A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization

Zhang, Tao; Shen, Zhengwei

doi:10.1186/s13660-019-2080-0

Cited by 12 publications

(18 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While the convergence of ADMM is well-known for convex problems [18], convergence can also be proven in some other scenarios that often necessitate special proofs, such as classes of weakly convex problems [126], certain non-convex ADMMs with linear constraints [124,116,47], non-convex and non-linear, but equality-constrained problems [114], and bilinear constraints [123]. For specific non-linear non-convex ADMMs, specialized proofs exist [36,121].…”

Section: Alternating Direction Methods Of Multipliersmentioning

confidence: 99%

See 1 more Smart Citation

A Splitting Scheme for Flip-Free Distortion Energies

Stein¹,

Li²,

Solomon³

2021

Preprint

View full text Add to dashboard Cite

We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel Alternating Direction Method of Multipliers (ADMM) algorithm to deal with the non-convex, non-smooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.

show abstract

Section: Alternating Direction Methods Of Multipliersmentioning

confidence: 99%

“…Figure 3 shows how the gradient of f evolves for a few UV parametrization experiments. It is worthwhile mentioning that a boundedness-type condition is standard in the non-convex ADMM literature [123,126,36]. This is because the energy function f is not globally, but only locally Lipschitz continuous.…”

Section: Alternating Direction Methods Of Multipliersmentioning

confidence: 99%

A Splitting Scheme for Flip-Free Distortion Energies

Stein¹,

Li²,

Solomon³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…As f NNG is not strictly convex, the convergence in the context of general forward-backward splitting approaches is not straightforward. However, the convergence of general semi-convex (weakly convex) penalty functions inside an iterative shrinkage/thresholding algorithm (ISTA) has been shown in, 43,45 inside an Douglas-Rachford splitting in 46 and inside the ADMM algorithm recently in 47 Author to whom correspondence should be addressed. Electronic mail: florian.lieb@th-ab.de…”

Section: Conflict Of Interestmentioning

confidence: 99%

A wavelet‐based sparse row‐action method for image reconstruction in magnetic particle imaging

Lieb

Knopp

2021

Medical Physics

View full text Add to dashboard Cite

Purpose: Magnetic particle imaging (MPI) is a preclinical imaging technique capable of visualizing the spatio-temporal distribution of magnetic nanoparticles. The image reconstruction of this fast and dynamic process relies on efficiently solving an ill-posed inverse problem. Current approaches to reconstruct the tracer concentration from its measurements are either adapted to image characteristics of MPI but suffer from higher computational complexity and slower convergence or are fast but lack in the image quality of the reconstructed images. Methods: In this work we propose a novel MPI reconstruction method to combine the advantages of both approaches into a single algorithm. The underlying sparsity prior is based on an undecimated wavelet transform and is integrated into a fast row-action framework to solve the corresponding MPI minimization problem. Results: Its performance is numerically evaluated against a classical FISTA (Fast Iterative Shrinkage-Thresholding Algorithm) approach on simulated and real MPI data. The experimental results show that the proposed method increases image quality with significantly reduced computation times. Conclusions: In comparison to state-of-the-art MPI reconstruction methods, our approach shows better reconstruction results and at the same time accelerates the convergence rate of the underlying row-action algorithm.

show abstract

“…Another issue is the convergence of f NNG in the context of general forward-backward splitting approaches as it is not strictly convex. However, the convergence of general semi-convex (weakly-convex) penalty functions inside an iterative shrinkage/thresholding algorithm (ISTA) has been shown in [19,33], inside an Douglas-Rachford splitting in [34] and inside the alternating direction method of multipliers (ADMM) recently in [35].…”

Section: Note That In Practice γmentioning

confidence: 99%

A Wavelet Based Sparse Row-Action Method for Image Reconstruction in Magnetic Particle Imaging

Lieb,

Knopp

2020

Preprint

View full text Add to dashboard Cite

Magnetic Particle Imaging (MPI) is a preclinical imaging technique capable of visualizing the spatiotemporal distribution of magnetic nanoparticles. The image reconstruction of this fast and dynamic process relies on efficiently solving an ill-posed inverse problem. Current approaches to reconstruct the tracer concentration from its measurements are either adapted to image characteristics of MPI but suffer from higher computational complexity and slower convergence or are fast but lack in the image quality of the reconstructed images. In this work we propose a novel MPI reconstruction method to combine the advantages of both approaches into a single algorithm. The underlying sparsity prior is based on an undecimated wavelet transform and is integrated into a fast row-action framework to solve the corresponding MPI minimization problem. Its performance is numerically evaluated against a classical FISTA approach on simulated and real MPI data. We also compare the results to the state-of-the-art MPI reconstruction methods. In all cases, our approach shows better reconstruction results and at the same time accelerates the convergence rate of the underlying row-action algorithm.

show abstract

A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization

Cited by 12 publications

References 25 publications

A Splitting Scheme for Flip-Free Distortion Energies

A Splitting Scheme for Flip-Free Distortion Energies

A wavelet‐based sparse row‐action method for image reconstruction in magnetic particle imaging

A Wavelet Based Sparse Row-Action Method for Image Reconstruction in Magnetic Particle Imaging

Contact Info

Product

Resources

About