Threshold semidefinite program for the largest eigenvalues

The sum of the largest eigenvalues has the semidefinite program representationFor every feasible $Y$ in the fantope, positive semidefinite trace nonnegativity gives $\mathrm{tr}(XY)\le kt+\mathrm{tr}S$. To attain equality, use an orthonormal eigenbasis of $X$ and choose $S=\mathrm{diag}(({\lambda }_i-t{)}_{+})$ 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.