General optimization problems

Give an explicit solution of the following LP.

Give an explicit solution of the following LP.
where

Give an explicit solution of the following LP.
where

Give an explicit solution of the following LP.
This problem can be considered as a simplest portfolio optimization problem.

Give an explicit solution of the following LP.
where is an integer between and . What happens if is not an integer (but satisfies )? What if we change the equality to an inequality ?

Give an explicit solution of the following QP.
where . What is the solution if the problem is not convex (Hint: consider eigendecomposition of the matrix: ) and different cases of ?

Give an explicit solution of the following QP.
where .

Give an explicit solution of the following QP.
where .

Consider the equality constrained leastsquares problem
where with , and with . Give the KKT conditions, and derive expressions for the primal solution and the dual solution .

Derive the KKT conditions for the problem
where and are given with . Verify that the optimal solution is given by

Supporting hyperplane interpretation of KKT conditions. Consider a convex problem with no equality constraints
Assume, that satisfy the KKT conditions
Show that
for all feasible . In other words the KKT conditions imply the simple optimality criterion or defines a supporting hyperplane to the feasible set at .