If $\max_a ~~ a^Tx - tr(a^TVe) $ is less than $+\infty$, can we say $x=Ve$?

Question

If $\max\limits_a\{a^Tx - tr(a^TVe)\}< +\infty$, where $a\in\mathbb{R}^n, V\in \mathbb{S}_n, $ and $e$ is a column vector with all components valued $1$.

can we say $x=Ve$ ?

Answer 1

Yes! This problem is unbounded above, unless $x=Ve$.

By the way, if $a$, $x$, and $e$ are all column vectors, then $a^Tx - tr(a^TVe)=a^Tx - a^TVe=a^T(x - Ve)$.