Nearest Point Projection Research Articles

Convergence To Approximate Solutions of Multivalued Operators

The goal of this study is to provide a new explicit iterative process method approach for solving maximal monotone(M.M )operators in Hilbert spaces utilizing a finite family of different types of mappings as( nonexpansive mappings,resolvent mappings and projection mappings. The findings given in this research strengthen and extend key previous findings in the literature. Then, utilizing various structural conditions in Hilbert space and variational inequality problems, we examine the strong convergence to nearest point projection for these explicit iterative process methods Under the presence of two important conditions for convergence, namely closure and convexity. The findings reported in this research strengthen and extend key previous findings from the literature

Forcing Strong Convergence of a Mann‐Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann‐based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.

Optimization-based subdivision algorithm for reachable sets

<p style='text-indent:20px;'>Reachable sets for nonlinear control systems can be computed via the use of solvers for optimal control problems. The paper presents a new improved variant which applies adaptive concepts similar to the framework of known subdivision techniques by Dellnitz/Hohmann. Using set properties of the nearest point projection, the convergence and rigorousness of the algorithm can be proved without the assumption of diffeomorphism on a nonlinear mapping. The adaptive method is demonstrated by two nonlinear academic examples and for a more complex robot model with box constraints for four states, two controls and five boundary conditions. In these examples adaptive and non-adaptive techniques as well as various discretization methods and optimization solvers are compared. <p style='text-indent:20px;'>The method also offers interesting features, like zooming into details of the reachable set, self-determination of the needed bounding box, easy parallelization and the use of different grid geometries. With the calculation of a 3d funnel in one of the examples, it is shown that the algorithm can also be used to approximate higher dimensional reachable sets and the resulting box collection may serve as a starting point for more sophisticated visualizations or algorithms.

A novel curvature circle iterative algorithm for contour error control of multi-axis CNC machine tools

High-precision real-time estimation of contouring errors is a prerequisite for contouring errors control of multi-axis CNC machine tools. This paper focuses on developing a nearest point projection curvature circle iterative (NPP–CCI) algorithm to achieve real-time estimation of multi-axis contouring errors. It is found that the traditional curvature circle iterative (CCI) method has two major shortcomings. The first is that the iterative process may terminate incorrectly at the local contour position, and the other is that the actual tool position and local curvature circle are not necessarily coplanar in three-dimensional space, which would lead to inaccurate calculation of the delay time parameter and eventually affect the estimation accuracy. In order to address the problem of false termination, an index method is used to find the closest reference position with respect to the actual position. At the same time, the projection technology is proposed to overcome the problem met in extending the planar curvature circle iterative method to the spatial applications. The proposed NPP-CCI algorithm is more suitable for spatial contouring errors estimation in tracking complex trajectories and has higher estimation accuracy than the traditional CCI algorithm. Various experiments with different tool paths are conducted on an in-house developed multi-axis experimental platform to verify the effectiveness of the proposed algorithm. The experimental results show that the NPP-CCI algorithm can estimate the contouring errors with higher accuracy than the traditional CCI algorithm, and with the help of real-time computation and compensation, the contouring errors are reduced by more than 44% in terms of the MAX and RMS values.

Symmetric orthogonality and non-expansive projections in metric spaces

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it is shown that every space having non-positive curvature in the sense of Busemann is a CAT(0)-space provided that its tangent cones are uniquely geodesic and their nearest point projections onto convex sets are non-expansive.

Projections of the sphere graph to the arc graph of a surface

Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.

Projections in the curve complex arising from covering maps

Let $P : \Sigma \rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $\Pi : \mathcal{C}(S) \rightarrow \mathcal{C}(\Sigma)$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}(\Sigma)$ using minimal intersection number conditions and prove that it approximates a nearest point projection to $\Pi(\mathcal{C}(S))$. We also approximate hulls of finite sets of vertices in the curve complex, together with their corresponding nearest point projections, using intersection numbers.

Shadows of Teichmüller Discs in the Curve Graph

We consider several natural sets of curves associated to a given Teichm\"uller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to uniformly bounded Hausdorff distance. Furthermore, we describe two operations on curves and show that they approximate nearest point projections to their respective targets. Our techniques can be used to prove a bounded geodesic image theorem for a natural map from the curve graph to the filling multi-arc graph associated to a Teichm\"uller disc.

Weak convergence of infinite products of operators in Hadamard spaces

We first prove the weak convergence of iterates of both products and convex combinations of strongly nonexpansive operators in complete CAT(0) spaces. Then we establish the weak convergence of infinite products of possibly discontinuous approximations to such operators. Our main focus is on nearest point projections.

Optimization‐based subdivision algorithm for reachable sets

AbstractThis paper shows how an optimization‐based approach to calculate reachable sets can be improved by using a subdivision‐algorithm. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A Cyclic Douglas–Rachford Iteration Scheme

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas-Rachford scheme, are promising.

A note on alternating projections in Hilbert space

We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.

Approximating fixed points for nonself mappings in CAT(0) spaces

AbstractSuppose K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T : K → X be a nonself mapping, satisfying Condition (E) with F(T): = {x ∈ K : Tx = x} ≠ ∅. Suppose {x n } is generated iteratively by x1 ∈ K, x n +1 = P ((1 - α n )x n ⊕ α n TP [(1 - β n )x n ⊕ β n Tx n ]),n ≥ 1, where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε ∈ (0, 1). Then, {x n } Δ-converges to some point x⋆ in F(T). This extends a result of Laowang and Panyanak [Fixed Point Theory Appl. 367274, 11 (2010)] for nonself mappings satisfying Condition (E).

Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces

AbstractSuppose that "Equation missing" is a nonempty closed convex subset of a complete CAT(0) space "Equation missing" with the nearest point projection "Equation missing" from "Equation missing" onto "Equation missing". Let "Equation missing" be a nonexpansive nonself mapping with "Equation missing". Suppose that "Equation missing" is generated iteratively by "Equation missing", "Equation missing", "Equation missing", where "Equation missing" and "Equation missing" are real sequences in "Equation missing" for some "Equation missing". Then "Equation missing""Equation missing"-converges to some point "Equation missing" in "Equation missing". This is an analog of a result in Banach spaces of Shahzad (2005) and extends a result of Dhompongsa and Panyanak (2008) to the case of nonself mappings.

Nonexpansive retractions in Hilbert spaces

Let H be a Hilbert space and C H be closed and convex. The mapping P : H ! C known as the nearest point projection is nonexpansive (1-lipschitzian). We observed that, the natural question: Are there nonex- pansive projections Q : H! C other than P? is neglected in the literature. Also, the answer is not often present in the folklore of the Hilbert space theory. We provide here the answer and discuss some facts connected with the subject.

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold Cartesian product of the max-plus semiring: It is known that one can separate a vector from a closed subsemimodule that does not contain it. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem.

A critical global convergence analysis of recurrent neural networks with general projection mappings

In this paper, we present some general analysis on global convergence of the recurrent neural networks (RNNs) with projection mappings in the critical case when M ( L , Γ ) , a matrix related to the weight matrices and the activation mappings of the networks, is nonnegative definite for some positive diagonal matrix Γ . Considerable stability results have been obtained for the RNNs in the noncritical case when M ( L , Γ ) is positive definite. In contrast, only a few conclusions have been conducted under the critical conditions. Comparing with the existing critical studies, the present critical stability results in this paper require no additional assumption on the weight matrices, can be applied to the RNNs with general projection mappings other than nearest point projection mappings, and can serve for both two fundamental RNN models. The results established for several typical RNN models unify, sharpen or generalize most of the existing stability assertions. Two examples are given to show both theoretical importance and practical feasibility of the critical results obtained.

Implicit iteration process of nonexpansive non‐self‐mappings

Suppose C is a nonempty closed convex subset of real Hilbert space H. Let T : C → H be a nonexpansive non‐self‐mapping and P is the nearest point projection of H onto C. In this paper, we study the convergence of the sequences {xn}, {yn}, {zn} satisfying xn = (1 − αn)u + αnT[(1 − βn)xn + βnTxn], yn = (1 − αn)u + αnPT[(1 − βn)yn + βnPTyn], and zn = P[(1 − αn)u + αnTP[(1 − βn)zn + βnTzn]], where {αn}⊆(0, 1), 0 ≤ βn ≤ β &lt; 1 and αn → 1 as n → ∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.

An approximation of solutions of variational inequalities

We use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.

A Dual Approach to Linear Inverse Problems with Convex Constraints

A simple constraint qualification is developed and used to derive an explicit solution to a constrained optimization problem in Hilbert space. A finite parameterization is obtained for the minimum norm element in the intersection of a linear variety of finite co-dimension and a closed convex constraint set. The result extends previous duality theorems for convex cone set constraints. A fixed point iteration is presented for computing the parameters and yields a least-squares solution when the variety and constraint set have empty intersection. Proofs rely on nearest-point projections onto convex sets and the properties of monotone, firmly nonexpansive, and averaged mappings.