Analysis of the Convergence Rate for the Cyclic Projection Algorithm Applied to Basic Semialgebraic Convex Sets

Borwein, Jonathan M.; Li, Guoyin; Yao, Liangjin

doi:10.1137/130919052

Cited by 52 publications

(54 citation statements)

References 32 publications

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“…Notable methods of this kind include the alternating projection algorithm [5,29,17], the Douglas-Rachford (DR) algorithm [37,43,38], and many extensions and variants [19,20,10,48]. Even in settings without convexity [1,2,3,16,41,42], such methods remain a popular choice due largely to their simplicity, ease-of-implementation and relatively -often surprisingly -good performance.…”

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confidence: 99%

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Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Borwein¹,

Li²,

Tam³

2017

SIAM J. Optim.

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In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Hölder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for many important iterative methods in solving broad mathematical problems such as convex feasibility problems and variational inequality problems. These include Krasnoselskii-Mann iterations, the cyclic projection algorithm, forward-backward splitting and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Hölder regularity properties are automatically satisfied, from which sublinear convergence follows.

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confidence: 99%

“…In the following lemma, B(n) denotes the central binomial coefficient with respect to n given by n [n/2] where [ · ] denotes the integer part of a real number. Lemma 2.11 (Hölder regularity of basic semi-algebraic convex sets in R n [17]). Let C i be basic convex semi-algebraic sets in R n given by C i = {x ∈ R n | g ij (x) ≤ 0, j = 1, .…”

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confidence: 99%

Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Borwein¹,

Li²,

Tam³

2017

SIAM J. Optim.

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“…It is worth notice that error bound results with explicit exponents are indeed important for both theory and applications since they can be used, e.g., to establish explicit convergence rates of the proximal point algorithm as demonstrated in [9], [37], [38].…”

Section: Cd(x S) ≤ [F (X)]mentioning

confidence: 99%

Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems

Đinh

Phạm

2017

Acta Math Vietnam

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“…The APM has been the subject of several works in itself [15,18,19]. The authors of [20] provide general results on the convergence rate of APM for semi-algebraic sets. They show that the convergence is geometric for polyhedra.…”

Section: Introductionmentioning

confidence: 99%

A Privacy-preserving Disaggregation Algorithm for Non-intrusive Management of Flexible Energy

Jacquot¹,

Beaude²,

Benchimol³

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimizing a global, possibly non-convex, cost while satisfying the agents' constraints. We focus on the practical case of the management of energy consumption flexibilities by the operator of a microgrid. This paper provides a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we maintain a global allocation of resources, and gradually disaggregate this allocation to enforce the satisfaction of private contraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projections method à la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Our theoretical and numerical results show that the method scales well as the number of agents gets large, and thus can be used to solve the allocation problem in high dimension, while addressing privacy issues.

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Analysis of the Convergence Rate for the Cyclic Projection Algorithm Applied to Basic Semialgebraic Convex Sets

Cited by 52 publications

References 32 publications

Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems

A Privacy-preserving Disaggregation Algorithm for Non-intrusive Management of Flexible Energy

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