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
your use.
(a) Download the m-file maxstep.m and place it in your Matlab working folder. Make
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
cTp < 0 and Aa p ≥ 0.
Matlab #4
5
Exercise 4.4.∗ Consider the linear program
minimize
cTx subject to Ax ≥ b.
n
x∈R
Consider a particular problem with objective vector c =
feasible point x̄, the active constraint matrix is given by
−6
5
1
0 −1
7 −6 −2
3
Aa = −2
−4 −2
7
2 −4
(−10, 3, 8, 2, −5, 6, 2)T . At a
3
0
3
1
0 .
1
(a) Is x̄ a vertex? If x̄ is a vertex, is it degenerate or nondegenerate? Justify your answer.
(b) Determine whether or not x̄ is optimal.
6
Mathematics 171A
Exercise 4.5.∗ Load the diet problem of Assignment 1 and compute the point x̄ at which
the constraints corresponding to rows 2, 3, 5, 7, 9, 11, 12, 14, and 16 of A are active. Using
x̄ as initial point, solve the diet problem using the simplex method. Show your work.
Math 171A: Linear Programming
Instructor: Philip E. Gill © 2021 (Not to be Reposted)
Winter Quarter 2021
Homework Assignment #4
Due Wednesday February 24, 2021
2
Mathematics 171A
Exercise 4.1. Consider the feasible region F of points in Rn satisfying the constraints
Ax ≥ b, where A is a nonzero m × n matrix. Assume that F contains at least one point.
Show that, if a nonzero vector p exists such that Ap ≥ 0, then F is unbounded.
Homework #4
3
Exercise 4.2. Show that the feasible region for the equality constraints Ax = b is either
empty or convex.
4
Mathematics 171A
Exercise 4.3. Let Aa be a nonzero m × n matrix, and let CN denote the dual cone
CN = {w : w = ATa λ,
λ ≥ 0}.
(a) Show that CN is a convex set.
(b) Is CN a subspace of Rn ? Explain why or why not.
Homework #4
5
Exercise 4.4. Suppose that the objective vector of a linear program is c = (2, 1)T , and
that the matrix of active constraints at a feasible point x̄ is
1
1
1 −2
Aa =
1 −3 .
1
2
(a) Is x̄ a vertex? If so, is it degenerate or nondegenerate? Explain your answer.
(b) Graph the dual cone. Use your graph to either verify that x̄ is optimal or find a
direction p such that cTp < 0 and Aa p ≥ 0.
6
Mathematics 171A
Exercise 4.5. Consider the following five constraints
x1 + 2x2 ≤ 3, x1 − x2 ≥ 0, 2x1 + x2 ≤ 3, x1 + 5x2 ≤ 6, x1 − 2x2 ≥ −1.
(a) Sketch the feasible region and find the degenerate vertex x0 .
(b) How many possible working sets are there at x0 ?
(c) Suppose that we wish to minimize x1 + x2 subject to these constraints, starting at x0
and using the simplex method. Find a working set A0 for which the Lagrange multiplier
vector λ (the solution of AT0 λ = c) contains at least one negative component λs , but
the simplex search direction satisfying A0 p = es is not a feasible descent direction.
Draw a picture showing p emanating from x0 . What are the blocking constraints?
(d) Under the same conditions as in part (c), find a working set Ā0 for which the Lagrange
multiplier vector contains at least one negative component, but the associated search
direction p̄ is a feasible descent direction. Draw a picture showing p̄ emanating from
x0 .
(e) Can you find a feasible descent direction at x0 if we wish instead to minimize −x1 −x2 ?
Explain your answer.
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