Math 164,Homework #5, due on Friday, February 10, 2006
Remarks:
•REMINDER: midterm on Friday, February 10, 12-12.50pm (MS 5137). This will be a
closed note and closed book written examination.
•Sections covered for the midterm: 1.2-1.5, 2.2, 2.3 (except 2.3.1), 3.1, 4.1, 4.2, 4.3, 4.4,
5.2 (except 5.2.1 and 5.2.2).
•Office hours with the instructor Vese before the midterm: Monday Feb. 6, time 2-4pm,
and Wednesday Feb. 8, time 2-4pm (MS 7620-D).
•There are no office hours with the instructor Vese on Thursday Feb. 9 or on Friday
Feb. 10. For questions on Thursday or Friday, please contact the teaching assistant Salazar
(MS 2951).
•Sample practice problems with solutions for the midterm posted on the class webpage.
[1] Convert the following linear program to standard form:
minimize z=x1−5x2−7x3, subject to
5x1−2x2+ 6x3≥5
3x1+ 4x2−9x3= 3
7x1+ 3x2+ 5x3≤9
x1≥ −2, x2, x3free.
[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] Consider the linear program: minimize z=−5x1−7x2, subject to
−3x1+ 2x2≤30
−2x1+x2≤12
x1, x2≥0.
(a) Draw a graph of the feasible region and determine two linearly-independent directions
of unboundedness.
(b) Represent the point x= (6,12)Tas a convex combination of extreme points, plus if
applicable, a direction of unboundedness.
(c) Convert the linear program to standard form and determine two linearly-independent
directions of unboundedness for this version of the problem. Verify that the directions of
unboundedness satisfy Ad =~
0 and d≥~
0.
[5] Solve the following linear program using the simplex method (graph the feasible region,
and outline the progress of the solution).
maximize z= 7x1+ 8x2, subject to
4x1+x2≤100
x1+x2≤80
x1≤40
x1, x2≥0.
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