${\bf F} \in {\mathbb C}^{N \times N}$ is positive semidefinite matrix and satisfies ${\text {tr}} ({\bf F}) \leq P$. $f({\bf F}): {\mathbb C}^{N \times N} \rightarrow {\mathbb R}$ is a real-valued and continuous differentiable function. Consider the following problem \begin{array}\ \begin{align} \mathop {\min }\limits_{\bf F} \quad & f({\bf F}) \tag{1.a}\\ \text{s.t.} \quad & {\text {tr}}({\bf F}) \leq P, \tag{1.b}\\ & {\bf F} \succeq {\bf 0}. \tag{1.c} \end{align} \end{array} Let $g({\bf V}) = f({\bf V} {\bf V}^H)$, where ${\bf V} \in {\mathbb C}^{N \times N}$ satisfies ${\text {tr}} ({\bf V} {\bf V}^H) \leq P$. We get problem \begin{array}\ \begin{align} \mathop {\min }\limits_{\bf V} \quad & g({\bf V}) \tag{2.a}\\ \text{s.t.} \quad & {\text {tr}}({\bf V} {\bf V}^H) \leq P. \tag{2.b}\\ \end{align} \end{array} Obviously, $f({\bf F})$ and $g({\bf V})$ have the same value domain. A solution of problem $(1)$ can thus be obtained from $\bf V$ by setting ${\bf F} = {\bf V} {\bf V}^H$, and a solution of problem $(2)$ can also be obtained from solving $(1)$ and decomposing $\bf F$.
Assume that ${\bf V}_0$ is a locally optimal solution of problem $(2)$. Hence, for any ${\bf V} \in {\mathbb C}^{N \times N}$ satisfying $(2.b)$, \begin{equation} {\text{tr}} \left[ \nabla_{\bf V} g({\bf V}_0)^H \left( {\bf V} - {\bf V}_0 \right) \right] \geq 0, \end{equation} which can be rewritten as \begin{equation} 2 ~{\text{tr}} \left[ \nabla_{\bf F} f({\bf V}_0 {\bf V}_0^H)^H \left( {\bf V} - {\bf V}_0 \right) {\bf V}_0^H \right] \geq 0. \end{equation} Let ${\bf F}_0 = {\bf V}_0 {\bf V}_0^H$. Then, is ${\bf F}_0$ a locally optimal solution of problem $(1)$, i.e., for any ${\bf F} \in {\mathbb C}^{N \times N}$ satisfying $(1.b)$ and $(1.c)$, does the following inequation hold? \begin{equation} {\text{tr}} \left[ \nabla_{\bf F} f({\bf F}_0)^H \left( {\bf F} - {\bf F}_0 \right) \right] = {\text{tr}} \left[ \nabla_{\bf F} f({\bf V}_0 {\bf V}_0^H)^H \left( {\bf F} - {\bf V}_0 {\bf V}_0^H \right) \right] \geq 0. \end{equation}
I have checked some simple scalar cases and found it is true. But I don't know if it is correct for the genereal matrix case. Could anyone provide some hints or references? I would be very appreciated.