1) Prove : If F is k-dimensional face of a polytope P in R^d, then F 
is also a polytope, and furthermore every vertex of F is also a vertex of P.
 	
Hint: In a 3 dimensional space, the convex polygon on the plane which is 
tangent to the polytope is called a face.


2) Given is the following LP. let us call it (P).
min x_1 + x_2
    x_1 + 2x_2 <= 5
	x_2 + 2x_3  = 6
    x_1,x_2,x_3   => 0

Call the problem as (P)

Give in the following steps:-
(a) Bring the LP to standard form
(b) Solve using the revised SM (i.e., the two phasaes)
(c) Suppose that the rhs is (6,6); is the solution still feasible? 
optimal?
(d) Suppose that c_1 were -1; is the solution still optimal? If not, find
     the new optimal solution
(e) Suppose that a new activity P_4 = (1,-1)' becomes available with 
c_4 = -3. Is the current solution still optimal? If not find the new
optimal solution.
(f) For what values of a_11 is the current solution still optimal?
(g) Write the dual of the given problem (P) and from your solution
     in (b), obtain its optimal solution. Verify complementary slackness
     condition.

3) Another form of Farkas' lemma and the duality theorem is as follows:
  A system of linear inequalities Ax <= b is called inconsistent if there exists a y such 
that y'A=0,y'b<0 and y => 0. Show that the system Ax<=b has no solution if and only if it 
is inconsistent.

4) Suppose a standard form LP has a unique optimal solution. Does it follow that the dual 
has a non-degenerate optimal solution? Does the converse hole true?


5) P is a linear program.
max 7x_1 + 2x_2
  -x_1 + 2x_2 <= 4
  5x_1 +  x_2 <= 20
 -2x_1 - 2x_2 <= -7

   Write  its dual.

6) Starting with vectors P1,P2,P7 as the initial basis, will the following 
problem cycle when regular simplex rules are applied?

Minimize:
  x_1 + 2x_3 - 3x_4 - 5x_5 + 6x_6 = 0
  x_2 + 6x_3 - 5x_4 - 3x_5 + 2x_6 = 0
  3x_3 + x_4  + 2x_5 + 4x_6 + x_7  = 1

 all variables >= 0