I think Optimization toolbox might be helpful, but I am not sure about which method I should use.
I am trying to update the value of $\phi$ (a matrix), given all the other variables, matrices and operators:
I think differentiation might not be suitable in my case, since H is an operator rather than a variable. The one below is same as the one above but in linear step from. It's actually quite hard to solve for $H^{T}H$.
How does your function look like? If it is simple enough you can use the symbolic toolbox to differentiate your function with respect to x with the diff(f(x),x) command and than use the solve(f'(x),x) command to get the extreme point(s).
If you apply diff(diff(f(x),x),x) and than insert the extreme point(s) you can check whether its a minimum or maximum...Also make sure to check the asymptotic behavior of your function to find out whether your points are global or local extreme points.
Edit: Ok I just saw you edited your post and wrote down the function, you can't use my method for this.