Please help with 2 and 3 of this linear programming exercises!

( LP) from ex I min . 2 st Ax= b T 2 2Cix IX20Exercise 1. Rather than minimizing a single linear function, suppose we want to minimize a maximum of linear functions: min max {cfx, cox, ..., ckx} (P) s.t. Ax = b x 20 Let us introduce a new variable z, and consider the following linear program: min z (LP) s.t. Conditions 2 0, z unrestricted Find a collection of linear equality and inequality conditions on z and a such that the problem (P) is equivalent to the problem (LP), and prove the equivalence. Exercise 2. Find the dual program for the (LP) you found in Exercise 1. Show that a necessary condition for the dual program to be feasible is the existence of a point y E RM such that Aly a convex combination of c1, . . ., Ck Exercise 3. The 41 norm is defined for vectors in R" via i= 1 By writing x 1 as a maximum of linear functions in x, use Exercise 1 to find an (LP) which is equivalent to the following problem: min |all1 s.t. Ax = b x 20