  We propose a novel approach to solve K-adaptability problems with convex
objective and constraints and integer first-stage decisions. A logic-based
Benders decomposition is applied to handle the first-stage decisions in a
master problem, thus the sub-problem becomes a min-max-min robust combinatorial
optimization problem that is solved via a double-oracle algorithm that
iteratively generates adverse scenarios and recourse decisions and assigns
scenarios to K subsets of the decisions by solving p-center problems.
Extensions of the proposed approach to handle parameter uncertainty in both the
first-stage objective and the second-stage constraints are also provided. We
show that the proposed algorithm converges to an optimal solution and
terminates in finite number of iterations. Numerical results obtained from
experiments on benchmark instances of the adaptive shortest path problem, the
regular knapsack problem, and a generic K-adaptability problem demonstrate the
performance advantage of the proposed approach when compared to
state-of-the-art methods in the literature.
