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 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 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 Global convergence of the alternating projection method for the Max-Cut relaxation problem

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 737-746
Author(s):  
Suliman Al-Homidana
Keyword(s):  
Convex Sets ◽  
Alternating Projections ◽  
Correlation Matrices ◽  
Von Neumann ◽  
Relaxation Problem ◽  
Max Cut Problem ◽  
Np Hard Problem ◽  
Convergent Method ◽  
Alternating Projection Method ◽  
Alternating Projections Method

The Max-Cut problem is an NP-hard problem [15]. Extensions of von Neumann?s alternating projections method permit the computation of proximity projections onto convex sets. The present paper exploits this fact by constructing a globally convergent method for the Max-Cut relaxation problem. The feasible set of this relaxed Max-Cut problem is the set of correlation matrices.

CONVERGENCE OF MANN’S ALTERNATING PROJECTIONS IN CAT() SPACES

2018 ◽  
Vol 98 (1) ◽  
pp. 134-143 ◽  
Author(s):  
BYOUNG JIN CHOI
Keyword(s):  
Strong Convergence ◽  
Projection Method ◽  
Convex Feasibility Problem ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Normed Spaces ◽  
Closed Sets ◽  
Convex Feasibility ◽  
Alternating Projection ◽  
Iterative Projection Method

We study the convex feasibility problem in$\text{CAT}(\unicode[STIX]{x1D705})$spaces using Mann’s iterative projection method. To do this, we extend Mann’s projection method in normed spaces to$\text{CAT}(\unicode[STIX]{x1D705})$spaces with$\unicode[STIX]{x1D705}\geq 0$, and then we prove the$\unicode[STIX]{x1D6E5}$-convergence of the method. Furthermore, under certain regularity or compactness conditions on the convex closed sets, we prove the strong convergence of Mann’s alternating projection sequence in$\text{CAT}(\unicode[STIX]{x1D705})$spaces with$\unicode[STIX]{x1D705}\geq 0$.

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 Conformal radiotherapy computation by the method of alternating projections onto convex sets

1997 ◽  
Vol 42 (6) ◽  
pp. 1065-1086 ◽  
Author(s):  
Shinhak Lee ◽  
Paul S Cho ◽  
Robert J Marks ◽  
Seho Oh
Keyword(s):  
Convex Sets ◽  
Conformal Radiotherapy ◽  
Alternating Projections ◽  
Method Of Alternating Projections

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.

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