Indicate whether each of the following statements is true or false (brie y explain your reason). (a) [1pt] Consider a standard LP with four variables and three constraints. Then two basic solutions (0; 0; 0; 4; 0; 12; 18) and (3; 0; 0; 1; 0; 2; 0) are adjacent. (b) [1pt] If a linear program has no optimal

Indicate whether each of the following statements is true or false (brie y explain your reason). (a) [1pt] Consider a standard LP with four variables and three constraints. Then two basic solutions (0; 0; 0; 4; 0; 12; 18) and (3; 0; 0; 1; 0; 2; 0) are adjacent. (b) [1pt] If a linear program has no optimal solution, then it must have an unbounded feasible region. (c) [1pt] Consider the shadow prices of a standard form of LP. The vector formed by the shadow prices is a feasible solution of the dual problem of this LP. (d) [1pt] A linear program can have exactly 10 feasible solutions. (e) [1pt] Consider a primal problem of maximizing c^Tx and a dual problem of minimizing b^Ty (both subject to some constraints). If for a primal feasible solution x and a dual solution y, we have c^Tx > b^Ty, then y must be dual infeasible. (i.e not a feasible solution for the dual problem). (f) [1pt] In a two player zero sum game, there exists at least one Nash equilibrium.

Step-by-step explanation:

a. true

Because two basic feasible solution stands to be adjacent in case they possess basic variable in common. Two distinct basic solutions with respect to set related with linear constraint under is considered to be adjacent.

b.False.

If a linear problem has no solution it may have null feasible region not important to have unbounded feasible region.

c.True.

If Shadow price is feasible for standard form of LP then it will be feasible solution of dual problem of this LP.

d. False.

As there will be 'n' variables 'm' constraints having nCm feasible solutions.

e.True.

As stated in weak duality theorem

f.True

For every zero-sum 2-player normal-form game, a Nash equilibrium exists. Moreover, a pair of mixed strategies (p,q)(p,q) for the two players is a Nash equilibrium if and only if each strategy is a maximin strategy.