On certain classes of curve singulariries with reduced tangent cone

We study a class of rational curves with an ordinary singular point, which was introduced in [GO]. We find some conditions under which the tangent cone is reduced and we show that the tangent cone is not always reduced. We construct another class of rational curves with an ordinary singular point satisfying the condition required in [GO] and whose tangent cone is always reduced.     

Geometry of the copositive and completely positive cones

The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NP-complete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays.     

Optimization over structured subsets of positive semidefinite matrices via column generation

We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.     

When do several linear operators share an invariant cone?

We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented.     

On the existence of efficient points in locally convex spaces

We study the existence of efficient points in a locally convex space ordered by a convex cone. New conditions are imposed on the ordering cone such that for a set which is closed and bounded in the usual sense or with respect to the cone, the set of efficient points is nonempty and the domination property holds.     

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.     

Inverse Mean Curvature Flow for Star-Shaped Hypersurfaces Evolving in a Cone

For a given convex cone we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicular. The evolution of those hypersurfaces inside the cone yields a nonlinear parabolic Neumann problem. We show that one can use the convexity of the cone to prove long time existence of this flow. Finally, we show that the hypersurfaces converge smoothly to a piece of the round sphere.     

Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.

We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

     

A Characterization of Generalized Monotone Normed Cones

Let C be a cone and consider a quasi-norm p defined on it. We study the structure of the couple （C, p） as a topological space in the case where the function p is also monotone. We characterize when the topology of a quasi-normed cone can be defined by means of a monotone norm. We also define and study the dual cone of a monotone normed cone and the monotone quotient of a general cone. We provide a decomposition theorem which allows us to write a cone as a direct sum of a monotone subcone that is isomorphic to the monotone quotient and other particular subcone.     

Transmission of progressing waves across a moving interface

We describe global time existence nd uniqueness results for the wave equations with boundary conditions of Dirichlet type on a characteresitc cone and either Dirichlet or Neumann type on a timelike tube. We find that the solution is in general only half as regular as the data and we provide estimates which describe the differing differentiabilities of the solution in directions which are either tangent or transvers to the characteristic cone.     

Propriétés Asymptotiques des Groupes Linéaires

Let G be a reductive linear real Lie group and be a Zariski dense subgroup. We study asymptotic properties of through the set of logarithms of the radial components of the elements of : we prove that the asymptotic cone of this set is a convex cone with non empty interior and is stable by the Cartan involution. Reciprocally any closed convex cone of the positive Weyl chamber whose interior is non empty and which is stable by the opposition involution can be obtained this way.?We relate this limit cone and the limit set of to the set of open semigroups of G which meet .?We also prove similar results over any local fields.

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Propriétés Asymptotiques des Groupes Linéaires

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Submitted: January 1996     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.     

Quadratic factorization heuristics for copositive programming

Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.     

Elliptic cone optimization and primal–dual path-following algorithms

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

Haar锥逼近的交错定理及其在系数有界逼近中的应用

本文首先推广了Ｐ．Ｐｅｉｓｋｅｒ １９８３年给出的Ｈａａｒ锥的定义及Ｈａａｒ锥一致逼近的交错定量，然后得到了Ｈａａｒ锥根数的一种求法。利用这些结果，讨论了系数有界限逼近的特征问题，特别是给出了系数有界限的代数多项式逼近与广义Ｂｅｒｎｓｔｅｉｎ多项式逼近的使用十分方便的交错定理。     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.     

Commutative extended BCK-algebras

We introduce an extended cone algebra, which generalises a Bosbach’s cone algebra within the framework of extended BCK-algebras and show that every such an algebra is a direct product of an ℓ-group and a cone algebra of Bosbach.