Vtyjkyuk

I have a somewhat complicated function of $M+1$ variables, which looks as follows.

$$f (x_0, x_1, x_2, dots, x_M) = sum_i=1^N_A ln left[1 - texterfleft(x_0 + sum_j=1^M a_ij x_jright) right] + sum_i=1^N_B ln left[1 + texterfleft(x_0 + sum_j=1^M b_ij x_jright) right].$$

All is real here, but in in principle other than that there are no restrictions on the possible values of the coefficients $a_ij$ and $b_ij$. $N_A$ and $N_B$ are in general some "larger" numbers (say, at least on the order of 100 or 1000), while $M$ tends to be pretty small, say for example 5 to 10 or so. Not that it should matter, but to add some context, this is actually a likelihood expression for some model.

Now my question is, is this a concave function with a unique maximum? Intuitively, the answer seems to be yes, and "numerical evidence" hints to the same direction, but I have a hard time proving it rigorously. I tried to calculate the Hessian, and eventually a general expression for its eigenvalues (which should all be negative in case my assumptions holds true), but it was just too much.

Any suggestions would be appreciated!

Thanks,
Lennex

A sum of concave functions is concave, and $ln(1+texterf(t))$ and $ln(1-texterf(t))$ are easily seen to be concave. So your function is concave.
Moreover, since $ln(1+texterf(t))$ and $ln(1-texterf(t))$ are strictly concave,
the only way for a maximum of your function (assuming it exists) to be non-unique would be for all the $x_0 + sum_j a_ij x_j$ and all the $x_0 + sum_j b_ij x_j$ to be equal
at two different points, i.e. the matrix $pmatrix1 & Acr 1 & Bcr$ to have rank $< M+1$, where $A$ and $B$ are the matrices of coefficients $a_ij$ and $b_ij$, and the $1$'s are column vectors of all $1$'s.