Gaussian Mixture Model in CVX


I am trying to solve the gaussian mixture modelling problem in CVX but the thing is that one of my constraints includes log of weighted sum of gaussian PDF’s. So it looks something like

\sum_{n=1}^N log(\sum_{i=1}^M w_i * Gauss(x_n,\Sigma_i) ) >= t

where t is my objective function which I am maximizing over (along with \Sigma_i and w_i)
w_i is the weight associated with each gaussian and 0 <= w_i <= 1
and Gauss() is the gaussian PDF.

So I am taking everything inside the argument of the exponential of the gaussian pdf (ignoring other positive constants):

\sum_{n=1}^N log(\sum_{i=1}^M exp(-matrix\_frac(x_n, \Sigma_i)+log(w_i)) ) >= t

and then using log_sum_exp() to do the “sum and exponentiation” part which gives me:

\sum_{n=1}^N log\_sum\_exp([-matrix\_frac(x_n, \Sigma_i)+log(w_i)] ) >= t

where [-matrix\_frac(x_n, \Sigma_i)+log(w_i)] is vector now with number of rows = M.

But I am getting an error
"Disciplined convex programming error:
Illegal operation: log_sum_exp( {concave} )."

which means that the input to log_sum_exp has to be convex. So, is there some other way to go about doing this?