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CAAM 471/571 - Fall 2014
INTRODUCTION TO LINEAR AND INTEGER PROGRAMMING
Homework 3, February 4, 2014
Due: February 11, 2014
Note: All M ATLAB or AMPL functions/files mentioned in this homework assignment can be
found on the CAAM471/571 homepage, or come with M ATLAB . You can use the M ATLAB
or AMPL files posted on the CAAM471/571 web-page. If you modify these codes, please turn
in the modified files. Otherwise you do not have to turn in printouts of the codes posted on
the CAAM471/571 web-page. Turn in all M ATLAB / AMPL code/files that you have written
and turn in all output generated by your M ATLAB functions/scripts. M ATLAB and AMPL
functions/files/scripts must be commented, output must be formatted nicely, and plots must be
labeled. In all the assignments, BT refers to the textbook by Bertsimas and Tsitsiklis.
Problem 1 (5 points) BT Exercise 3.1.
Problem 2 (10 points) BT Exercise 3.2.
Problem 3 (5 points) BT Exercise 3.3.
Problem 4 (10 points) BT Exercise 3.7.
Problem 5 (15 points) BT Exercise 3.12.
Problem 6 (571 only. 10 points) BT Exercise 3.13.
Problem 7 (15 points)
Klee and Minty (1972) constructed an example with n variables which requires 2n iterations for the simplex algorithm to solve when using the “most-negative” pivot rule. This example is often used to illustrate
the worst-case behavior of the Simplex algorithm. In practice, the observed convergence behavior of the
Simplex algorithm is typically much better.
The problem constructed by Klee and Minty (1972) is given by
n
max
∑ 10n−i xi ,
i=1
i−1
s.t.
2 ∑ 10i− j x j + xi ≤ 100i−1
for i = 1, . . . , n,
xi ≥ 0
for i = 1, . . . , n.
j=1
(Note that the above matrix A is a Teoplitz matrix with constant diagonals.)
CAAM 471/571
Homework 3
(generated February 4, 2014)
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(a) (10 points) Solve the Klee Minty problem for n = 2, 3, . . . , 8 using the script solve an LP.m that calls
the simplex solver yzsimplex.p (provided as an executable file for now) with maxiter no less than 300.
First read and run the demo file test randomLP.m. Copy it to test KleeMinty.m, and modify it for
solving the Klee-Minty problems (keeping prule = 1).
Generate a table
n 2 3 ··· 8
iter ? ? · · · ?
and then deduce a general formula that you think should relate the two quantities.
(b) (5 points) Then solve the same problems again using prule = 2. Here we change the so-called pivot
rule that determines which edge to follow to the next vertex out of multiple edges that are descending.
Present you results and observation in one sentence.
Submission: your Matlab code printout and results for (a)-(b).
Problem 8 (40 points) We will solve the standard form LP for given data (A, b, c):
min cT x, s.t. Ax = b, x ≥ 0,
where A ∈ Rm×n with rank(A) = m < n, b ∈ Rm and c ∈ Rn .
(a) (25 points) Implement the simplex method, given in TB book pages 90-91, into a Matlab function
mysimplex.m to solve the standard LP starting from a given vertex. We only do a “naive implementation” (as described in the TB book).
For more details and specifications (and an alternative algorithm description), see
http://www.caam.rice.edu/~yzhang/caam471/assignments/hw3/simplex_specs.pdf
(http://www.caam.rice.edu/~yzhang/caam471/assignments/hw3/Simplex_alg.pdf).
A Matlab function for selecting entering variable (including 2 pivot rules) is provided online. It is
important to be aware of an index’s reference range (local in B or N, or global in 1 to n). Also keep in
mind of rounding error effects.
(b) (10 points) Run the scripts test randomLP.m and test KleeMinty.m (from Problem 7) using both
your code mysimplex.m and the handout code yzsimplex.p. Modify the former script so that it can
run mysimplex.m and yzsimplex.p side by side on the same set of data for a comparison purpose.
(c) (5 points) Run the handout script test cycle.m (that uses yzsimplex.p and cycle.mat provided together), and give your observations in a couple of sentences. Then you also test your code
mysimplex.m on the problem (in place of yzsimplex.p).
Submission: your code mysimplex.m and the output from running the two scripts.
CAAM 471/571
Homework 3
(generated February 4, 2014)