Thanks for your replying, Mr. Lichtblau! Your suggestion completely solves my problem and I can finally relieve a little bit now. Since I am beginner to Mathematica, I did not realize the difference between 0.01 and 1/100. Really learnt something from you.

Btw, this LP is not intended as an integer linear programming. The parameter n is given before the optimization start. I just want to observe how the optimal value changes when n goes to large. So I can plot a 2-D picture to show this change. As for p, it is just a fixed parameter which lies between zero and one in my problem.

Thanks again!

According to the documentation, the arguments are supposed to be real numbers. That means Reals in Wolfram Language. If you give it integers or rationals (like your code is doing) the documentation says it tries to give an exact rational result. For example, instead of

Binomial[19, 9] 2^19 (*48432676864*)

you would use

N[Binomial[19, 9] 2^19]

(better to just use N on the arguments ob, M) but you'd want to make sure the Precision is high enough. See the documentation for N and try using the Tolerance option for LinearProgramming.

I cannot say for certain what the underlying algorithm is, but it is possible that the numbers in your second argument are too big for it when n=19.

For example, even for the small example you gave with n=3,

Det[Transpose[M].M] == 4971003904000