I'm trying to solve the problem $$\min \{ \langle A(x),y\rangle + f(y) \mid y \in S^m, \operatorname{tr}(y) =1, y \geq 0\}$$ where $x \in \mathbb{R}^n$, $y \in S^M$, that is, it's a symmetric $m$ by $m$ matrix. The operator $A(x)$ is given by $A(x) = \sum_{i=1}^n x_iA_i$ where $A_i$ is a $m$ by $m$ symmetric matrix for every $i$.
Of course, it will depend on the convex function $f(y)$, but let's say I have a way of viewing the operator $A(x)$ as a matrix $A \in \mathbb{R}^{m^2 \times n}$ and see the matrixes $y$ as a vector in $\mathbb{R}^{m^2}$ and also let's say that I have a method for calculating the minimum $$\min \{ \langle Ax,z\rangle + f(z) \mid z \in R^p, \sum_{i=1}^p(z_i) = 1, z_i \geq 0\}$$ Is there a way I can make the problem above be a particular case of this and use the method for this problem instead?
On the problem above the function $f(y)$ is seen on a way that $y$ is already considered in $R^{m^2}$ so setting $z$ as the vector representation of $y$ and $p = m^2$ I get the same objective function, however how about the constraints?