arXiv subjects
Publications and source records attributed to A. B. Németh.
It is proved that the linearity of metric projections on subspaces and the convexity of the polars of the convex cones in the uniformly convex and uniformly smooth Banach space are equivalent, and both of them is equivalent with the fact that the space is an inner product space.
Using the notion of the normalized duality map it is characterized the class of the uniformly convex uniformly smooth Banach spaces in which each convex cone has a convex polar.
This paper explores a natural generalization of Euclidean projection through the lens of strongly quasiconvex functions, as developed in prior works. By establishing a connection between strongly quasiconvex functions and the theory of mutually polar mappings on convex cones, we integrate this generalized projection concept into the duality framework of Riesz spaces, vector norms, and Euclidean metric projections. A central result of this study is the identification of conditions under which the null space of a projection onto a closed convex cone forms a closed convex cone. We provide a comprehensive characterization of such cones and projections, highlighting their fundamental role in extending the duality theory to generalized projection operators.
Two retractions $M$ and $N$ on convex cones $\bf M$ and respectively $\bf N$ of a real vector space $X$ are called mutually polar if $M+N=I$ and $MN=NM=0.$ In this note it is shown, that if the cones $\bf M$ and $\bf N$ are generating, $σ$-monotone complete, $M$ and $N$ are $σ$-monotone continuous, then the subadditivity of $M$ and $N$ (with respect to the order relation induced by $\bf M$ and respectively $\bf N$) imply that $\bf M$ and $\bf N$ are lattice cones. $(X,\bf M)$ and $(X,\bf N)$ become $σ$-Dedekind complete Riesz spaces, $M$ and $-N$ are the positive part , respectively the negative part mappings in $(X,\bf M)$; $N$ and $-M$ are the positive part, respectively the negative part mappings in $(X,\bf N)$.
There is given the geometric characterization of an asymmetric norm $q$ on the real vector space $X$, for which exists an $u\in X$ such that $q(x-q(x)u)=0$, for each $x\in X$. The result is used in the theory of mutually polar retractions on cones.
Two retractions Q and R on closed convex cones M and respectively N of a Banach space are called mutually polar if Q+R=I and QR=RQ=0. This note investigates the existence of a pair of mutually polar retractions for given cones M and N. It is shown that if dim N=1 (or dim M=1) then the retractions are subadditive with respect to the order relation their cone range endow.
Asymmetric vector norms are generalizations of asymmetric norms, where the subadditivity inequality is understood in ordered vector space sense. This relation imposes strong conditions on the ordering itself. This note studies on these conditions in the general case, and in the case when the asymmetric vector norm is the metric projection onto the cone engendering the order relation.
The congruence orbit of a matrix has a natural connection with the linear complementarity problem on simplicial cones formulated for the matrix. In terms of the two approaches -- the congruence orbit and the family of all simplicial cones -- we give equivalent classification of matrices from the point of view of the complementarity theory.
Proper cones with the property that the projection onto them is isotone with respect to the order they induce are called isotone projection cones. Isotone projection cones and their extensions have been used to solve complementarity problems and variational inequalities. Q-matrices are matrices with the property that all classical linear complementarity problems defined by them are solvable. This note will use the isotone projection cones to generate a large class of Q-matrices. More specifically, it will be shown that the product between a non-negative matrix with positive diagonal elements and a Stieltjes matrix is a Q-matrix.
The basic tool for solving problems in metric geometry and isotonic regression is the metric projection onto closed convex cones. Isotonicity of these projections with respect to a given order relation can facilitate finding the solutions of the above problems. In the recent note "A. B. Németh and S.Z. Németh: Isotonic regression and isotonic projection. Linear Algebra and its Applications, 494: 80-89 (2016)" this problem was studied for the coordinate-wise ordering. This study was the starting point for further investigations, such as the ones presented here. The order relation in the Euclidean space endowed by a proper cone is considered and the proper cones admitting isotone metric projections with respect to this order relation are investigated.
A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their intersection is empty; (ii) the intersection of any proper subfamily of them is non-empty. In the case of caps (i) and (ii) are also sufficient for the family to be a covering of the sphere.
It is considered a special, convex variant of Sperner lemma type .
The note describes the cones in the Euclidean space admitting isotonic metric projection with respect to the coordinate-wise ordering. As a consequence it is showed that the metric projection onto the regression cone (the cone defined by the general isotonic regression problem) admits a projection which is isotonic with respect to the coordinate-wise ordering.
While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by metods for proving the existence of solutions of variational inequalities and methods for finding these solutions in a recursive way. It turns out, that the closed convex sets admitting isotone projections are exactly the sets which are invariant with respect to these lattice-like operations, called lattice-like sets. In this paper it is shown that the Jordan subalgebras are lattice-like sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least three the lattice-like property is rather restrictive, e.g., there are no lattice-like proper closed convex sets with interior points.
By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.
By using the metric projection onto a closed self-dual cone of the Euclidean space, M. S. Gowda, R. Sznajder and J. Tao have defined generalized lattice operations, which in the particular case of the nonnegative orthant of a Cartesian reference system reduce to the lattice operations of the coordinate-wise ordering. The aim of the present note is twofold: to give a geometric characterization of the closed convex sets which are invariant with respect to these operations, and to relate this invariance property to the isotonicity of the metric projection onto these sets. As concrete examples the Lorentz cone and the nonnegative orthant are considered. Old and recent results on closed convex Euclidean sublattices due to D. M. Topkis, A. F. Veinott and to M. Queyranne and F. Tardella, respectively are obtained as particular cases. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac, H. Nishimura and E. A. Ok.
A wedge (i.e., a closed nonempty set in the Euclidean space stable under addition and multiplication with non-negative scalars) induces by a standard way a semi-order (a reflexive and transitive binary relation) in the space. The wedges admitting isotone metric projection with respect to the semi-order induced by them are characterized. The obtained result is used to show that the monotone wedge (called monotone cone in regression theory) admits isotone projection.
The metric projection onto an order nonnegative cone from the metric projection onto the corresponding order cone is derived. Particularly, we can use Pool Adjacent Violators-type algorithms developed for projecting onto the monotone cone for projecting onto the monotone nonnegative cone too.