For example, we have F(t) = \begin{bmatrix}
M^*M & t\overline{M_s} \\
tM_s& \overline{M^*M}
\end{bmatrix} where M_s = M+M^T and t \in [0, \infty) and \psi(t) = \lambda_2(F(t)), where \lambda_2 denotes second largest eigenvalue. we have to find
\inf_{t \in [0, \infty)} \psi(t).
lambda_sum_largest(X,2) is the sum of the 2 largest eigenvalues, and is convex. If you are trying to to maximize spectral gap for a matrix whose largest eigenvalue is known to be 1, then it will serve the purpose to minimize lambda_sum_largest(X,2)
But in general,. the 2nd largest eigenvalue by itself is not convex.