  Many problems require the selection of a subset of variables from a full set
of optimization variables. The computational complexity of an exhaustive search
over all possible subsets of variables is, however, prohibitively expensive,
necessitating more efficient but potentially suboptimal search strategies. We
focus on sparse variable selection for generalized Rayleigh quotient
optimization and generalized eigenvalue problems. Such problems often arise in
the signal processing field, e.g., in the design of optimal data-driven
filters. We extend and generalize existing work on convex optimization-based
variable selection using semidefinite relaxations toward group-sparse variable
selection using the $\ell_{1,\infty}$-norm. This group-sparsity allows, for
instance, to perform sensor selection for spatio-temporal (instead of purely
spatial) filters, and to select variables based on multiple generalized
eigenvectors instead of only the dominant one. Furthermore, we extensively
compare our method to state-of-the-art methods for sensor selection for
spatio-temporal filter design in a simulated sensor network setting. The
results show both the proposed algorithm and backward greedy selection method
best approximate the exhaustive solution. However, the backward greedy
selection has more specific failure cases, in particular for ill-conditioned
covariance matrices. As such, the proposed algorithm is the most robust
currently available method for group-sparse variable selection in generalized
eigenvalue problems.
