Construct an equivalent convex-composition formulation for regularized LDA

Determine whether the LDA trace-ratio objective with $\theta=1$ and $D=0$, namely $\operatorname{tr}(P^{}A_2P)/\operatorname{tr}(P^{}A_1P)$ on the Stiefel manifold, can be represented as a convex composition of $\operatorname{tr}(P^{}A_2P)$ and $\operatorname{tr}(P^{}A_1P)$ such that maximizing the composition is equivalent to maximizing the original objective.

Background

The paper’s NEPv framework requires the smooth learning objective to be expressible as a suitable convex composition of atomic trace functions. LDA is the special case of the $\Theta$-trace-ratio objective with $\theta=1$ and $D=0$, but the corresponding composition does not fall within the convexity range directly established for the framework. The authors therefore replace the original ratio by the modified objective $$[\operatorname{tr}(P^{}A_2P)]^2/\operatorname{tr}(P^{}A_1^2P)$$, rather than resolving whether an equivalent convex-composition representation of the original LDA objective exists.