I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ and $N=n_1+...+n_m$ are the (positive) observed value and we want to find optimum value of $\alpha$. It can be shown that the above function is (non-strictly) convex function e.g. if $\alpha=(\alpha_1, ..., \alpha_m)$ be the optimum value then $\alpha=(\alpha_1+C, ..., \alpha_m+C)$ is optimum too, for any C. My problem is that any hessian free optimization method like steepest descent and LBFGS takes a lot of time to do the optimization and actually I need to limit the maximum number of iterations to get an answer which is not a very good one.also for some (far)initial values it freezes in line-search step. I really would be appreciated if some one provide me description of shape of my function and the source of them problem and how I can solve it.
Edit: this is an example of steepest descent algorithm with line search for 30 iterations. the first column is grad. norm and the second column is function value. as you see norm value decreases very slowly.
