$-$Tucker

$($KKT$)$ conditions are first$-$order conditions for a solution in nonlinear programming to be optimal, provided

that some regularity conditions are satisfied. In this question, you will be solving the following optimization

problem