Maximizing strictly increasing convex function Let the objective be to maximize the sum of $f_i(x_i)$ where all $f_i$ are strictly increasing convex functions. Maximizing a convex function is hard as a local maximum might not be a global one. However for the special case described above, since $f_i$ are strictly increasing, from what I understand, the local maximum should be the global one. 

If so, are there any special techniques that can optimize the objective (or maybe a piecewise approximation of it) subject to linear inequality constraints quickly?

Thanks
I'm sorry I don't fully understand the question, but if the point is to find the maximum, considering the local maximum is the global, then any gradient based method is going to take you directly to it. For instance artificial neural networks (gradient backpropagation).