The regularity properties of a family of closed convex sets with nonempty intersection are investigated in the frame of a real Hilbert space. The significant role of these properties in solving convexfeasibility prob...
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The regularity properties of a family of closed convex sets with nonempty intersection are investigated in the frame of a real Hilbert space. The significant role of these properties in solving convex feasibility problems with projection algorithms is pointed out. (C) 2008 Elsevier Ltd. All rights reserved.
The weak and strong convergence of a sequence generated by a Mann-type iteration are investigated in the frame of a real Hilbert space. Some applications to the projection method for the convex feasibility problem are...
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The weak and strong convergence of a sequence generated by a Mann-type iteration are investigated in the frame of a real Hilbert space. Some applications to the projection method for the convex feasibility problem are given. (C) 2007 Elsevier B.V. All rights reserved.
The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets A and B in a normed space X. More generally, we can consider the problem of finding (if possible) two points ...
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The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets A and B in a normed space X. More generally, we can consider the problem of finding (if possible) two points in A and B, respectively, which minimize the distance between the sets. In the present paper, we study some stability properties for the convex feasibility problem: we consider two sequences of sets, each of them converging, with respect to a suitable notion of set convergence, respectively, to A and B. Under appropriate assumptions on the original problem, we ensure that the solutions of the perturbed problems converge to a solution of the original problem. We consider both the finite-dimensional and the infinite-dimensional case. Moreover, we provide several examples that point out the role of our assumptions in the obtained results.
Let us consider two sequences of closed convex sets {A(n)} and {B-n} converging with respect to the Attouch-Wets convergence to A and B, respectively. Given a starting point a(0), we consider the sequences of points o...
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Let us consider two sequences of closed convex sets {A(n)} and {B-n} converging with respect to the Attouch-Wets convergence to A and B, respectively. Given a starting point a(0), we consider the sequences of points obtained by projecting onto the "perturbed" sets, i.e., the sequences {a(n)} and {b(n)} defined inductively by b(n) = PBn (a(n-1)) and an = P-An (b(n)). Suppose that An 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 {a(n)} and {b(n)} as above we have dist(a(n), A boolean AND B) -> 0 and dist(bn, A boolean AND B) -> 0. Similar results are obtained also in the case A boolean AND B = 0, considering the set of best approximation pairs instead of A boolean AND B.
In this semi-tutorial paper, the positioning problem is formulated as a convex feasibility problem (CFP). To solve the CFP for non-cooperative networks, we consider the well-known projection onto convex sets (POCS) te...
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In this semi-tutorial paper, the positioning problem is formulated as a convex feasibility problem (CFP). To solve the CFP for non-cooperative networks, we consider the well-known projection onto convex sets (POCS) technique and study its properties for positioning. We also study outer-approximation (OA) methods to solve CFP problems. We then show how the POCS estimate can be upper bounded by solving a non-convex optimization problem. Moreover, we introduce two techniques based on OA and POCS to solve the CFP for cooperative networks and obtain two new distributed algorithms. Simulation results show that the proposed algorithms are robust against non-line-of-sight conditions.
In this paper, we consider a class of convex feasibility problem, which arises in quantum computation. Based on the matrix equation theory, the feasible sets are characterized by exploiting the special structure of th...
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In this paper, we consider a class of convex feasibility problem, which arises in quantum computation. Based on the matrix equation theory, the feasible sets are characterized by exploiting the special structure of the linear constraints, and its analytic expression is given. By making use of the nice structure properties and the KKT condition, we derive the projection formulas of a matrix onto the feasible sets. The relaxed alternating projection method is designed to solve the convex feasibility problem. Numerical experiments show that the new method is feasible and effective. (C) 2018 Elsevier B.V. All rights reserved.
In this paper, we present a successive centralization process for the circumcentered-reflection method with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a stan...
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In this paper, we present a successive centralization process for the circumcentered-reflection method with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Moreover, under additional smoothness assumptions on the target sets, we establish the superlinear convergence. Numerical experiments confirm the efficiency of our method.
In this paper, we present a variant of the circumcenter method for the convex feasibility problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex set...
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In this paper, we present a variant of the circumcenter method for the convex feasibility problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex sets with projections onto separating half-spaces, perturbed by positive exogenous parameters that decrease to zero along the iterations. If the perturbation parameters decrease slowly enough, such as the terms of a diverging series, finite convergence is achieved. To the best of our knowledge, this is the first circumcenter method for CFP that guarantees finite convergence.
The classical problem of finding a point in the intersection of countably many closed and convex sets in a Hilbert space is considered. Extrapolated iterations of convex combinations of approximate projections onto su...
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The classical problem of finding a point in the intersection of countably many closed and convex sets in a Hilbert space is considered. Extrapolated iterations of convex combinations of approximate projections onto subfamilies of sets are investigated to solve this problem. General hypotheses are made on the regularity of the sets and various strategies are considered to control the order in which the sets are selected. Weak and strong convergence results are established within this broad framework, which provides a unified view of projection methods for solving hilbertian convex feasibility problems.
The existing methods of projection for solving convex feasibility problem may lead to slow conver- gence when the sequences enter some narrow＂corridor＂ between two or more convex sets. In this paper, we apply a tech...
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The existing methods of projection for solving convex feasibility problem may lead to slow conver- gence when the sequences enter some narrow＂corridor＂ between two or more convex sets. In this paper, we apply a technique that may interrupt the monotonity of the constructed sequence to the sequential subgradient pro- jection algorithm to construct a nommonotonous sequential subgradient projection algorithm for solving convex feasibility problem, which can leave such corridor by taking a big step at different steps during the iteration. Under some suitable conditions, the convergence is *** also compare the numerical performance of the proposed algorithm with that of the monotonous algorithm by numerical experiments.
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