Math 164, Lecture 2, Vese
Homework #4, due on Wednesday, October 26, 2005
Remarks:
•Each time, please solve as many problems as you can from the textbook
•Please review Sections 4.2 and 4.3 from the textbook.
[1] Consider the problem
minimize f(x)
subject to x1+ 2x2+ 3x3= 6, x1≥0, x2≥0, x3≥0,
and the point xc= (1,1,1)T.
(a) Show that xcis a feasible point.
(b) Verify that p= (3,0,−1)Tis a feasible direction for xc(see also your
previous homework).
(c) Find an upper bound on the step length αso that xc+αp is a feasible
point, with p= (3,0,−1)T.
[2] Consider the following linear program (compare with Problem [3] from
Hw#3):
minimize
z= 3x1+x2,
subject to
−x1+x2≥ −1,
3x1+ 2x2≤12,
2x1+ 3x2≤3,
x1, x2≥0.
(a) Solve the problem graphically.
(b) Write the problem in standard form.
[3] Consider the system of linear constraints
2x1+x2≤100,
x1+x2≤80,
x1≤40,
x1, x2≥0.
(a) Write this system in standard form, and determine all the basic solu-
tions (feasible and infeasible).
(b) Determine the extreme points of the feasible region (corresponding to
both the standard form, as well as the original version).
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