Assume that we have the following two optimization problems with exactly the same constraints but different objective:
P1: $\min_{{\bf u}} {\bf u}^\text{T} {\bf T}_1 {\bf u}$ subject to C1: ${\bf 1}^\text{T}{\bf u}=1$; C2: $u_k \ge 0$, $\forall k$
P2: $\min_{{\bf v}} {\bf v}^\text{T} {\bf T}_2 {\bf v}$ subject to C1: ${\bf 1}^\text{T}{\bf v}=1$; C2: $v_k \ge 0$, $\forall k$
where ${\bf u}=[u_1,u_2,⋯,u_K]^\text{T}$ and ${\bf v}=[v_1,v_2,⋯,v_K]^\text{T}$ are two column vectors to be optimized.
My question is: If we have ${{\bf T}_2}={\bf A}^\text{T} {{\bf T}_1} {\bf A}$, where $\bf A$ is a known matrix, then is there any hidden mathematical/analytical relationships between the optimal solutions ${\bf u}^*$ and ${\bf v}^*$ for the above two problems?
Are there any existing mathematical tools or optimization theory that I can use for the above question?
Thanks!