Home Work Set 2 MSIS 685, Linear Programming Fall1998 Due : November 5th 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 non-degenerate 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 either there is a probability vector p=(p1, p2,…, pn )T such that RTp=0, or there exist arbitrage (sure bet) wager x=(x1,x2,…,xm)T such that Rx> 0. Note: By definition a probability vector p=(p1, p2,…, pn )T satisfies pi³ 0 and p1+ p2+…+ pn=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. Either there exist x³ 0 such that Ax³ b, or there exist y³ 0, ATy³ 0, and bTy< 0. Either there exist x such that Ax³ b, or there exist y³ 0 such that ATy=0 and bTy< 0.

 Question 5 Let f(x) be a function Ân® Â . Then f is convex if for all real numbers 0£ a ,b £ 1, a +b =1 f(a x1+b x2) £ a f(x1)+b f(x2) Prove that the function f(x)=Max{a1Tx+b1, a2Tx+b2,…, amTx+bm}, where x, ai 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: "Maxk A" represent "sum of the largest k elements of set A" (thus Max2 {2,5,3,7}=12.) Prove that the function fk(x)=Maxk{a1Tx+b1, a2Tx+b2,…, amTx+bm} is convex. Formulate the optimization problem Minimize fk(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. cTx¢ ³ cTx, or conclude that the problem is unbounded.