Math 164, Lecture 2, Vese
Homework #5, due on Wednesday, November 2nd, 2005
Remarks:
•Please review Sections 4.4 and 5.2 from the textbook.
•Reminder: midterm exam on Wednesday, November 9, 1-2pm (MS 6229). This will be a
closed note and closed book written examination.
•Sections covered for the midterm: 1.2-1.5, 2.2-2.3, 3.1, 4.1-4.4, 5.2, and 6.1.
•Office hours with the instructor before the midterm: Monday Nov. 7, time 2-4pm, in MS
7620-D. (please note: on Tuesday Nov. 8, the office hours will be only with the T.A. Tristan Roy).
[1] Consider a linear program with the constraints in standard form
Ax =band x≥~
0.
(a) Recall the definition of a direction of unboundedness.
(b) Prove that if d6=~
0 satisfies Ad =~
0 and d≥~
0,then dis a direction of unboundedness for
these constraints.
(c) Prove that, if dis a direction of unboundedness for these constraints, then −dcannot be a
direction of unboundedness.
(d) 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.
[2] Suppose that a linear program in standard form, with bounded feasible region, has loptimal
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}.
[3] 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.
[4] 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.
[5] 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) Convert the linear program to standard form and determine two linearly-independent direc-
tions of unboundedness for this version of the problem. Verify that the directions of unboundedness
satisfy Ad =~
0 and d≥~
0.
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