Use "fmincon".
The primary optimization variable is k.
Specify the lower bound for k to be 0 in the variable "lb".
The objective function of the primary optimization problem is
f(k)=(q(k))^2*H(Y(k))+F(Y(k)) -> max
In order to specify the value of the objective function for a given value of k, you have to determine q(k).
Thus within the objective function, call "fmincon" (or "fminunc") again with the secondary optimization variable q and the objective function
H(X(q)) + q^2*k*F(X(q)) -> max
In the primary optimization problem, use the function "nonlcon" to impose the constraint
c(1) = 0.01 - q(k)
Here, the value of q(k) has to be determined by again solving the secondary optimization problem
H(X(q)) + q^2*k*F(X(q)) -> max
using "fmincon" (or "fminunc") with the given value for k.
Maybe within the primary optimization you don't need to call "fmincon" (or "fminunc") twice in the objective function and in the constraint function by using the strategy explained here:
