My questions are:
- Are generalized linear models (GLMs) guaranteed to converge to a global maximum? If so, why?
- Furthermore, what constraints are there on the link function to insure convexity?
My understanding of GLMs is that they maximize a highly nonlinear likelihood function. Thus, I would imagine that there are several local maxima and the parameter set you converge to depends on the initial conditions for the optimization algorithm. However, after doing some research I have not found a single source which indicates that there are multiple local maxima. Furthermore, I am not so familiar with optimization techniques, but I know the Newton-Raphson method and IRLS algorithm are highly prone to local maxima.
Please explain if possible both on an intuitive and mathematical basis!
EDIT: dksahuji answered my original question, but I want to add the followup question  above. ("What constraints are there on the link function to insure convexity?")