Determine whether each of the following LP relaxation optima x  is optimal in the corresponding ILP over the specified variable type constraints.

(a) xj = 0 or 1, j = 1,c, 4 x 
= a1, 0, 13 , 23 b

(b) x1, x2 = 0 or 1, x3, x4 Ú 0 x 
= a0, 1, 32 , 12 b

(c) x1, x2, x3 = 0 or 1, x4 Ú 0 x 
= a1, 0, 1, 23 7 b

(d) xj Ú 0 and integer, j = 1,c, 4 x 
= a0, 3, 32 , 1b