I would like to consider how to solve the following optimization problem:
$\max x^{T}\mathbf{a}-b\sqrt{x^{T}\Sigma x}\;s.t.x^{T}\mathbf{1}=1$, provided that $b>0$ and $\Sigma$ is positive definite.
I want to solve KKT condition to solve the problem, however, I do not obtain a nice result.
Are you asking how you can solve this analytically through the KKT conditions? My guess is you cannot, but you use a numerical solver to solve the problem instead.
This is a fairly easy convex optimization problem, and almost any nonlinear solver will solve it without problems. If you want to go fancy, you use a solver specially devoted to problems of this kind (it can be written as a so called SOCP problem)