Optimal convergence rates for generalized alternating projections

Fält, Mattias; Giselsson, Pontus

doi:10.1109/cdc.2017.8263980

Cited by 13 publications

(21 citation statements)

References 27 publications

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“…Substituting this into (24), we also obtain (25). Now, in view of Remark 3.5, 1 + 2γα + γ 2 ℓ 2 ≥ 1 + 2γα + γ 2 α 2 = (1 + γα) 2 > 0.…”

Section: Lemma 36 (Resolvents Of Lipschitz α-Monotone Operators) Letmentioning

confidence: 86%

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

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The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm has been well understood when the involved operators are monotone or strongly monotone, the convergence theory for weakly monotone settings is far from being complete. In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With appropriately chosen parameters, the algorithm converges globally to a fixed point from which we derive a solution of the problem. When one operator is Lipschitz continuous, we prove global linear convergence which sharpens recent known results. entire mathematical model. It is worth mentioning (see, e.g., [23]) that several splitting methods such as the method of partial inverses [35] and the alternating direction method of multipliers (ADMM) [25] can be written in the form of the DR algorithm, which itself can be transformed into the proximal point algorithm [34]. Other splitting schemes can be found in [13,14,16] and the references therein.When applied to two normal cone operators, the DR algorithm can be used to solve the feasibility problem of finding a common point of two sets. In this context, the DR algorithm possesses many good properties, for example, it finds a best approximation point when the intersection of sets is empty [3,5,8], it finds an exact solution after only a finite number of iterations under verifiable conditions [1,4,7], and it converges globally in some nonconvex settings [9,19] while converges locally with linear or sublinear rate under some regularity assumptions [11,29,33]. In the absence of constraint qualifications, [6] suggests that the DR algorithm outperforms the well-known method of alternating projections. In attempting to generalize the DR algorithm for feasibility problems, several parameters were added to its formulation [12,18,17,24]. In this case, one has the freedom to modify the parameters that are associated with the projections without giving up the solution. This approach is possible because the underlying normal cone operators have homogeneous values, which allows for scaling them independently. The situation changes completely when working with general problems where two involved operators may no longer have such homogeneity. In this case, a naive scaling may destroy the ability to solve the original problem. Therefore, we aim to overcome this hurdle by proposing an adaptive approach.The paper is devoted to the convergence analysis of the adaptive DR algorithm for finding a zero of the sum of α-and β-monotone operators, in which α-monotonicity is a unification of strong and weak monotonicity (see Definition 3.1). This situation arises in various important applications; see [27] for a brief discussion. The main contribution is summarized below.(R1) We incorporate parameters into the DR algorith...

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“…Substituting this into (24), we also obtain (25). Now, in view of Remark 3.5, 1 + 2γα + γ 2 ℓ 2 ≥ 1 + 2γα + γ 2 α 2 = (1 + γα) 2 > 0.…”

Section: Lemma 36 (Resolvents Of Lipschitz α-Monotone Operators) Letmentioning

confidence: 86%

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

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“…Firstly we demonstrate how, under the assumption of strong convexity and smoothness, PDMM is contractive over a certain subspace. We then show how, for such "partially contractive" operators, a global convergence bound can be found by linking PDMM with the generalised alternating method of projections (GAP) [44] allowing us to derive the aforementioned γ and .…”

Section: A a Primal Geometric Convergence Bound For Strongly Convex mentioning

confidence: 99%

“…While the "partially contractive" nature of the PDMM updates suggests its geometric convergence, it is unclear what this convergence rate may be. For this reason, in the following we derive a geometrically primal convergence bound by connecting two-step PDMM with the GAP algorithm [44].…”

Section: B Partially Contractive Nature Of Pdmm Over a Subspacementioning

confidence: 99%

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Derivation and Analysis of the Primal-Dual Method of Multipliers Based on Monotone Operator Theory

Sherson

Heusdens

Kleijn

2019

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

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In this paper we present a novel derivation for an existing node-based algorithm for distributed optimisation termed the primal-dual method of multipliers (PDMM). In contrast to its initial derivation, in this work monotone operator theory is used to connect PDMM with other first-order methods such as Douglas-Rachford splitting and the alternating direction method of multipliers thus providing insight to the operation of the scheme. In particular, we show how PDMM combines a lifted dual form in conjunction with Peaceman-Rachford splitting to remove the need for collaboration between nodes per iteration. We demonstrate sufficient conditions for strong primal convergence for a general class of functions while under the assumption of strong convexity and functional smoothness, we also introduce a primal geometric convergence bound. Finally we introduce a distributed method of parameter selection in the geometric convergent case, requiring only finite transmissions to implement regardless of network topology.Index Terms-Primal-Dual method of multipliers (PDMM), alternating direction method of multipliers (ADMM), distributed optimisation, monotone operator, optimisation over networks.

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“…Otherwise,

0 < σ_{β} < 1

. According to the generalized Rayleigh quotient, we can obtain from (A9) and (A12) that:

0 < β < 1

Thus, the iteration of the ISO-CA has linear convergence rates [32]. …”

mentioning

confidence: 99%

A Novel Method for Designing General Window Functions with Flexible Spectral Characteristics

Sun

Liu

Cai

et al. 2018

Sensors

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In the field of sensor signal processing, windows are time-/frequency-domain weighting functions that are widely applied to reduce the well-known Gibbs oscillations. Conventional methods generally control the spectral characteristics of windows by adjusting several of the parameters of closed-form expressions. Designers must make trade-offs among the mainlobe width (MW), the peak sidelobe level (PSL), and sometimes the sidelobe fall-off rate (SLFOR) of windows by carefully adjusting these parameters. Generally, not all sidelobes need to be suppressed in specified applications. In this paper, a novel method, i.e., the inverse of the shaped output using the cyclic algorithm (ISO-CA), for designing window functions with flexible spectral characteristics is proposed. Simulations are conducted to test the effectiveness, flexibility and versatility of the method. Some experiments based on real measured data are also presented to demonstrate the practicability. The results show that the window functions generated using the cyclic algorithm (CA) yield better performance overall than the windows of conventional methods, achieving a narrower MW, a lower PSL, and a controllable SLFOR. In addition, steerable sidelobes over specified regions can be acquired both easily and flexibly while maintaining the original properties of the initial window as much as possible.

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Optimal convergence rates for generalized alternating projections

Cited by 13 publications

References 27 publications

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Derivation and Analysis of the Primal-Dual Method of Multipliers Based on Monotone Operator Theory

A Novel Method for Designing General Window Functions with Flexible Spectral Characteristics

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