My code is as follows. (In my real code, the indentation is observed well, but is not shown here.)
variable x nonnegative
maximize( (44+41x-3x*x) * inv_pos(x+4))
x <= 10
Actually I want to use CVX to solve the following problem:
maximize w1 * log(1 + P1/n1) + w2 * log(1 + P2 / (beta*P1 + n2))
subject to P1+P2 <= Pmax ; P1, P2 >= 0.
where w1, w2, beta, n1, n2, Pmax are all given parameters and nonnegative.
I tried to solve the above problem, but I got some errors while using CVX.
Thus, I tried to solve a simplified problem of the above problem, by setting w1 = w2 = n1 = n2 = 1, beta = 0.25, and Pmax = 10, and by changing the first constraint form inequality to equality constraint.
The simplified problem is as follows.
maximize log(1 + x) + log(1 + (40-4x) / (x+4))
subject to 0 <= x <= 10
I tried to solve the above problem using the following code, but I got some errors again.
Using the fact that log is concave and increasing, I changed the problem as follows.
maximize (1+x) * (44-3x) / (x+4)
subject to 0 <= x <=10
The code is as follows.
maximize( log(1 + x) * log(1+(10-x)/(0.25*x+1)) )
0 <= x <= 10
The error message is as follows.
Please note that when
f = (1+x) * (44-3x) / (x+4),
its second derivative is given as
f’’(x) = -336 / (x+4)^3.
Thus, for nonnegative x, f’’(x) is always negative, so f(x) is concave and max f(x) is a convex problem.