Home Work Set 2 MSIS 685, Linear Programming Fall1998 Due : November 5^{th} 

Question 1 
Solve the following problem by the dual simplex method:

Question 2 
Let in the pair of primal and dual problems: the primal optimal be x* the dual optimal y*. Prove that if x* is nondegenerate then y* is unique. Formulate and prove the similar statement for the dual nondegenerate y*. 
Question 3 
Prove the arbitrage theorem as stated in class using Farkas lemma: Suppose there are n outcomes possible and there are m wagers allowed. Let R be an m´ n matrix whose ij entry is the payoff (or loss) to be paid if one bets on wager i and outcome j occurs. Then
Note: By definition a probability vector p=(p_{1}, p_{2},…, p_{n })^{T} satisfies p_{i³ }0 and p_{1}+ p_{2}+…+ p_{n}=1.

Question 4 
The Farkas lemma is an example of a class of theorems called "theorems of alternatives". Here are some other variants. Prove them either by using the separating hyperplane theorem or by reducing to the Farkas lemma proved in class.



Question 5 
Let f(x) be a function Â^{n}® Â^{ }. Then f is convex if for all real numbers 0£ a ,b £ 1, a +b =1 f(a x_{1}+b x_{2}) £ a f(x_{1})+b f(x_{2}) Prove that the function f(x)=Max{a_{1}^{T}x+b_{1}, a_{2}^{T}x+b_{2},…, a_{m}^{T}x+b_{m}}, where x, a_{i Î Â }^{n}, is convex.

Question 6 
Let f(x) be the function defined in question 5. Formulate the optimization problem Minimize f(x) As a linear program.

Question 7 
Let the notation: "Max_{k} A" represent "sum of the largest k elements of set A" (thus Max_{2} {2,5,3,7}=12.)
f_{k}(x)=Max_{k}{a_{1}^{T}x+b_{1}, a_{2}^{T}x+b_{2},…, a_{m}^{T}x+b_{m}} is convex. Minimize f_{k}(x) As a linear program. Hint: This problem is challenging. You will need to use duality to formulate the minimization problem.

Question 8 
For the linear programming problem Suppose x is a feasible but not basic solution. Describe a procedure by which you either find another feasible solution x¢ which is both basic and has a better objective function value, i.e. c^{T}x¢ ³ c^{T}x, or conclude that the problem is unbounded.
