First-Order Primal–Dual Methods for Nonsmooth Non-convex Optimisation

Valkonen, Tuomo

doi:10.1007/978-3-030-03009-4_93-1

Cited by 5 publications

(2 citation statements)

References 57 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

View full text Add to dashboard Cite

show abstract

“…This choice is not practical in non-Hilbert spaces, as a crucial (Pythagoras') three-point identity does not hold. Such an identity, however, holds for Bregman divergences [6]; see [34]. The Frank-Wolfe method can, moreover, be seen a forward-backward method for a modified but equivalent problem 1 with zero as the proximal penalty.…”

Section: Introductionmentioning

confidence: 99%

Proximal methods for point source localisation

Valkonen

2023

Journal of Nonsmooth Analysis and Optimization

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Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.

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An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function

Gao

Zhang

2023

Comput Optim Appl

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First-Order Primal–Dual Methods for Nonsmooth Non-convex Optimisation

Cited by 5 publications

References 57 publications

Generalized variational framework with minimax optimization for parametric blind deconvolution

Generalized variational framework with minimax optimization for parametric blind deconvolution

Proximal methods for point source localisation

An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function

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