Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.     

Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x ⁺. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.     

Isotone retraction cones in Hilbert spaces

A mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in Isac and Németh (1986, 1990, 1992) [1], [2], [3], [4] and [5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial (Isac (1990) [2]). This problem is extended by replacing the projection mapping with continuous retractions onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp. Several particular cases are considered and examples are given.     

Inequalities Characterizing Coisotone Cones in Euclidean Spaces

The isotone projection cone, defined by G. Isac and A. B. Németh, is a closed pointed convex cone such that the order relation defined by the cone is preserved by the projection operator onto the cone. In this paper the coisotone cone will be defined as the polar of a generating isotone projection cone. Several equivalent inequality conditions for the coisotonicity of a cone in Euclidean spaces will be given. Thanks are due to A. B. Németh who draw the author’s attention on the relation of latticial cones generated by vectors with pairwise non-accute angles with the theory of isotone cones.     

Projections onto convex sets on the sphere

In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited.     

Isotonicity of the Metric Projection and Complementarity Problems in Hilbert Spaces

In this paper, as the extension of the isotonicity of the metric projection, the isotonicity characterizations with respect to two arbitrary order relations induced by cones of the metric projection operator are studied in Hilbert spaces, when one cone is a subdual cone and some relations between the two orders hold. Moreover, if the metric projection is not isotone in the whole space, we prove that the metric projection is isotone in some domains in both Hilbert lattices and Hilbert quasi-lattices. By using the isotonicity characterizations with respect to two arbitrary order relations of the metric projection, some solvability and approximation theorems for the complementarity problems are obtained. Our results generalize and improve various recent results in the field of study.     

Polar Transform of Conformally Flat Metrics

In the theory of convex subsets in a Euclidean space, an important role is played by Minkowski duality (the polar transform of a convex set, or the Legendre transform of a convex set). We consider conformally flat Riemannian metrics on the n-dimensional unit sphere and their embeddings into the isotropic cone of the Lorentz space. For a given class of metrics, we define and carry out a detailed study of the Legendre transform.     

Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems

Conditions for the non-existence of a regular exceptional family of elements with respect to an isotone projection cone in a Hilbert space will be presented. The obtained results will be used for generating existence theorems for a complementarity problem with respect to an isotone projection cone in a Hilbert space.     

Iterative methods for nonlinear complementarity problems on isotone projection cones

In this paper we present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a Hilbert space and a continuous mapping defined on the cone. If the recursion is convergent, then its limit is a solution of the nonlinear complementarity problem. In the case of isotone projection cones sufficient conditions are given for the mapping so that the recursion to be convergent.     

Motzkin predecomposable sets

We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which hold also for the class of Motzkin decomposable sets (i.e., those for which the convex cone in the decomposition is requested to be closed), while others are specific of the new family.     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

Centers of sets with symmetry or cyclicity properties

We introduce an axiomatic formalism for the concept of the center of a set in a Euclidean space. Then we explain how to exploit possible symmetries and possible cyclicities in the set in order to localize its center. Special attention is paid to the determination of centers in cones of matrices. Despite its highly abstract flavor, our work has a strong connection with convex optimization theory. In fact, computing the so-called “incenter” of a solid closed convex cone is a matter of solving a nonsmooth convex optimization program. On the other hand, the concept of the incenter of a solid closed convex cone has a bearing on the complexity analysis and design of algorithms for convex optimization programs under conic constraints.     

How to project onto an isotone projection cone

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 P_K°(I-f), where P_K 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 they generate) and f satisfying certain monotonicity properties, the solution can be obtained by iterative processes (see G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990) 258-275 and S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350(1) (2009) 340-347). These algorithms require computing at each step the projection onto the cone K. In general, computing the projection mapping onto a cone K is a difficult and computationally expensive problem. In this note it is shown that the projection of an arbitrary point onto an isotone projection cone in Rⁿ can be obtained by projecting recursively at most n-1 times into subspaces of decreasing dimension. This emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones, as for example the non-negative monotone cones occurring in reconstruction problems (see e.g. Section 5.13 in J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, 2005, v2009.04.11).     

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Graph rigidity via Euclidean distance matrices

Let G=(V,E,ω) be an incomplete graph with node set V, edge set E, and nonnegative weights ω_ij's on the edges. Let each edge (v_i,v_j) be viewed as a rigid bar, of length ω_ij, which can rotate freely around its end nodes. A realization of a graph G is an assignment of coordinates, in some Euclidean space, to each node of G. In this paper, we consider the problem of determining whether or not a given realization of a graph G is rigid. We show that each realization of G can be epresented as a point in a compact convex set ; and that a generic realization of G is rigid if and only if its corresponding point is a vertex of Ω, i.e., an extreme point with full-dimensional normal cone.     

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.     

Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone

The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in the mathematical and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections.In this paper, the case where one of the constraints is an obtuse cone is considered. Because the nonnegative orthant as well as the set of positive-semidefinite symmetric matrices form obtuse cones, we cover a large and substantial class of feasibility problems. Motivated by numerical experiments, the method of reflection-projection is proposed: it modifies the method of cyclic projections in that it replaces the projection onto the obtuse cone by the corresponding reflection.This new method is not covered by the standard frameworks of projection algorithms because of the reflection. The main result states that the method does converge to a solution whenever the underlying convex feasibility problem is consistent. As prototypical applications, we discuss in detail the implementation of two-set feasibility problems aiming to find a nonnegative [resp. positive semidefinite] solution to linear constraints in ⁿ [resp. in , the space of symmetric n×n matrices] and we report on numerical experiments. The behavior of the method for two inconsistent constraints is analyzed as well.     

Proximal Point Algorithm On Riemannian Manifolds

Abstract

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.