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MA/IE/OR 505-001: Linear Programming
Homework 2
Instructor: Dr. Kartik Sivaramakrishnan
INSTRUCTIONS
Due in class on Thursday, the 22nd of February 2007. No late homeworks will be accepted without prior instructor approval. All problems are from Chv´atal unless otherwise specified. Please read Chapters 5 and 9 in Chv´atal before beginning the assignment.
- Problem 5.3 (a), page 69 (do part (a) only!)
- Problem 9.5, page 147.
- Consider the following linear program
max cT^ x s.t. Ax ≥ bl Ax ≤ bu x ≥ l x ≤ u.
where A is a m × n matrix; c, x, l, u ∈ IRn; and bl, bu ∈ IRm.
(a) Write down the dual linear program to (1). You must give the correct dimensions of all the dual variables. (b) Show that the dual linear program is always feasible by exhibiting a feasible solution. If the primal problem is also feasible, what conclusions can you draw from duality theory? (c) What are the complementary slackness conditions for (1) and its dual? (d) What is the dual linear program if one is minimizing cT^ x instead of maximizing it in (1)?
- Let A ∈ IRm×n, x ∈ IRn^ and y ∈ IRm. Derive the following theorem of the alternative
Theorem 1 The system Ax = 0, x > 0 is unsolvable if and only if the system AT^ y ≥ 0 , AT^ y 6 = 0 is solvable.
using LP duality (along the lines of the proof of Farkas’ theorem in class). Note that the strict vector inequality y > 0 implies that yi > 0 for all i. Proceed as follows:
(a) Show that if the 1st system is solvable then the 2nd system is unsolvable. (b) Show that if the 1st system is unsolvable then the 2nd system is solvable (this is the part that involves LP duality!) (Hint: Rewrite the 1st system as Ax = 0, x ≥ e (where e is the n dimensional vector of all ones) since LPs deal with inequalities rather than strict inequalities!)
- Consider an LP in standard form
max cT^ x s.t. Ax ≤ b, x ≥ 0.
(a) Form the dual problem and convert it into an equivalent maximization problem. (b) Derive a set of conditions on the matrix A and the vectors b and c under which the dual is identical to the primal (such LPs are known as self-dual linear programs). (c) Construct an example in which these conditions are satisfied. (d) Can a self dual LP (i) be unbounded? (ii) be infeasible? (iii) have an optimal solution? Give reasons or examples to justify each of these claims.
- Two players are involved in a competitive game. One player, called the row player has two strategies available; the other player, called the column player, has three strategies available. If the row player selects strategy i and the column player chooses strategy j, the payoff to the row player is cij and the payoff to the column player is −cij. Thus, the column player loses what the row player wins and vice versa; this is called a two person zero-sum game (Chapter 15 in Chv´atal has a nice description of this problem). The matrix in Table 1 gives the payoffs to the row player: Let x 1 , x 2 , and x 3 be the
2 -1 0 -3 2 1
Table 1: Payoff matrix
probabilities with which the column player will select the various strategies over various plays of the game. Thus x 1 + x 2 + x 3 = 1, x 1 , x 2 , x 3 ≥ 0. If the column player applies these probabilities to the selection of his strategy for any play of the game, consider the row player’s options: If the row player selects row 1, then his expected payoff is 2 x 1 − x 2. If he chooses row 2, his payoff is − 3 x 1 + 2x 2 + x 3. Wishing to minimize the maximum payoff to the row player, the column player solves the following problem
min x 1 ,x 2 max{ 2 x 1 − x 2 , − 3 x 1 + 2x 2 + x 3 } s.t. x 1 + x 2 + x 3 = 1 , x 1 , x 2 , x 3 ≥ 0.
(a) Write down the above problem as a linear program.
