Optimization in the Heisenberg group

In this article, the local unconstrained and the constrained optimization problems in the Heisenberg group H are investigated. The framework on which we work is given by the class of weakly H-convex functions recently introduced in the literature. This geometric notion of convexity, that is strictly related to the stratified structure of the group and has not an analogous in a Euclidean setting, requires a condition of convexity to be satisfied at the points of the horizontal planes. We find second-order sufficient conditions for a local extremum in both the unconstrained and the constrained optimization problems exploiting the weak H-convexity and the geometric behaviour of the horizontal planes in H.