Home Work Set 1

MSIS 685, Linear Programming

Fall 1998

Due : October 15

Question 1

Solve the following problem by the two-phase simplex method (that is find an initial feasible dictionary and then using that solve to optimality. Show all intermediate dictionaries.

 

Question 2

Consider the following dictionary:

  1. Use the lexicographic method to find the optimal solution
  2. Use Blands' rule to find the optimal solution.

Question 3

Let C be a convex set which is both bounded and closed (that is it contains all of the points on its boundary. Prove that the set of all optimal points for the problem:

is a convex set.

Question 5

Give an example of a linear program which is infeasible and its dual is also infeasible.

Question 6

For the problem in Question 1 write the dual problem and extract the optimal dual solution form the optimal dictionary.

Question 7

Find the dual of the following problem

Then write down the complementary slackness relations between the primal and dual problems.

Question 8

Show that finding any feasible solution for a linear program in canonical form is equivalent to finding the optimal solution. To do so Prove that any optimal solution of

is feasible for the system of inequalities

and vice-versa.