On The Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces

Butnariu, Dan; Censor, Yair; Gurfil, Pini; Hadar, Ethan

doi:10.1137/070689127

Cited by 31 publications

(14 citation statements)

References 35 publications

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“…When each set C j is linear (i.e., hyperplanes or half-spaces) or otherwise "simple" to orthogonally project onto (like balls), then there is no advantage in using subgradient projections. But in other cases the subgradient projections are easier to compute than orthogonal projections since they do not call for the, computationally demanding, inner-loop of least Euclidean distance minimization, but rather employ the "subgradient projection" which is merely a step in the negative direction of a calculable subgradient of g j at the current iteration; see, e.g., [22,34,37,74]. For a general review on projection algorithms for the CFP see [8] and consult the recent work [29].…”

Section: Subgradient Projection Methodsmentioning

confidence: 99%

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Censor

Reem

2014

Math. Program.

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The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and nonconvex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

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Section: Subgradient Projection Methodsmentioning

confidence: 99%

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Censor

Reem

2014

Math. Program.

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“…Hence, the goal of a CFP is to identify a point inside the intersection of a collection of closed convex sets in a Euclidean (or, in general, Hilbert) space. In feasibility problems, a commonly pursued approach is to perform projections onto the individual constraint sets in a sequential manner, rather than projecting onto their intersection due to analytical intractability [44]. The work in [27] formulates the problem of acoustic source localization as a CFP and employs the well-known projections onto convex sets (POCS) technique for convergence to true source locations.…”

Section: B Literature Survey On Set-theoretic Estimationmentioning

confidence: 99%

“…In this section, we design iterative subgradient projections based algorithms to solve Problem 1. The idea of using subgradient projections is to approach a convex set defined as a lower contour set of a convex/quasiconvex function by moving in the direction that decreases the value of that function at each iteration, i.e., in the opposite direction of the subgradient of the function at the current iterate [44], [62]. First, the definition of the gradient projector is presented as follows:…”

Section: Gradient Projections Algorithmsmentioning

confidence: 99%

“…To analyze (51) and (52), assume that Algorithm 2 requires O(N 2 ) trials for determining a non-negative integer m. At each trial, (44) requires O(M ) operations, where M is the number of functions at the input of Algorithm 2. Then, the computational complexity of step size selection in (51) and (52)…”

Section: ) Complexity Analysis For Algorithmmentioning

confidence: 99%

“…On the other hand, Assumption A2 represents a realistic scenario through the Armijo rule in (43) and (44), which ensures a certain level of decline in the Lambertian functions at each iteration and generates a nonincreasing sequence of step sizes.…”

Section: Convergence Analysismentioning

confidence: 99%

See 2 more Smart Citations

Cooperative Localization in Hybrid Infrared/Visible Light Networks: Theoretical Limits and Distributed Algorithms

Keskin

Erdem

Gezici

2019

IEEE Trans. on Signal and Inf. Process. over Networks

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Light emitting diode (LED) based visible light positioning networks can provide accurate location information in environments where the global positioning system (GPS) suffers from severe signal degradation and/or cannot achieve high precision, such as indoor scenarios. In this paper, we propose to employ cooperative localization for hybrid infrared/visible light networks that involve multiple LED transmitters having known locations (e.g., on the ceiling) and visible light communication (VLC) units equipped with both LEDs and photodetectors (PDs) for the purpose of cooperation. In the considered scenario, downlink transmissions from LEDs on the ceiling to VLC units occur via visible light signals, while the infrared spectrum is utilized for deviceto-device communications among VLC units. First, we derive the Cramér-Rao lower bound and the maximum likelihood estimator (MLE) for the localization of VLC units in the proposed cooperative scenario. To tackle the nonconvex structure of the MLE, we adopt a set-theoretic approach by formulating the problem of cooperative localization as a quasiconvex feasibility problem, where the aim is to find a point inside the intersection of convex constraint sets constructed as the sublevel sets of quasiconvex functions resulting from the Lambertian formula. Next, we devise two feasibility-seeking algorithms based on iterative gradient projections to solve the feasibility problem. Both algorithms are amenable to distributed implementation, thereby avoiding high-complexity centralized approaches. Capitalizing on the concept of quasi-Fejér convergent sequences, we carry out a formal convergence analysis to prove that the proposed algorithms converge to a solution of the feasibility problem in the consistent case. Numerical examples illustrate the improvements in localization performance achieved via cooperation among VLC units and evidence the convergence of the proposed algorithms to true VLC unit locations in both the consistent and inconsistent cases.

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Introduction

Cegielski

2012

Lecture Notes in Mathematics

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On The Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces

Cited by 31 publications

References 35 publications

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Cooperative Localization in Hybrid Infrared/Visible Light Networks: Theoretical Limits and Distributed Algorithms

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