4.7 (Duality in piecewise linear convex optimization) Consider the problem of minimizing maxi=1jmjm (agx bi) over all x E R\". Let U be the value of the optimal cost, assumed nite. Let A be the matrix with rows a1, . . . ,am, and let b be the vector with components b1,...,bm. (3) Consider any vector p E Rm that satises p'A 2 0'7 p 2 0 and 2:1391' : 1. Show that p'b g i). (b) In order to obtain the best possible lower bound of the form considered in part (a), we form the linear programming problem Formulate the dual: maximize subject to subject to minimize subject to subject to p'b p/AZOI p'e:1 P20 12 p/A:0/ p'e= xfree