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There are two files in which one is written homework and the other one requires Matlab. Please do the questions the way before. Thank you!Math 171A: Linear Programming
Instructor: Philip E. Gill © 2021 (Not to be Reposted)
Winter Quarter 2021
Matlab Assignment #4
Due Wednesday February 24, 2021
Starred exercises require the use of Matlab.
2
Mathematics 171A
Exercise 4.1.∗ A completed m-file maxstep.m has been placed on the class web-site for
sure that you understand how it works—later you will be required to use it in an m-file
of your own that solves a linear program.
(b) The diet problem of Assignment 1 has inequality constraints Ax ≥ b, where
28
 15

 6

 30

 20

510

 34

 1
A=
 0

 0

 0

 0

 0

 0

 0
0

24
15
10
20
20
370
35
0
1
0
0
0
0
0
0
0
25
6
2
25
20
500
42
0
0
1
0
0
0
0
0
0
14
2
0
15
10
370
38
0
0
0
1
0
0
0
0
0
31
8
15
15
8
400
42
0
0
0
0
1
0
0
0
0
3
0
15
0
2
220
26
0
0
0
0
0
1
0
0
0
15
4
0
20
15
345
27
0
0
0
0
0
0
1
0
0
9
10
4
30
0
110
12
0
0
0
0
0
0
0
1
0

55
1
 100 
2 

 100 
120

 100 
2 

 100 
2 

2000
80 

 350 
20 

0 
 , and b =  0  .

0 
 0 
 0 
0 

 0 
0 

 0 
0 

 0 
0 

 0 
0 

 0 
0 
1
0
Load the diet problem and compute the point x̄ at which the constraints corresponding
to rows 2, 3, 5, 7, 9, 11, 12, 14, and 16 of A above are active. Use the m-file maxstep.m
to find the maximum feasible step from x̄ along the direction
 
−1
 1
 
−1
 
 1
 

p=
 1 .
−1
 
 1
 
 1
1
Matlab #4
3
Exercise 4.2.∗
(a) Consider the matrix of active constraints

0
0
5
2
7
−4
2
0
4 −2
.
Aa = 
−6
0 −3
3
5
−1 −8
4 −7 −3
Use mysolve.m to construct a direction p along which the residual of the third constraint increases, but all the other residuals remain the same. If the rows of Aa are
labeled 1 through 4, what are the indices of the active set after a positive step along
this direction?
(b) Repeat part (a) for the vertex

0
 9

 0

Aa = 
−3
 3

−8
−1

3
6 −5 −4
1
6 −4
5

1
6 −5 −4

−4 −5
3
2
.
5 −6
3
2

−3
0
7 −1
0
0
0
0
Comment on your results. Is the vertex degenerate or nondegenerate?
4
Mathematics 171A
Exercise 4.3.∗ Consider the linear program
minimize
x1 ,x2 ,x3
subject to
3×1 − x2 + 2×3
− 2×1 + 4×2 + 4×3
x1 + 4×2 + x3
− 2×1 + x2 + 2×3
2×1 − 2×2
− 3×2 + x3
x1
x2
x3
≥ 6
≥ 5
≥ 1
≥ 0
≥ −2
≥ 0
≥ 0
≥ 0,
and the point x̄ = (1, 1, 1)T . Find the active set at x̄ and determine if the point x̄ is
optimal. If x̄ is not optimal, find a direction p such that

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