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CO 255 - Fall ’24

Homework assignment #2: (Due on Friday, Oct 11th, 11:59pm)

Question 1 (20 points)

Take the dual of the following LPs:

(a) (5 points)

(b) (5 points)

(c) (10 points) Let S1, . . . , Sk be subsets of {1, . . . , n}. Let K = {1, . . . , k}.

Note: For this part, write the dual in summation form. Do NOT define a big matrix A and take the dual.

Question 2 (20 points)

(a) (5 points) Let (P) be the following linear program.

Write down the dual (D) of (P) using y as the vector of dual variables. (No justifications required.)

(b) (2 points) Use the Weak Duality Theorem to prove that ¯x = (3, 0, −1, 0, 4)T and ¯y = (0, 1, 0, 5)T are optimal for (P) and (D) respectively.

(c) (5 points) Write down all the complementary slackness conditions for (P) and (D).

(d) (8 points) Use the Complementary Slackness Theorem to determine all possible optimal solutions to (P) and (D).

Question 3 (20 points)

Consider the following pair of linear programming problems:

We say a set S is bounded if there exists a real number γ such that ||x|| ≤ γ for all x ∈ S. We say a set S is unbounded if it is NOT bounded.

Suppose that at least one of (P) and (D) is feasible. Prove that the feasible region of at least one of (P) and (D) is unbounded.

Hint: Consider changing the objective function / right-hand sides of either (P) or (D) using the all ones vector e or the zero vector

Question 4 (20 points)

Let d1 , . . . , dq be vectors in R n and let ¯d ∈ R n. Prove, using linear programming, that either ¯d can be written as a convex combination of at most n+ 1 of the vectors d 1 , . . . , dq or there exists α ∈ R n, αo ∈ R such that α T d k ≥ αo, ∀k = 1, . . . , q and α T ¯d < αo.

Question 5 (20 points)

Consider the following pair of LPs:

and assume that both (P) and (D) are feasible.

(a) (8 points) Suppose that every optimal solution to (P) satisfies xj = 0 for some j. Show that there exists an optimal solution to the dual such that ATjy > cj (Aj is the j-th column of A).

(b) (5 points) Use a similar argument to argue that if every optimal solution to the dual satisfies yi = 0 for some i, then there exists an optimal solution to the primal such that a Tix < bi .

(c) (7 points) Show that there exist optimal solutions to the primal and the dual satisfying strict complementary slackness, that is,

i. For every j, either xj > 0 or ATjy > cj .

ii. For every i, either yi > 0 or a Tix < bi .