How can i build a global optimization problem with lower bounds only on the optimal variable?

I have to solve a global optimization problem regarding portfolio optimization, so i have as variable the weights of the assets and as objective function a custumized risk/return measure (non smooth). So i have only linear constraints: the sum of the weigths must be 1 and the weights have LB=0 and UB=1. Is it possible to put a lower bound only on the optimal weight that the solver will choose. For example i'd like that if the solver will choose the assets 1 and 3 both have LB=0.01 and the others have LB=0. That was the first question. The second is: what's the best optimization solver (the one with the lower obj function and with positive exitflag) for this kind of problem? I'm trying fmincon with internal-point, with sqp, with active-set, Globalsearch with fmincon with internal-point, with sqp, with active-set, Multistart with fmincon with internal-point, with sqp, with active-set in order to see the best one, and also the speedest as i have to optimize hundreds of times.