Home Work Set 2   6

Home Work Set 2
MSIS 685, Linear Programming
Question 1
Due : November 5th
Solve the following problem by the dual simplex method:
Question 2
- 7 x1  7 x2  2 x3  x4  6 x5
3x1  x2  x3  2 x4  3
2 x1  x2  x4  x5  4
 x1  3 x2  3 x4  x6  12
xi  0, i  1,...,6
Let in the pair of primal and dual problems:
cT x
Ax  b
Max b T y
AT y  c
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
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 pi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 x0 such that Axb, or there exist y0, ATy0, and bTy0.
 Either there exist x such that Axb, or there exist y0 such that ATy=0 and bTy0.
Question 5
Let f(x) be a function n. Then f is convex if for all real numbers 0,1,
f(x1+x2)  f(x1)+f(x2)
Prove that the function
f(x)=Max{a1Tx+b1, a2Tx+b2,…, amTx+bm},
where x, ai 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.)
a) Prove that the function
fk(x)=Maxk{a1Tx+b1, a2Tx+b2,…, amTx+bm}
is convex.
b) 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
Max c T x
Ax  b
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  cTx, or conclude that the problem is unbounded.