Other Math Answers

Matrix Games:

If we are given an arbitrary I by J matrix A, there exists a greater than 0 so that the matrix B with entries Bij = Aij + a has only positive entries. Show that any optimal randomized probability vectors for the game with pay-off matrix B are also optimal for the game with pay-off matrix A.

Let i_c(x) be the indicator function of the closed convex set C. Show that the sub-differential of the function i_c at a point c in C is the normal cone to C at the point c.

From the first day of this month Paltu has started saving one 2 taka coin each day in a
box. The box will turn on a red light if it contains 50 taka or more. On the ninth day,
Paltu’s father secretly put 5 coins of 2 taka in that box. On the other hand, Paltu forgot to
save coins on the twelfth and the fifteenth day. In which date will the red light turn on
just after putting the coin?

Poly, Mili and Lily are three sisters. Lily tells truth in 6 days of a week. Poly tells
truth in one day per week and Mili tells truth in 2 days per week. If Poly tells truth
on a day, Mili also tells truth on that day. At Friday, Lily told Mili, “Today Poly
will tell truth.” But Mili said, “No. Tomorrow Poly will tell truth.” Then Poly said,
“Lily is right.”. On which day of week, Poly tells truth?

LP (linear programming): refers to the problem of optimizing a linear function of several variables over linear equality or inequality constraints.

Consider the LP problem of maximizing the function f(x,y) = x + 2y subject to

-2x + y <= 2,
-x + 2y <= 7,
x <= 3,

and x >= 0, y >= 0. Start at x = 0, y = 0. You will find that you have a choice for the entering variable; try it both ways.

Definition: Given any non-empty closed convex set C and an arbitrary vector x in R^j, there is a unique member Pcx of C closest, in the sense of the two norm, to x. The vector Pcx is called the orthogonal projection of x onto C and the operator Pc the orthogonal projection onto C.

Question: Given a point s in a convex set C, where are the points x for which s = Pcx?

Use Farka's Lemma directly to prove that, if p* is finite, then PS has a feasible solution.

Let z and a be as in the Support Theorem, let lamda > 0, and let x = z + lamda a.Show that z = Pcx.