Minimizing $f(X) + \langle Y,X+X^T \rangle +\|X+X^T\|^2$ The optimization problem is as follows:
\begin{equation}
\min_{X \in \mathbb{R}^{n \times n}} ~f(X) + \langle Y, X+X^T \rangle + \|X+X^T\|^2,
\end{equation}
where $f \colon \mathbb{R}^{n \times n} \to \mathbb{R}$ is convex function and $Y$ is given parameter matrix. 
The inner product $\langle \cdot, \cdot \rangle$ is Frobenius inner product.
I want to know could I obtain the closed-form solution of this problem. 
I am stuck for a long time but have no idea at all. Any help will be appreciated.
 A: Your function $f$ is unknown (assume it is differentiable?). 
Let us use a colon for the Frobenius product, i.e.,
$$A:B={\operatorname{Trace}}(A^TB) \equiv \langle A, B\rangle.$$
The cyclic property of the Frobenius product, e.g.,
$$\eqalign{
A:B &= A^T:B^T &= B:A 
}$$
Let us define the following with differential.
\begin{align} 
\phi_1 := Y : \left( X + X^T \right) \Rightarrow d\phi_1 = \left( Y + Y^T \right): dX
\end{align}
and
\begin{align} 
\phi_2 := \left( X + X^T \right) : \left( X + X^T \right) \Rightarrow d\phi_1 = 2\left( X + X^T \right): \left( dX + dX^T \right) = 4\left( X + X^T \right): dX.
\end{align}
The composite function can be expressed as
\begin{align} 
\theta = f + \phi_1 + \phi_2.
\end{align}
Take the differential of the composite by plugging in the differentials of $\phi_i$, i.e.,
\begin{align} 
&d\theta = df + d\phi_1 + d\phi_2 = df + \left( Y + Y^T \right):dX + 4\left( X + X^T \right): dX 
\end{align}
Then, obtain the gradient and I think you know what to do?
\begin{align} 
0 \in \frac{\partial \theta}{\partial X} = \frac{\partial f}{\partial X} + \left( Y + Y^T \right) + 4\left( X + X^T \right).
\end{align}
