Suppose I have an algorithm $\mathit{A}$ that can solve the optimization problem over certain functions $g: Sym(n) \to \mathbb{R}$. Now given $g$, let $f: \mathbb{R}^{k ,n} \to \mathbb{R}$ be another function such that $f(X) = g(X^TX)$, how can I solve the following problem?
$$\begin{array}{ll} \underset{X \in \mathbb{R}^{k, n}}{\text{minimize}} & \displaystyle f(X) \\ \text{subject to} & \displaystyle X \geq \bf{0}\end{array}$$
suppose I also know the basic info of $g$ like its gradient, hessian, and also that $g$ is sufficiently nice (in a sense that the program is solvable).