  We study two-stage robust optimization problems with mixed
discrete-continuous decisions in both stages. Despite their broad range of
applications, these problems pose two fundamental challenges: (i) they
constitute infinite-dimensional problems that require a finite-dimensional
approximation, and (ii) the presence of discrete recourse decisions typically
prohibits duality-based solution schemes. We address the first challenge by
studying a $K$-adaptability formulation that selects $K$ candidate recourse
policies before observing the realization of the uncertain parameters and that
implements the best of these policies after the realization is known. We
address the second challenge through a branch-and-bound scheme that enjoys
asymptotic convergence in general and finite convergence under specific
conditions. We illustrate the performance of our algorithm in numerical
experiments involving benchmark data from several application domains.
