Math 164: Homework #5, due on Wednesday, May 6
[1] Consider the linear program in problem [5] from previous homework.
(a) Represent the point x= (6,12)Tas a convex combination of extreme points, plus if
applicable, a direction of unboundedness.
(b) Show by the method of your choice that this problem has no finite optimal solution.
[2] Consider a linear program with the constraints in standard form
Ax =band x≥~
0.
(a) Prove that, if dis a direction of unboundedness for these constraints, then −dcannot
be a direction of unboundedness.
(b) Let {d1, ..., dk}be directions of unboundedness for these constraints. Prove that a
nonzero vector d=Pk
i=1 αidi, with αi≥0 is also a direction of unboundedness.
[3] Suppose that a linear program in standard form, with bounded feasible region, has l
optimal extreme points {v1, v2, ..., vl}. Prove that a point is optimal for the linear program
if, and only if, it can be expressed as a convex combination of {v1, v2, ..., vl}.
[4] Solve the following linear program using the simplex method (graph the feasible region,
and outline the progress of the algorithm).
maximize z= 7x1+ 8x2, subject to
4x1+x2≤100
x1+x2≤80
x1≤40
x1, x2≥0.
[5] Consider the linear program: Minimize z=x1−x2subject to
−x1+x2≤1
x1−2x2≤2
x1, x2≥0.
Derive an expression for the set of optimal solutions to this problem, and show that this
set is unbounded.
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