/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

A large number of imaging problems reduce to the optimization of a cost function , with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.

/pdf/an-introduction-to-continuous-optimization-for-imaging-3lzw7asbyh.pdf

An introduction to continuous optimization for imaging

Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

/pdf/solving-inverse-problems-using-data-driven-models-3jeyo4a896.pdf

Solving inverse problems using data-driven models

We revisit the proofs of convergence for a first order primal---dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms.

/pdf/on-the-ergodic-convergence-rates-of-a-first-order-primal-36yz68m0vs.pdf

On the ergodic convergence rates of a first-order primal---dual algorithm

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. However, deriving efficient strategies that jointly bring into play the primal and dual problems is a more recent idea that has generated many important new contributions in recent years. These novel developments are grounded in the recent advances in convex analysis, discrete optimization, parallel processing, and nonsmooth optimization with an emphasis on sparsity issues. In this article, we aim to present the principles of primal?dual approaches while providing an overview of the numerical methods that have been proposed in different contexts. Last but not least, primal?dual methods lead to algorithms that are easily parallelizable. Today, such parallel algorithms are becoming increasingly important for efficiently handling high-dimensional problems.

https://hal.archives-ouvertes.fr/hal-01010437/document

Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems

Inertial Douglas-Rachford splitting for monotone inclusion problems

In this paper we propose two different primal-dual splitting algorithms for solving inclusions involving mixtures of composite and parallel-sum type monotone operators which rely on an inexact Douglas--Rachford splitting method, but applied in different underlying Hilbert spaces. Most importantly, the algorithms allow one to process the bounded linear operators and the set-valued operators occurring in the formulation of the monotone inclusion problem separately at each iteration, the latter being individually accessed via their resolvents. The performance of the primal-dual algorithms is emphasized via some numerical experiments on location and image denoising problems.

/pdf/a-douglas-rachford-type-primal-dual-method-for-solving-2ggoj4pgud.pdf

A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators

We present two modified versions of the primal-dual splitting algorithm relying on forward---backward splitting proposed in V $$\tilde{\mathrm{u}}$$ u ~ (Adv Comput Math 38(3):667---681, 2013) for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of $$\mathcal{{O}}(\frac{1}{n})$$ O ( 1 n ) and $$\mathcal{{O}}(\omega ^n)$$ O ( ? n ) , for $$\omega \in (0,1)$$ ? ? ( 0 , 1 ) , respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and pattern recognition in cluster analysis.

/pdf/on-the-convergence-rate-improvement-of-a-primal-dual-ecmc25emp3.pdf

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

In this paper we propose two different primal-dual splitting algorithms for solving inclusions involving mixtures of composite and parallel-sum type monotone operators which rely on an inexact Douglas-Rachford splitting method, however applied in different underlying Hilbert spaces. Most importantly, the algorithms allow to process the bounded linear operators and the set-valued operators occurring in the formulation of the monotone inclusion problem separately at each iteration, the latter being individually accessed via their resolvents. The performances of the primal-dual algorithms are emphasized via some numerical experiments on location and image deblurring problems.

In this paper we investigate the convergence behavior of a primal-dual splitting method for solving monotone inclusions involving mixtures of composite, Lipschitzian and parallel sum type operators proposed by Combettes and Pesquet (in Set-Valued Var. Anal. 20(2):307---330, 2012). Firstly, in the particular case of convex minimization problems, we derive convergence rates for the partial primal-dual gap function associated to a primal-dual pair of optimization problems by making use of conjugate duality techniques. Secondly, we propose for the general monotone inclusion problem two new schemes which accelerate the sequences of primal and/or dual iterates, provided strong monotonicity assumptions for some of the involved operators are fulfilled. Finally, we apply the theoretical achievements in the context of different types of image restoration problems solved via total variation regularization.

/pdf/convergence-analysis-for-a-primal-dual-monotone-skew-3qcldu0kmr.pdf

Christopher Hendrich

Papers

Inertial Douglas-Rachford splitting for monotone inclusion problems

A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators

Convergence Analysis for a Primal-Dual Monotone + Skew Splitting Algorithm with Applications to Total Variation Minimization