Math 171A:
Mathematical Programming
Instructor: Philip E. Gill
Winter Quarter 2009
Homework Assignment #2
Due Friday January 23, 2009
The starred exercises require the use of
Matlab
. Remember that it is necessary to do all
the
Matlab
assignments to obtain credit for the class.
Exercise 2.1.
(a)
Write the constraints
x
1
+
x
2
≥
4,
x
1
+ 3
x
2
≥
6, 6
x
1

x
2
≤
18, 3
≤
x
2
≤
6 and
x
1
≥ 
1 in matrix form
Ax
≥
b
.
(b)
Draw the graphical representation of the feasible region defined by the constraints of
part
(a)
. Mark the
infeasible
side of the halfspace defined by each constraint.
(c)
Consider the linear function
ℓ
(
x
) =
c
T
x
where
c
T
= ( 2

3 ). On your graph from
part
(a)
, show the halfspace of
descent directions
emanating from the point ¯
x
= (2
,
4).
(d)
Use the graphical method to minimize 2
x
1

3
x
2
subject to the constraints of part
(a)
.
Give the optimal values of
x
1
,
x
2
and the minimum value of the objective function.
(e)
Compute the residual vector
r
(
x
) for the constraints at the point ¯
x
and find the con
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 Winter '08
 staff
 Math, Linear Algebra, matlab, Optimization, ax, matrix form AX

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