Convex Set In Hilbert Space Research Articles

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

The difference vectors for convex sets and a resolution of the geometry conjecture

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.

On angles between convex sets in Hilbert spaces

The notion of the angle between two subspaces has a long history, dating back to Friedrichs's work in 1937 and Dixmier's work on the minimal angle in 1949. In 2006, Deutsch and Hundal studied extensions to convex sets in order to analyze convergence rates for the cyclic projections algorithm. In this work, we characterize the positivity of the minimal angle between two convex cones. We show the existence of, and necessary conditions for, optimal solutions of minimal angle problems associated with two convex subsets as well. Moreover, we generalize a result by Deutsch on minimal angles from linear subspaces to cones. This generalization yields sufficient conditions for the closedness of the sum of two closed convex cones. This also relates to conditions proposed by Beutner and by Seeger and Sossa. Furthermore, we investigate the relation between the intersection of two cones (at least one of which is nonlinear) and the intersection of the polar and dual cones of the underlying cones. It turns out that the two angles involved cannot be positive simultaneously. Various examples illustrate the sharpness of our results.

On the asymptotic behaviour of the Aragón Artacho–Campoy algorithm

Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.

Projections onto closed convex sets in Hilbert spaces

Let X be a real Hilbert space. We give necessary and sufficient algebraic conditions for a mapping $${F\colon X \to X}$$ with a closed image set to be the metric projection mapping onto a closed convex set. We provide examples that illustrate the necessity of each of the conditions. Our characterizations generalize several results related to projections onto closed convex sets.

Approximating solutions of generalized pseudocontractive variational inequalities by admissible perturbation type iterative methods

In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let \( \mathcal{H} \) be a separable Hilbert space, \( \mathcal{U} \subseteq \mathcal{H} \) an open convex subset, and f: \( \mathcal{U} \to \mathcal{H} \) a smooth map. Let Ω be an open convex set in \( \mathcal{H} \) with \( \bar \Omega \subseteq \mathcal{U} \), where \( \bar \Omega \) denotes the closure of Ω in \( \mathcal{H} \). We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): \( \mathcal{H} \to \mathcal{H} \) is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ \( \mathcal{U} \), where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L2-space under the flow of a class of kinetic transport equations so called BGK model.

A ball separation property of bounded convex sets in Hilbert spaces

In this paper we provide a ball separation property of bounded convex sets in a Hilbert space. As a consequence, we obtain a representation form of convex closures and two results about convex functionals.

Properties of typical bounded closed convex sets in Hilbert space

For a nonempty separable convex subset X of a Hilbert space ℍ(Ω), it is typical (in the sense of Baire category) that a bounded closed convex set C ⊂ ℍ(Ω) defines an m‐valued metric antiprojection (farthest point mapping) at the points of a dense subset of X, whenever m is a positive integer such that m ≤ dimX + 1.

Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses.

The composition of projections onto closed convex sets in Hilbert space is asymptotically regular

The composition of finitely many projections onto closed convex sets in Hilbert space arises naturally in the area of projection algorithms. We show that this composition is asymptotically regular, thus proving the so-called “zero displacement conjecture” of Bauschke, Borwein and Lewis. The proof relies on a rich mix of results from monotone operator theory, fixed point theory, and convex analysis.

Strongly nonlinear variational inequalities and generalized pseudocontractions

The solvability of a class of generalized strongly nonlinear variational inequality problems on nonempty closed convex sets in Hilbert spaces is presented.

Some geometric properties of typical compact convex sets in Hilbert spaces

Some geometric properties of typical compact convex sets in Hilbert spaces

Finding projections onto the intersection of convex sets in Hilbert spaces. II

We present a parallel iterative algorithm to find the shortest distance projection of a given point onto the intersection of a finite number of closed convex sets in a real Hilbert space; the number of sets used at each iteration step, corresponding to the number of available processors, may be smaller than the total number of sets. The relaxation coefficient at each iteraction step is determined by a geometric condition in an associated Hilbert space, while for the weights mild conditions are given to assure norm convergence of the resulting sequence. These mild conditions leave enough flexibility to determine the weights more specifically in order to improve the speed of convergence.

Finding projections onto the intersection of convex sets in hilbert spaces

We present a parallel iterative method to find the shortest distance projection of a given point onto the intersection of a finite number of closed convex sets in a real Hilbert space. In the method use is made of weights and a relaxation coefficient which may vary at each iteration step, and which are determined at each step by geometrical conditions.

TOPOLOGICAL PROOFS OF KELLER'S THEOREM AND AN EQUIVARIANT VERSION OF IT

The known proofs of Keller's theorem that any infinite-dimensional compact convex set in Hilbert space is homeomorphic to the Hilbert cube are analytic. Here a topological proof of this theorem is given. A new approach to the old theorem leads to a proof of an equivariant version of Keller's theorem.

Noncontractive Uniformly Lipshitzian Semigroups in Hilbert Space

It is shown that any $k$-Lipshitzian, $k < \pi /2$, noncontractive commutative semigroup acting on a closed bounded convex set in Hilbert space has a common fixed point.

Noncontractive uniformly Lipshitzian semigroups in Hilbert space

It is shown that any k k -Lipshitzian, k &gt; π / 2 k &gt; \pi /2 , noncontractive commutative semigroup acting on a closed bounded convex set in Hilbert space has a common fixed point.

Image Restoration by the Method of Convex Projections: Part 1ߞTheory

A projection operator onto a closed convex set in Hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms. Moreover, it minimizes distance and is nonexpansive, and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds. In this paper, we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically. Their common conceptual basis is as follows. Every known property of an original image f is envisaged as restricting it to lie in a well-defined closed convex set. Thus, m such properties place f in the intersection E(0) = E(i) of the corresponding closed convex sets E(1),E(2),...EE(m). Given only the projection operators PE(i) onto the individual E(i)'s, i = 1 --> m, we restore f by recursive means. Clearly, in this approach, the realization of the P(i)'s in a Hilbert space setting is one of the major synthesis problems. Section I describes the geometrical significance of the three main theorems in considerable detail, and most of the underlying ideas are illustrated with the aid of simple diagrams. Section II presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal-processing applications, and the Appendix contains proofs of all the major results.

How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities

How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities