Understanding the Convergence of the Preconditioned PDHG Method: A View of Indefinite Proximal ADMM

Ma, Yumin; Cai, Xingju; Jiang, Bo; Han, Deren

doi:10.1007/s10915-023-02105-9

Cited by 6 publications

(6 citation statements)

References 43 publications

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“…The primal-dual hybrid gradient (PDHG) method has continued its success in solving nonlinear SPP (see, e.g. [52][53][54][55][56] and references therein). For instance, given an initial point (x 0 , y 0 ), the recursion of PDHG for solving nonlinear SPP reads…”

Section: Solver For Parametric Bd Model (17)mentioning

confidence: 99%

Generalized variational framework with minimax optimization for parametric blind deconvolution

Cao,

Han,

Wang

et al. 2024

Inverse Problems

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Blind deconvolution (BD), which aims to separate unknown colvolved signals, is a fundamental problem in signal processing. Due to the ill-posedness and underdetermination of the convolution system, it is a challenging nonlinear inverse problem. This paper is devoted to the algorithmic studies of parametric BD, which is typically applied to recover images from ad hoc optical modalities. We propose a generalized variational framework for parametric BD with various priors and potential functions. By using the conjugate theory in convex analysis, the framework can be cast into a nonlinear saddle point problem. We employ the recent advances in minimax optimization to solve the parametric BD by the nonlinear primal-dual hybrid gradient method, with all subproblems admitting closed-form solutions. Numerical simulations on synthetic and real datasets demonstrate the compelling performance of the minimax optimization approach for solving parametric BD.

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Section: Solver For Parametric Bd Model (17)mentioning

confidence: 99%

Generalized variational framework with minimax optimization for parametric blind deconvolution

Cao,

Han,

Wang

et al. 2024

Inverse Problems

Self Cite

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“…where f : H → (−∞, ∞] and g : G → (−∞, ∞] denote proper lower semicontinuous convex functions, L : H → G and J : K → G are linear continuous operators, and G, H and K are real Hilbert spaces. Unlike the indefinite proximal ADMM (iPADMM) proposed in [32,34], NP-ADMM can handle more general situations where both L and J are not Id, meaning that NP-ADMM can solve more general problems than iPADMM.…”

Section: Algorithmmentioning

confidence: 99%

“…Next, for the more general case where h * is not necessarily linear, the global convergence of PDHG in finite-dimensional Hilbert spaces can be demonstrated under condition (1.11) by Li and Yan [31]. The convergence of P-PDHG in finitedimensional Hilbert spaces was proved by Jiang et al [29] and Ma et al [34] under two conditions similar to (1.11).…”

Section: Introductionmentioning

confidence: 96%

“…Motivation and our contribution. It is widely recognized that the convergence conditions of P-PDHG can be enhanced [29,34], and SDR can be reduced to P-PDHG for problem (1.2) [6]. This leads to some natural questions: Can the convergence condition of SDR be improved?…”

Section: Introductionmentioning

confidence: 99%

“…Our main contributions are as follows. Building upon the insights from [29,31,34], we refine the convergence condition (1.13) of SDR to…”

Section: Introductionmentioning

confidence: 99%

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Convergence Analysis of Split-Douglas-Rachford Algorithm and a Novel Preconditioned ADMM with an Improved Condition

Maoran Wang,

Xingju Cai,

Yongxin Chen

2024

NMTMA

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For the primal-dual monotone inclusion problem, the split-Douglas-Rachford (SDR) algorithm is a well-known first-order splitting method. Similar to other first-order algorithms, the efficiency of SDR is greatly influenced by its step parameters. Therefore, expanding the range of stepsizes may lead to improved numerical performance. In this paper, we prove that the stepsize range of SDR can be expanded based on a series of properties of the product Hilbert space. Moreover, we present a concise counterexample to illustrate that the newly proposed stepsize range cannot be further enhanced. Furthermore, to bridge the theoretical gap in the convergence rate of SDR, we analyze the descent of SDR's fixed point residuals and provide the first proof of a sublinear convergence rate for the fixed point residuals. As an application, we utilize SDR to solve separable convex optimization problems with linear equality constraints and develop a novel preconditioned alternating direction method of multipliers (NP-ADMM). This method can handle scenarios where two linear operators are not identity operators. We propose strict convergence conditions and convergence rates for the preconditioned primal-dual split (P-PDS) method and NP-ADMM by demonstrating the relationship between SDR, P-PDS, and NP-ADMM. Finally, we conduct four numerical experiments to verify the computational efficiency of these algorithms and demonstrate that the performance of these algorithms has been significantly improved with the improved conditions.

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Indefinite Linearized Augmented Lagrangian Method for Convex Programming with Linear Inequality Constraints

He,

Xu,

Yuan

2024

Vietnam J. Math.

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Understanding the Convergence of the Preconditioned PDHG Method: A View of Indefinite Proximal ADMM

Cited by 6 publications

References 43 publications

Generalized variational framework with minimax optimization for parametric blind deconvolution

Generalized variational framework with minimax optimization for parametric blind deconvolution

Convergence Analysis of Split-Douglas-Rachford Algorithm and a Novel Preconditioned ADMM with an Improved Condition

Indefinite Linearized Augmented Lagrangian Method for Convex Programming with Linear Inequality Constraints

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