In this paper we establish necessary as well as sufficient conditions for a given feasible point to be a global minimizer of smooth minimization problems with mixed variables. These problems, for instance, cover box constrained smooth minimization problems and bivalent optimization problems. In particular, our results provide necessary global optimality conditions for difference convex minimiza...

Building on recent results for submodular minimization with combinatorial constraints, and on online submodular minimization, we address online approximation algorithms for submodular minimization with combinatorial constraints. We discuss two types of algorithms and outline approximation algorithms that integrate into those.