Multiobjective optimization problems with a variable ordering structure instead of a partial ordering have recently gained interest due to several applications. In the last years a basic theory has been developed for such problems. The difficulty in their study arises from the fact that the binary relations of the variable ordering structure, which are defined by a cone-valued map which associa...

The Coulomb interaction among massless Dirac fermions in graphene is unscreened around the isotropic Dirac points, causing a logarithmic velocity renormalization and a cone reshaping. In less symmetric Dirac materials possessing anisotropic cones with tilted axes, the Coulomb interaction can provide still more exotic phenomena, which have not been experimentally unveiled yet. Here, using site-s...

Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linear programming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The r...

We study the orbits of G = GL(V ) in the enhanced nilpotent cone V ×N , where N is the variety of nilpotent endomorphisms of V . These orbits are parametrized by bipartitions of n = dimV , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition a...

We use the idea of a scalarization of a cone metric to prove that the topology generated by any cone metric is equivalent to a topology generated by a related metric. We then analyze the case of an ordering cone with empty interior and we provide alternative definitions based on the notion of interior cone. Finally we discuss the implications of such cone metrics in the theory of iterated funct...

In this paper, we consider an approximation of the ordering cone by a sequence of convex well-based cones that are contained inside the original cone. We derive related results on approximating minimal point sequences and propose a density result being evocative of the well-known ABB theorem.

Efficient points are obtained for cone-ordered maximizations in n R using the method of scalarization. Various scalarizations are presented for ordering cones in general and then for the important special case of polyhedral cones. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented.

In this paper, we present the notion of multimodular triangulation under a new geometrical point of view. We also show the link with multimodular functions by a new proof of the convexity theorem. This is used to de ne a partial ordering compatible with multimodularity called the cone ordering. An application in admission control in queues is then presented.

We propose a new data mining approach in ranking documents based on the concept of cone-based generalized inequalities between vectors. A partial ordering between two vectors is made with respect to a proper cone and thus learning the preferences is formulated as learning proper cones. A pairwise learning-to-rank algorithm (ConeRank) is proposed to learn a non-negative subspace, formulated as a...