scholarly journals Circumcentering approximate reflections for solving the convex feasibility problem

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
G. H. M. Araújo ◽  
R. Arefidamghani ◽  
R. Behling ◽  
Y. Bello-Cruz ◽  
A. Iusem ◽  
...  
Keyword(s):  
Convex Sets ◽  
Classical Method ◽  
Computational Cost ◽  
Linear Convergence ◽  
Feasibility Problem ◽  
Reflection Method ◽  
Convex Feasibility ◽  
Method Of Alternating Projections ◽  
Low Computational Cost ◽  
Approximate Version

AbstractThe circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP), and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

scholarly journals A variational approach to the alternating projections method

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Convex Sets ◽  
Nonempty Intersection ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Limit Sets ◽  
Von Neumann ◽  
Convex Feasibility ◽  
Starting Point ◽  
Method Of Alternating Projections ◽  
Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

scholarly journals Stochastic quasi-subgradient method for stochastic quasi-convex feasibility problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Gang Li ◽  
Minghua Li ◽  
Yaohua Hu
Keyword(s):  
Convex Functions ◽  
Computational Cost ◽  
Subgradient Method ◽  
Feasibility Problem ◽  
Rate Theory ◽  
Physical Sciences ◽  
Index Set ◽  
Convex Feasibility ◽  
Low Computational Cost ◽  
Random Control

<p style='text-indent:20px;'>The feasibility problem is at the core of the modeling of many problems in various disciplines of mathematics and physical sciences, and the quasi-convex function is widely applied in many fields such as economics, finance, and management science. In this paper, we consider the stochastic quasi-convex feasibility problem (SQFP), which is to find a common point of infinitely many sublevel sets of quasi-convex functions. Inspired by the idea of a stochastic index scheme, we propose a stochastic quasi-subgradient method to solve the SQFP, in which the quasi-subgradients of a random (and finite) index set of component quasi-convex functions at the current iterate are used to construct the descent direction at each iteration. Moreover, we introduce a notion of Hölder-type error bound property relative to the random control sequence for the SQFP, and use it to establish the global convergence theorem and convergence rate theory of the stochastic quasi-subgradient method. It is revealed in this paper that the stochastic quasi-subgradient method enjoys both advantages of low computational cost requirement and fast convergence feature.</p>

scholarly journals Applications of Fixed-Point and Optimization Methods to the Multiple-Set Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yonghong Yao ◽  
Rudong Chen ◽  
Giuseppe Marino ◽  
Yeong Cheng Liou
Keyword(s):  
Fixed Point ◽  
Inverse Problems ◽  
Linear Operator ◽  
Linear Transformation ◽  
Convex Sets ◽  
Split Feasibility Problem ◽  
Optimization Methods ◽  
Feasibility Problem ◽  
Convex Feasibility ◽  
Split Feasibility

The multiple-set split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. It generalizes the convex feasibility problem as well as the two-set split feasibility problem. In this paper, we will review and report some recent results on iterative approaches to the multiple-set split feasibility problem.

scholarly journals A New Generalized Projection and Its Application to Acceleration of Audio Declipping

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 105
Author(s):  
Pavel Rajmic ◽  
Pavel Záviška ◽  
Vítězslav Veselý ◽  
Ondřej Mokrý
Keyword(s):  
Signal Processing ◽  
Convex Optimization ◽  
Linear Operator ◽  
Explicit Formula ◽  
Convex Sets ◽  
State Of The Art ◽  
Computational Cost ◽  
The Other ◽  
Speed Up ◽  
Low Computational Cost

In convex optimization, it is often inevitable to work with projectors onto convex sets composed with a linear operator. Such a need arises from both the theory and applications, with signal processing being a prominent and broad field where convex optimization has been used recently. In this article, a novel projector is presented, which generalizes previous results in that it admits to work with a broader family of linear transforms when compared with the state of the art but, on the other hand, it is limited to box-type convex sets in the transformed domain. The new projector is described by an explicit formula, which makes it simple to implement and requires a low computational cost. The projector is interpreted within the framework of the so-called proximal splitting theory. The convenience of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two.

scholarly journals Iterative algorithms with seminorm-induced oblique projections

2003 ◽  
Vol 2003 (7) ◽  
pp. 387-406 ◽  
Author(s):  
Yair Censor ◽  
Tommy Elfving
Keyword(s):  
Existence And Uniqueness ◽  
Convex Sets ◽  
Iterative Algorithms ◽  
Convex Feasibility Problem ◽  
Feasibility Problem ◽  
Proximity Function ◽  
Oblique Projections ◽  
Convex Feasibility ◽  
Definition Of ◽  
Inconsistent Case

A definition of oblique projections onto closed convex sets that use seminorms induced by diagonal matrices which may have zeros on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal matrices involved. A block-iterative algorithmic scheme for solving the convex feasibility problem, employing seminorm-induced oblique projections, is constructed and its convergence for the consistent case is established. The fully simultaneous algorithm converges also in the inconsistent case to the minimum of a certain proximity function.

scholarly journals Regularity and Stability for a Convex Feasibility Problem

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Best Approximation ◽  
Convex Sets ◽  
Convex Feasibility Problem ◽  
Feasibility Problem ◽  
Convex Feasibility ◽  
Starting Point

AbstractLet us consider two sequences of closed convex sets {An} and {Bn} converging with respect to the Attouch-Wets convergence to A and B, respectively. Given a starting point a0, we consider the sequences of points obtained by projecting onto the “perturbed” sets, i.e., the sequences {an} and {bn} defined inductively by $b_{n}=P_{B_{n}}(a_{n-1})$ b n = P B n ( a n − 1 ) and $a_{n}=P_{A_{n}}(b_{n})$ a n = P A n ( b n ) . Suppose that A ∩ B is bounded, we prove that if the couple (A,B) is (boundedly) regular then the couple (A,B) is d-stable, i.e., for each {an} and {bn} as above we have dist(an,A ∩ B) → 0 and dist(bn,A ∩ B) → 0. Similar results are obtained also in the case A ∩ B = ∅, considering the set of best approximation pairs instead of A ∩ B.

scholarly journals Computing Expectiles Using k-Nearest Neighbours Approach

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 645
Author(s):  
Muhammad Farooq ◽  
Sehrish Sarfraz ◽  
Christophe Chesneau ◽  
Mahmood Ul Hassan ◽  
Muhammad Ali Raza ◽  
...  
Keyword(s):  
Computational Cost ◽  
Real Life ◽  
Distance Measures ◽  
Computational Time ◽  
High Dimensional ◽  
Test Error ◽  
Nearest Neighbours ◽  
Comparable Performance ◽  
Asymmetric Least Squares ◽  
Low Computational Cost

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-kNN, is proposed for computing expectiles. Firstly, the effect of various distance measures on ex-kNN in terms of test error and computational time is evaluated. It is found that Canberra, Lorentzian, and Soergel distance measures lead to minimum test error, whereas Euclidean, Canberra, and Average of (L1,L∞) lead to a low computational cost. Secondly, the performance of ex-kNN is compared with existing packages er-boost and ex-svm for computing expectiles that are based on nine real life examples. Depending on the nature of data, the ex-kNN showed two to 10 times better performance than er-boost and comparable performance with ex-svm regarding test error. Computationally, the ex-kNN is found two to five times faster than ex-svm and much faster than er-boost, particularly, in the case of high dimensional data.

scholarly journals Directional TGV-Based Image Restoration under Poisson Noise

2021 ◽  
Vol 7 (6) ◽  
pp. 99
Author(s):  
Daniela di Serafino ◽  
Germana Landi ◽  
Marco Viola
Keyword(s):  
Computed Tomography ◽  
Image Restoration ◽  
Gaussian Noise ◽  
Computational Cost ◽  
Data Fitting ◽  
Poisson Noise ◽  
Glass Fibres ◽  
Leibler Divergence ◽  
Generalized Variation ◽  
Low Computational Cost

We are interested in the restoration of noisy and blurry images where the texture mainly follows a single direction (i.e., directional images). Problems of this type arise, for example, in microscopy or computed tomography for carbon or glass fibres. In order to deal with these problems, the Directional Total Generalized Variation (DTGV) was developed by Kongskov et al. in 2017 and 2019, in the case of impulse and Gaussian noise. In this article we focus on images corrupted by Poisson noise, extending the DTGV regularization to image restoration models where the data fitting term is the generalized Kullback–Leibler divergence. We also propose a technique for the identification of the main texture direction, which improves upon the techniques used in the aforementioned work about DTGV. We solve the problem by an ADMM algorithm with proven convergence and subproblems that can be solved exactly at a low computational cost. Numerical results on both phantom and real images demonstrate the effectiveness of our approach.

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