A Proximal Minimization Algorithm for Structured Nonconvex and Nonsmooth Problems

Boţ, Radu Ioan; Csetnek, Ernö Robert; Nguyen, Dang-Khoa

doi:10.1137/18m1190689

Cited by 28 publications

(5 citation statements)

References 26 publications

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“…There is in fact a large number of nonconvex ADMM algorithms that work with nonsmooth, nonconvex composite h(K • ). For example, (Bot et al 2019) considered the following problem model…”

Section: Discussionmentioning

confidence: 99%

“…A full-splitting, ADMM algorithm was proposed in (Bot et al 2019), exploiting the proximal mapping of g, h, and the linear operator K, and the gradient ∇f (x, y), separately. The convergence of the proposed algorithm requires that K is full row rank (surjective), a common assumption shared by other ADMM algorithms for dealing with nonsmooth composite functions, see e.g., Pong 2015, Sun et al 2019).…”

Section: Discussionmentioning

confidence: 99%

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Convex optimization algorithms in medical image reconstruction—in the age of AI

Xu¹,

Noo²

2022

Phys. Med. Biol.

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The past decade has seen the rapid growth of model based image reconstruction (MBIR) algorithms, which are often applications or adaptations of convex optimization algorithms from the optimization community. We review some state-of-the-art algorithms that have enjoyed wide popularity in medical image reconstruction, emphasize known connections between different algorithms, and discuss practical issues such as computation and memory cost. More recently, deep learning (DL) has forayed into medical imaging, where the latest development tries to exploit the synergy between DL and MBIR to elevate the MBIR's performance. We present existing approaches and emerging trends in DL-enhanced MBIR methods, with particular attention to the underlying role of convexity and convex algorithms on network architecture. We also discuss how convexity can be employed to improve the generalizability and representation power of DL networks in general.

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“…There is in fact a large number of nonconvex ADMM algorithms that work with nonsmooth, nonconvex composite h(K • ). For example, (Bot et al 2019) considered the following problem model…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Convex optimization algorithms in medical image reconstruction—in the age of AI

Xu¹,

Noo²

2022

Phys. Med. Biol.

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“…Definition 2.5. (see [34,38], KL property) Let f : R n → R ∪ {+∞} be a proper lower semicontinuous function. If there exist η ∈ (0 , +∞], a neighborhood U of x * and a continuous concave function ϕ ∈ Φ η such that for all x…”

Section: Preliminariesmentioning

confidence: 99%

Multi-block relaxed-dual linear inertial ADMM algorithm for nonconvex and nonsmooth problems with nonseparable structures

Dang,

Chen,

Gao

2024

Numer Algor

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In this paper, we propose a multi-block relaxed-dual linear inertial alternating direction method of multipliers algorithm (MBRD-LIADMM) for solving the nonconvex and nonsmooth multi-block optimization problems with nonseparable structures, which combines the inertial technology, the relaxed-dual term and linear ADMM. The paper not only gives the simple condition to ensure the boundedness of the iterative sequence generated by the algorithm, but also proves the global convergence under the help of Kurdyka-Łojasiewicz property, which means that the generated sequence converges to a stationary point of the problem. Finally, two experiments are carried out to demonstrate the feasibility and effectiveness of the proposed algorithm.

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“…For non-convex separable objective functions with applications to consensus problem, the authors of [33] studied the properties of ADMM. Other related work can be found in for instance [38,51]. It should be noted that all these works are obtained for deterministic ADMM, i.e.…”

Section: Related Workmentioning

confidence: 99%

A stochastic alternating direction method of multipliers for non-smooth and non-convex optimization

Bian¹,

Liang²,

Zhang³

2021

Inverse Problems

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Alternating direction method of multipliers (ADMM) is a popular first-order method owing to its simplicity and efficiency. However, similar to other proximal splitting methods, the performance of ADMM degrades significantly when the scale of optimization problems to solve becomes large. In this paper, we consider combining ADMM with a class of variance-reduced stochastic gradient estimators for solving large-scale non-convex and non-smooth optimization problems. Global convergence of the generated sequence is established under the additional assumption that the object function satisfies Kurdyka-Łojasiewicz property. Numerical experiments on graph-guided fused lasso and computed tomography are presented to demonstrate the performance of the proposed methods.

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A Proximal Minimization Algorithm for Structured Nonconvex and Nonsmooth Problems

Cited by 28 publications

References 26 publications

Convex optimization algorithms in medical image reconstruction—in the age of AI

Convex optimization algorithms in medical image reconstruction—in the age of AI

Multi-block relaxed-dual linear inertial ADMM algorithm for nonconvex and nonsmooth problems with nonseparable structures

A stochastic alternating direction method of multipliers for non-smooth and non-convex optimization

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