Thank you for the question. Currently, this topic is not documented properly. I will work on it.
In fact, if you pass matrix arguments to solvers such as fmincon, the solver accepts these arguments, and internally converts the arguments to vector versions via the colon operator (so alfa gets changed internally to alfa(:)). At the end of the optimization, the solution is reshaped back to the original dimensions.
This internal conversion can cause some unexpected behavior. For example, a linear inequality constraint such as
A*alfa <= b
requires the matrix A to have as many columns as alfa(:) has rows.
For your case, you might not be aware that lower bounds must be given for each component separately. So if you want a lower bound of 1e-7 on each component, you need to give lb as 1e-7*ones(size(alfa0)).
Your question on bounding the gradient does not make sense to me. There is no built-in facility in Optimization Toolbox to bound the gradient of the objective function. Are you looking for a solution where the gradient is in a region? If so, you need to calculate separately where alfa must be in order to give the gradient that bound, and you need to put in that restriction as a constraint on alfa.
Good luck,
Alan Weiss
MATLAB mathematical toolbox documentation
