Optimization in the Heisenberg group

Abstract

In this article, the local unconstrained and the constrained optimization problems in the Heisenberg group 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 .

Keywords:

• Twisted convex combination
• Horizontal gradient
• Horizontal Hessian
• Weak H-convexity
• Optimality conditions

Keywords:

• Mathematics Subject Classifications 2000: 90C26
• 26A51
• 22E15
• 52C22

Keywords:

• OR/MS Subject Classifications: Primary: Programming
• Non-linear
• Theory
• Mathematics
• Convexity