A. B. Németh

Babeş-Bolyai University | UBB · Department of Mathematics and Computer Science in Hungarian

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, $\sigma$-monotone complete, $M$ and $N$ are $\sigma$-monotone continuous, then the subadditivity of $M$ and $N$ (w...

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 t...

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...

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 complement...

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 import...

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. N\'emeth and S.Z. N\'emeth: Isotonic regression and isotonic p...

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 isotonic regression cone (the cone defined by the general isotonic regression problem) admits a projection which is isotonic with respect to the coor...

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 in...

It is considered a special, convex variant of Sperner lemma type .

The isotone retraction onto the positive cone of an ordered Euclidean space is a continuous retraction onto this cone preserving the order relation this cone engenders. The existence of such a retraction imposes rather strong conditions on the cone. This chapter contains a comprehensive treatment of the problem, reviewing and completing the results...

In the present paper, we propose and analyze a novel method for estimating a univariate regression function of bounded variation. The underpinning idea is to combine two classical tools in nonparametric statistics, namely isotonic regression and the estimation of additive models. A geometrical interpretation enables us to link this iterative method...

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 importan...

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 obtai...

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...

If $K$ and $L$ are mutually dual pointed convex cones in $\mathbb R^n$ with the metric projections onto them denoted by $P_K$ and $P_L$ respectively, then the following two assertions are equivalent: (i) $P_K$ is isotone with respect to the order induced by $K$ (i.e. $v−u\in K$ implies $P_Kv−P_Ku\in K$; (ii) $P_L$ is subadditive with respect to the...

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 obtaine...

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.

The existence of continuous isotone retractions onto pointed closed convex cones in Hilbert spaces is studied. The cones admitting such mappings are called isotone retraction cones. In finite dimension, generating, isotone retraction cones are polyhedral. For a closed, pointed, generating cone in a Hilbert space the isotonicity of a retraction and...

The solution of the complementarity problem defined by a mapping f:Rn→Rn and a cone K⊂Rn consists of finding the fixed points of the operator PK∘(I-f), where PK is the projection onto the cone K and I stands for the identity mapping. For the class of isotone projection cones (cones admitting projections isotone with respect to the order relation th...

A very fast heuristic iterative method of projection on simplicial cones is presented. It consists in solving two linear systems at each step of the iteration. The extensive experiments indicate that the method furnishes the exact solution in more then 99.7 percent of the cases. The average number of steps is 5.67 (we have not found any examples wh...

It is defined the locally order convexity of an ordered uniform space and an ordered topological group and is investigated its relation to the existence of minimum points of denumerable, complete lower bounded sets.

The Fenchel–Young duality theory is revisited in order to reach an estimation on the sum of the dimensions of the subdifferential of a convex function and the subdifferential of its conjugate function. The obtained estimation is then used in deducing a relativised abstract Haar–Rubinstein theorem on best approximation.

In practically all Ekeland type variational problems we have to do with an “induced” order relation in some metric space or uniform space. The effective minimization takes place with respect to this order relation. Hence a natural approach is to start with ordered uniform spaces and to develop here a minimization theory. Then various Ekeland-type p...

The latticial cone K⊂ℝ n is called stiff if it has a proper face K ' of maximal dimension such that each of its element is comparable (with respect to the order relation induced by K) with each element of the set K∖K ' . The face K '' of the latticial cone K is a stiff face if it is a stiff cone in the subspace K '' -K '' ; it is a maximal stiff fa...

If H is a real Hilbert space, K is a closed, generating cone therein and P-K is the metric projection onto K, then the following two conditions 1 and 2 are equivalent: 1. (i) P-K is isotone: y - x is an element of K double right arrow P-K (y) - P-K (x) is an element of K and (ii) P-K is subadditive: P-K (x) + P-K (y) - P-K (x + y) is an element of...

The paper emphasizes the order theoretic aspects of the Ekeland's varia-tional principle. Behind the machinery of its various proofs stands the transformation of the ordering in a topological vector space into another one which assures that an order bounded, monotone net is convergent. This order is then "transported"in various structures, usually...

The aim of this note is to show that the problem of the augmentation of a function system consisting of the coordinate functions of a parameterization of a convex surface in ℝ n-1 with 0 in its interior, to a Chebyshev system of order n-3, see the paper by G. S. Rubinstein [Dokl. Akad. Nauk SSSR 102, 451-454 (1956; Zbl 0070.10904)], has its natural...

The subdifferentiability properties of convex operators are intimately related to the properties of the ordered (topological) vector spaces they range in. This relation is investigated for various types of subdifferentiability notions

One studies some iterative methods for solving complementarity problems (explicit or implicit) in Hilbert spaces. The iterative methods presented are based on the fixed point theory on the coincidence equations on convex cones and on some properties of projection operators onto a special cone named the «isotone projection cone». The ordering define...

One defines and studies the concept of isotone projection cone in a Hilbert space H. The isotone projection cone is a closed and convex cone K⊂H such that the metric projection P k is isotone with respect to the order defined by K. One gives some applications to the study of the complementarity problem.

By using an efficiency notion placed between Pareto efficiency and Pareto ε-efficiency it is shown that semi Archimedian ordered vector spaces and regular ordered locally convex spaces can be characterized by existence results concerning efficient points of this type for lower bounded sets. The technique developed in the paper allows us to obtain n...

0. Introduction. The distance function in finite dimensional spaces constitutes the object of various investigations. Some of these, which are more connected to the problems considered by us in the sequel are for example those considered in [1] and [2], which concerns some analytical properties of this function. In the notes [5] and [6] we have stu...

Definition 1. There will be said that the family ~T- of sets in the Euclidean space R n has a supporting sphere, ff there exists a sphere S in R n having common points with each member of the family ~ and the interior of S contains no point of any member of ~U.

Not available. MR0375195 (51 #11391) Zbl 0352.53002