Best way to describe sum log function

I am optimizing a function f

f(x) = \sum_i {log(a_i x + b)} - c x

where x is a vector (the optimization variable), a’s and c are row vectors. b is a scalar constant.

cvx shows the warning that using the log function will require more time as it uses the successive approximation method. Is there any other way to describe the objective function in a way that avoids this?

I am running so many instances, and with such delay, the overall runtime of the program will be very huge.

Due to the fact that you have both a linear and a log term, you will not be able to avoid CVX’s successive approximation approach. But if you’re doing an unconstrained minimization of f, and speed is what you need, you should certainly not be using CVX. Just construct your own Newton’s method.