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MA/IE/OR 505-001: Linear Programming
Homework 1
Instructor: Dr. Kartik Sivaramakrishnan
INSTRUCTIONS
Due in class on February 8, 2007. No late homeworks will be accepted without prior instruc- tor approval. All problems are from Chv´atal unless otherwise specified.
- Problem 2.2, Page 26.
- Find conditions on s and t to make the LP problem
min x 1 + x 2 s.t. sx 1 + tx 2 ≥ 1 x 1 ≥ 0 x 2 is free
(a) infeasible, (b) feasible, (c) have a unique optimal solution, (d) have multiple optimal solutions, (e) unbounded.
- Problem 3.9, Part (a), Page 44. Sketch the feasible region of linear program and circle the solutions obtained in the two phases of the simplex method here. Briefly summarize your findings.
- Consider the following problem
max 2 x 1 + 3x 2 s.t. x 1 + 2x 2 ≤ 10 −x 1 + 2x 2 ≤ 6 x 1 + x 2 ≤ 6 x 1 , x 2 ≥ 0.
(a) Draw the feasible region of the linear program in (x 1 , x 2 ) space and label the constraints. (b) Notice that including non-negativity, we have five constraints. What is the solu- tion corresponding to each extreme point of the feasible region?
(c) You will notice that one of the extreme points in the feasible region is the intersec- tion of three constraints, and any two of them will uniquely specify that extreme point. Show that there are three ways to do this. Such an extreme point is said to be degenerate. (d) Solve the problem graphically and verify that the optimal point is a degenerate extreme point. (e) Solve the problem by the simplex method. How many iterations do you need? Can you explain the discrepancy? (f) From Part (c), identify the constraint that causes degeneracy and resolve the problem after throwing this constraint away. Note that degeneracy disappears and the same optimal solution is obtained. How many iterations do you need? (g) Is it true in general that degenerate extreme points can be made nondegenerate by throwing some constraints away without affecting the feasible region?
- One of the most important problems in the field of statistics is the linear regression problem. Roughly speaking, this problem involves fitting a straight line to statistical data represented by points (x 1 , y 1 ), (x 2 , y 2 ),... , (xn, yn) on a graph. If we denote the line by y = a + bx, the objective is to choose the constants a, and b to provide the best fit according to some criterion. The criterion usually used is the method of least squares, but there are other interesting criteria where linear programming can be used to solve for the optimal value of a, and b (see Chapter 14 in Chv´atal). Formulate linear programming models for this problem under the following criteria:
(a) Minimize the sum of the absolute deviations of the data from the line; that is
min
∑^ n
i=
|yi − (a + bxi)|,
(b) Minimize the maximum absolute deviation; that is
min max i=1,...,n |yi − (a + bxi)|.
- A set S ⊂ IRn^ is convex if for any x, y ∈ S, and any λ ∈ [0, 1], we have λx+(1−λ)y ∈ S (see pages 262-263 in Chv´atal). In other words, a set is convex if the line segment joining any two of its elements is also contained in the set. Consider the LP in standard form max cT^ x s.t. Ax ≤ b x ≥ 0 Show the following:
(a) The feasible set of this LP is a convex set. (b) The optimal solution set of this LP is a convex set. Now, can you construct an LP that has exactly two optimal solutions? Why?
