I have to optimize a function $f(a,b,c_{ij})$ which consists of a terms like matrix $\mathrm{X = A + B + C}$ where $\mathrm{A}$ is a diagonal matrix with the diagonal elements equal to $a$. $\mathrm{B}$ is another diagonal matrix with the diagonal elements equal to $b$. $\mathrm{C}$ is matrix with elements $c_{ij}$. Also the function consists of terms like $ga + hb$ where $g$ and $h$ are constants, I mean the linear terms wrt $a$ and $b$ variables. Now I have the gradient of the function wrt variables $a,b,c_{ij}$. However, I want to update such that the function is maximized and also the matrix $\mathrm{X}$ is positive definite.
How can I accomplish this?