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D.: The PrimalDual Hybrid Gradient Method for Semiconvex Splittings. preprint http://arxiv.org/abs/1407.1723
, 2014
"... Abstract. This paper deals with the analysis of a recent reformulation of the primaldual hybrid gradient method, which allows one to apply it to nonconvex regularizers. Particularly, it investigates variational problems for which the energy to be minimized can be written as G(u) + F (Ku), where G ..."
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Abstract. This paper deals with the analysis of a recent reformulation of the primaldual hybrid gradient method, which allows one to apply it to nonconvex regularizers. Particularly, it investigates variational problems for which the energy to be minimized can be written as G(u) + F (Ku), where G is convex, F is semiconvex, and K is a linear operator. We study the method and prove convergence in the case where the nonconvexity of F is compensated for by the strong convexity of G. The convergence proof yields an interesting requirement for the choice of algorithm parameters, which we show to be not only sufficient, but also necessary. Additionally, we show boundedness of the iterates under much weaker conditions. Finally, in several numerical experiments we demonstrate effectiveness and convergence of the algorithm beyond the theoretical guarantees.
A Class of Randomized PrimalDual Algorithms for Distributed Optimization, arXiv preprint arXiv:1406.6404v3
, 2014
"... Abstract Based on a preconditioned version of the randomized blockcoordinate forwardbackward algorithm recently proposed in ..."
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Abstract Based on a preconditioned version of the randomized blockcoordinate forwardbackward algorithm recently proposed in
CONVERGENCE RATE ANALYSIS OF PRIMALDUAL SPLITTING SCHEMES∗
"... Abstract. Primaldual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple f ..."
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Abstract. Primaldual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings, gradient mappings, and multiplications by the linear maps. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain nearly stateoftheart performance. In this paper, we analyze a monotone inclusion problem that captures a large class of primaldual splittings as a special case. We introduce a unifying scheme and use some abstract analysis of the algorithm to prove convergence rates of the proximal point algorithm, forwardbackward splitting, PeacemanRachford splitting, and forwardbackwardforward splitting applied to the model problem. Our ergodic convergence rates are deduced under variable metrics, stepsizes, and relaxation. Our nonergodic convergence rates are the first shown in the literature. Finally, we apply our results to a large class of primaldual algorithms that are a special case of our scheme and deduce their convergence rates.
Performance of PrimalDual Algorithms for MultiChannel Image Reconstruction in Spectral XRay CT
"... Abstract—The development of spectral Xray CT using binned photoncounting detectors has enabled an imaging technique called Kedge imaging. This concept allows the selective and quantitative imaging of contrast agents loaded with Kedge materials. However, current limitations in detector hardware ..."
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Abstract—The development of spectral Xray CT using binned photoncounting detectors has enabled an imaging technique called Kedge imaging. This concept allows the selective and quantitative imaging of contrast agents loaded with Kedge materials. However, current limitations in detector hardware result in highnoise levels present in materialdecomposed sinograms. Recently, the spectral Xray CT reconstruction problem was formulated within a multichannel (MC) framework using a penalized weighted least squares (PWLS) estimator in which statistical correlations between the decomposed material sinograms can be exploited to improve image quality. Such an approach allows the use of any number of basis materials and is therefore applicable to photoncounting systems and Kedge imaging. The preliminary study results demonstrated the advantages of exploiting intersinogram correlations which are neglected in conventional (statisticallyprincipled) reconstruction methods where the materialdecomposed sinograms are treated individually. However, the utilization of intersinogram correlations, in particular in combination with modern sparsitypromoting penalties, results in numerical challenges in developing efficient reconstruction algorithms. In this work, the numerical performance of stateoftheart primaldual algorithms to minimize the MC PWLS objective function is investigated. The numerical schemes include the alternating direction method of multipliers (ADMM) and an ADMM related strategy, as well as the ChambollePock’s primaldual algorithm. A computersimulation phantom is conducted to simulate spectral Xray CT measurements and used to evaluate the performance of algorithms. I.