I have this optimization problem on the variables $\lambda_\ell^+, \lambda_\ell^-$ such that $ \lambda_\ell^+ \geq \lambda_\ell^-$ with $\ell=1,\ldots,n$ , and fixed $P\in [1/(n+1),1]$
\begin{align} \mbox{maximize}&\quad \lambda_{1}^+-\lambda_{1}^--2\sum_{\ell= 2}^n\sqrt{\lambda_\ell^+\lambda_\ell^-}\nonumber\\ \mbox{subject to}&\quad\sum_{\ell=1}^{n}{({\lambda_\ell^+}+{\lambda_\ell^-})}=1\,,\quad \sum_{\ell=1}^{n}{({\lambda_\ell^+}^2+{\lambda_\ell^-}^2)}\leq P\quad\mbox{and}\quad \lambda_\ell^+\geq\lambda_\ell^-\geq 0\quad \forall \ell=1,\ldots,n\,. \end{align}
If I make $\lambda_\ell⁻=0$ for every $\ell= 2,\ldots,n$, the objective function becomes linear and the resulting problem can be cast as a semidefinite program, which is nice. In fact, as it turns out, $\lambda_\ell⁻=0$ for every $\ell= 2,\ldots,n$ is a necessary condition for the optimal solution of the problem above (I know that because the physical problem that gives rise to this mathematical problem has been solved in some independent way elsewhere). I was hoping that someone could offer a mathematical argument that enables me to restrict my feasible set with $\lambda_{2,\ldots,n}^⁻=0$.
Let $\lambda_l^+,\lambda_l^-$ for $l=1,\ldots,n$ be a feasible solution such that there exists $l$ such that $\lambda_l^->0$. Setting
$$ \lambda_l^+ \leftarrow \lambda_l^ + + \lambda_l^-,\\ \lambda_l^- \leftarrow 0, $$
we obtain a new feasible solution. It is easy to see that the objective function cannot decrease.