Issues in optimization with positive definite constraints

Question

I have this function $f(\mathrm{X})$ such $\mathrm{X}$ is a positive definite matrix which is equal to $\mathrm{A+B+C}$. $\mathrm{A}$ is a diagonal matrix with variable $a$ on the diagonal elements. $\mathrm{B}$ is another diagonal matrix with $b$ on the diagonal elements and $\mathrm{C}$ is another matrix have a bunch of variables given by $c_{ij}$. Now I want to maximize this function $f(\mathrm{X})$ wrt variables $a,b$ and $c_{ij}$. However, I want to have this constraint of the matrix $\mathrm{X}$ being positive definite as well as symmetric. How can I do it.
I also have the gradient wrt individual variables. However, if I update the variables, it won't guarantee positive definiteness and symmetry. How can I do it?