Recently, a golden ratio primal-dual algorithm (GRPDA) was proposed by Chang and Yang for solving structured convex optimization problems. It is a new adaptation of the classical Arrow-Hurwicz method by using a convex...
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Recently, a golden ratio primal-dual algorithm (GRPDA) was proposed by Chang and Yang for solving structured convex optimization problems. It is a new adaptation of the classical Arrow-Hurwicz method by using a convex combination step, instead of the widely adopted extrapolation technique. The convex combination step is determined by a parameter psi, which, to guarantee global convergence, is restricted to (1, (1 + root 5)/2]. In this paper, by carrying out a refined analysis, we expand this region to (1, 1 + root 3). Moreover, we establish ergodic sublinear convergence rate results based on function value residual and constraint violation of an equivalently reformulated constrained optimization problem, rather than the previously adopted so-called primal-dual gap function that could vanish at nonstationary points. For linear equality constrained and regularized least-squares problems, we further show that GRPDA and Chambolle-Pock's primal-dual algorithm are equivalent provided that some parameters are chosen properly. Finally, experimental results on the LASSO and the basis pursuit problems are presented to demonstrate the performance of GRPDA with psi being chosen in the expanded region.
We generalize the well-known primal-dual algorithm proposed by Chambolle and Pock for saddle point problems and relax the condition for ensuring its convergence. The relaxed convergence -guaranteeing condition is effe...
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We generalize the well-known primal-dual algorithm proposed by Chambolle and Pock for saddle point problems and relax the condition for ensuring its convergence. The relaxed convergence -guaranteeing condition is effective for the generic convex setting of saddle point problems, and we show by the canonical convex programming problem with linear equality constraints that the relaxed condition is optimal. It also allows us to discern larger step sizes for the resulting sub-problems, and thus provides a simple and universal way to improve numerical performance of the original primal-dual algorithm. In addition, we present a structure-exploring heuristic to further relax the convergence-guaranteeing condition for some specific saddle point problems, which could yield much larger step sizes and hence significantly better performance. Effectiveness of this heuristic is numerically illustrated by the classic assignment problem.
This paper studies the decentralized stochastic optimization problem over an undirected network, where each agent owns its local private functions made up of two non-smooth functions and an expectation-valued function...
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This paper studies the decentralized stochastic optimization problem over an undirected network, where each agent owns its local private functions made up of two non-smooth functions and an expectation-valued function. A decentralized stochastic primal-dual algorithm is proposed, by combining the variance-reduced method and the stochastic approximation method. The local gradients are estimated by using the mean of a variable number of sample gradients and the stochastic error decreases with the number of samples in the stochastic approximation process. The highlight of this paper is the extension of the primal-dual algorithm to the stochastic optimization problems. The effectiveness of the proposed algorithm and the correctness of the theory are verified by numerical experiments.
This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel-Young inequ...
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This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel-Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel-Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas-Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm may even converge on manifolds of positive curvature.
In this paper, we consider the general first order primal-dual algo-rithm, which covers several recent popular algorithms such as the one proposed in [Chambolle, A. and Pock T., A first-order primal-dual algorithm for...
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In this paper, we consider the general first order primal-dual algo-rithm, which covers several recent popular algorithms such as the one proposed in [Chambolle, A. and Pock T., A first-order primal-dual algorithm for con-vex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011) 120-145] as a special case. Under suitable conditions, we prove its global con-vergence and analyze its linear rate of convergence. As compared to the results in the literature, we derive the corresponding results for the general case and under weaker conditions. Furthermore, the global linear rate of the linearized primal-dual algorithm is established in the same analytical framework.
Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non...
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Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non-smooth total variation (TV) functional. We analyze two aspects: Firstly, the convergence of the regularized solution of the minimization problem to the minimum norm solution. Second, the convergence of the iteratively regularized minimizer to the minimum norm solution by a primal-dual algorithm. For both aspects, we use the assumption of a variational source condition (VSC). This work emphasizes the impact of the choice of the parameters in stabilization of a primal-dual algorithm. Rates of convergence are obtained in terms of some concave, positive definite index function. The algorithm is applied to a simple two-dimensional image processing problem. Sufficient error analysis profiles are provided based on the size of the forward operator and the noise level in the measurement.
The total variation infimal convolution (TV-IC) model combining Kullback-Leibler and ������2-norm data fidelit y term works wel l for the inverse problem of mixed Poisson-Gaussian noise. Most existing algorithms for s...
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The total variation infimal convolution (TV-IC) model combining Kullback-Leibler and ������2-norm data fidelit y term works wel l for the inverse problem of mixed Poisson-Gaussian noise. Most existing algorithms for solving the TV-IC model rely on the Newton method to solve a nonlinear optimization subproblem, which inevitably increases the computation burden. In this study, we apply the first-order primal-dual Chambolle- Pock algorithm to solve the TV-IC model . In particular, we present an effective algorithm to solve the subproblem of the joint proximal operator with the Kullback-Leibler divergence and ������2-norm, which is based on the bilinear constraint alternating direction multiplier method. Numerical experiment results demonstrate that the proposed algorithm outperforms the state-of-the-art methods for mixed Poisson-Gaussian denoising and deblurring problems.
Demosaicking is a process to interpolate raw data subsampled by the color filter array into a full-resolution image. In this work, a new RGB-NIR demosaicking algorithm based on the primal-dual method is proposed. In t...
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ISBN:
(纸本)9781665408578
Demosaicking is a process to interpolate raw data subsampled by the color filter array into a full-resolution image. In this work, a new RGB-NIR demosaicking algorithm based on the primal-dual method is proposed. In the algorithm, we use total variance and guided image filters as the regularization terms, so that both flat and edge regions can be properly addressed. In addition, a two-phase optimization scheme is introduced to avoid problems caused by over-correlating the RGB and NIR channels. We also design an early-termination mechanism, which can stop iterations when it foresees the result could go wrong. As demonstrated in the paper, the proposed algorithm can preserve high frequency regions well while having chroma noises, halos, and artifacts effectively removed.
By time discretization of a second-order primal-dual dynamical system with damping alpha/t where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual ...
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By time discretization of a second-order primal-dual dynamical system with damping alpha/t where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach O(1/k(alpha-1)) at the most. We also study convergence properties of the corresponding primal-dual dynamical system to better understand the acceleration scheme. Finally, we report numerical experiments to demonstrate the effectiveness of the proposed algorithm. (c) 2022 Elsevier Ltd. All rights reserved.
In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed wi...
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In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.
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