4. Consider the standard problem ol sl { ;'_1 T by, E=1,...,m z;z0,j=1,...,n Assume that this problem has an optimal solution. A new problem is obtained by adding the constraint Z;'_l Bt 3% % Vg (a) What is the relationship between the feasible set of the new problem and the feasible set of the original problem? (b) Show that the optimal value in the new problem is less or equal than the optimal value in the original problem. (c) Assume that there is an optimal solution to the new problem that does not satisfy the new constraint at equality i.e. there is z** an optimal solution to the new problem, with o E mll._grj {:bmll F=1 Show that the optimal value in the new problem equals the optimal value in the original problem. Hint: (1) Show, using duality theory, that the optimal value in the new problem is greater or equal than the optimal value in the original problem. (2) Combine (1) and (b). (d) Assume that the original problem has a unique optimal solution z* and that =* does not satisfy the new constraint. Show that any optimal solution to the new problem must satisfy the new constraint at equality