Energetic Order for Optimization on Hierarchies of Partitions: Continuous hierarchy and Lagrange optimization

Jean Serra (1) , Bangalore Ravi Kiran (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

In the current technical note we provide a topological generalization of hierarchy of partitions(HOP) structure, and the implications over the axioms of h-increasingness and scale increasingness [13]. Further in this study we will explicit the Lagrange optimization in the optimal cuts problem and the conditions necessary on the energy to obtain a global optimum using the a dynamic program. Further a general multi-constraint optimization problem is considered with multiple Lagrangian multipliers, leading to a general version of scale increasingness that orders cuts, by ordered tuples of multipliers. The report also differentiates Inf-Modularity and Submodularity and their space of application. The final demonstration on wavering hierarchies show how one can relax conditions on the hierarchical structure.

Mots clés: Hierarchy of Partitions; Optimization; Lagrange; Optimal Cuts; Mathematical Morphology

Domaines: Optimisation et contrôle [math.OC]

Jean Serra, Bangalore Ravi Kiran. Energetic Order for Optimization on Hierarchies of Partitions: Continuous hierarchy and Lagrange optimization. 2014. ⟨hal-00947218⟩