  Sparse principal component analysis (PCA) is a popular dimensionality
reduction technique for obtaining principal components which are linear
combinations of a small subset of the original features. Existing approaches
cannot supply certifiably optimal principal components with more than $p=100s$
of variables. By reformulating sparse PCA as a convex mixed-integer
semidefinite optimization problem, we design a cutting-plane method which
solves the problem to certifiable optimality at the scale of selecting k=5
covariates from p=300 variables, and provides small bound gaps at a larger
scale. We also propose a convex relaxation and greedy rounding scheme that
provides bound gaps of $1-2\%$ in practice within minutes for $p=100$s or hours
for $p=1,000$s and is therefore a viable alternative to the exact method at
scale. Using real-world financial and medical datasets, we illustrate our
approach's ability to derive interpretable principal components tractably at
scale.
