  Sufficient dimension reduction (SDR) is a popular tool in regression
analysis, which replaces the original predictors with a minimal set of their
linear combinations. However, the estimated linear combinations generally
contain all original predictors, which brings difficulties in interpreting the
results, especially when the number of predictors is large. In this paper, we
propose a customized branch and bound algorithm, optimal sparse generalized
eigenvalue problem (Optimal SGEP), which combines a SGEP formulation of many
SDR methods and efficient and accurate bounds allowing the algorithm to
converge quickly. Optimal SGEP exactly solves the underlying non-convex
optimization problem and thus produces certifiably optimal solutions. We
demonstrate the effectiveness of the proposed algorithm through simulation
studies.
