= Threshold semidefinite program for the largest eigenvalues
The <sum of the largest eigenvalues> has the <semidefinite program> representation
$$
F_k(X)=\min_{t\in\mathbb R,\ S\succeq0}\{kt+\operatorname{tr}S:S\succeq X-tI\}.
$$
For every feasible $Y$ in the <fantope>, <positive semidefinite trace nonnegativity> gives $\operatorname{tr}(XY)\leq kt+\operatorname{tr}S$. To attain equality, use an <orthonormal eigenbasis> of $X$ and choose $S=\operatorname{diag}((\lambda_i-t)_+)$ in that basis, with the same threshold choice as in the <threshold formula for the sum of the largest components>. This argument also covers $k=n$, without requiring strict feasibility of the maximization program.
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