Maximize $\mbox{tr} (AX)$ subject to $\mbox{tr} (BX'CX) = 1$

Question

Suppose $A,B,C$ are given square matrices and $X$ is a matrix variable. Is there a nice way to express the following?

$$\begin{array}{ll} \text{maximize} & \mbox{tr} (AX)\\ \text{subject to} & \mbox{tr} (BX'CX) = 1\end{array}$$

Answer 1

Hint: This is a NLP with one constraint so LID CQ holds. Let $D=BC$.

Take $f(X)=tr(AX)$ and $G(X)=tr(BX'CX)= tr(BCX'X)=tr(DX'X)$ Now calculate the gradient of $F$ and $G$, for example $\nabla F(X) = A^T$, set KKT conditions, it gives you necessary optimality condition: $X$ is optimal then $G(X)=1$ and $$A = \lambda \nabla G(X) $$

for some scalar $\lambda \in R$.