Constrained Discrete Optimization via Dual Space Search

This paper proposes a novel algorithm for solving NP-hard constrained discrete minimization problems whose unconstrained versions are solvable in polynomial time such as constrained submodular function minimization. Applications of our algorithm include constrained MAP inference in Markov Random Fields, and energy minimization in various computer vision problems. Our algorithm assumes the existence of a polynomial time oracle for computing the Lagrangian dual of the constrained optimization problem. One of the key properties of our algorithm is its ability to compute minimizers for several different constraint instances simultaneously. We show that our algorithm isolates all the constraint instances for which strong duality holds, and provides a lower bound for any specific constraint instance. We also developed a variant of the algorithm that is able to efficiently compute a lower bound for a specific constraint instance using a cutting plane scheme. We demonstrated the efficacy of our approach by showing how it can be applied to the image segmentation problem in computer vision.