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VARIABLE METRIC forward-backward algorithm FOR COMPOSITE MINIMIZATION PROBLEMS

SIAM JOURNAL ON OPTIMIZATION 2021年第2期31卷 1215-1241页

作者： Repetti, Audrey Wiaux, Yves Heriot Watt Univ Sch Math & Comp Sci Edinburgh EH14 4AS Midlothian Scotland Heriot Watt Univ Sch Engn & Phys Sci Edinburgh EH14 4AS Midlothian Scotland Maxwell Inst Math Sci Bayes Ctr Edinburgh EH8 9BT Midlothian Scotland

We present a forward-backward-based algorithm to minimize a sum of a differentiable function and a nonsmooth function, both being possibly nonconvex. The main contribution of this work is to consider the challenging case where the nonsmooth function corresponds to a sum of nonconvex functions, resulting from composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function. The proposed variable metric composite function forward-backward (C2FB) algorithm circumvents the explicit, and often challenging, computation of the proximity operator of the composite functions through a majorize-minimize approach. Precisely, each composite function is majorized using a linear approximation of the differentiable function, which allows one to apply the proximity step only to the sum of the nonsmooth functions. We prove the convergence of the algorithm iterates to a critical point of the objective function leveraging the Kurdyka-Lojasiewicz inequality. The convergence is guaranteed even if the proximity operators are computed inexactly, considering relative errors. We show that the proposed approach is a generalization of reweighting methods, with convergence guarantees. In particular, applied to the log-sum function, our algorithm reduces to a generalized version of the celebrated reweighted l(1) method. Finally, we show through simulations on an image processing problem that the proposed C2FB algorithm necessitates fewer iterations to converge and leads to better critical points compared with traditional reweighting methods and classic forward-backward algorithms.

关键词： nonconvex optimization nonsmooth optimization proximity operator composite minimization problem majorize-minimize method forward-backward algorithm reweighting algorithm inverse problems

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Convergence of an asynchronous block-coordinate forward-backward algorithm for convex composite optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第1期86卷 303-344页

作者： Traore, Cheik Salzo, Saverio Villa, Silvia Univ Genoa Malga Ctr DIMA Genoa Italy Sapienza Univ Roma DIAG Rome Italy

In this paper, we study the convergence properties of a randomized block-coordinate descent algorithm for the minimization of a composite convex objective function, where the block-coordinates are updated asynchronously and randomly according to an arbitrary probability distribution. We prove that the iterates generated by the algorithm form a stochastic quasi-Fejer sequence and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type. Under the same condition strong convergence of the iterates is provided as well as their linear convergence rate.

关键词： Convex optimization Asynchronous algorithms Randomized block-coordinate descent Error bounds Stochastic quasi-Fejer sequences forward-backward algorithm Convergence rates

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Three-Dimensional Angle of Arrival Estimation in Dynamic Indoor Terahertz Channels Using a forward-backward algorithm

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2017年第5期66卷 3798-3811页

作者： Peng, Bile Kuerner, Thomas Tech Univ Carolo Wilhelmina Braunschweig Inst Nachrichtentech D-38106 Braunschweig Germany

A novel angle-of-arrival (AoA) estimation method for both azimuth and elevation based on Bayesian inference and statistical state transition probabilities is presented for the dynamic indoor terahertz (THz) channel. A precise AoA estimation is crucial for the deployment of a directive antenna, which can compensate for the high path loss and reduce the intersymbol interference. In many application scenarios, the user equipment is moved by the user during the data transmission, and the AoA is not constant. The novel algorithm exploits the fact that the AoA movement can be represented as a Markov process and that the Bayesian inference can be used to combine the likelihood and a priori information to provide a more precise estimate than using the likelihood alone. An indoor human movement model is developed to generate the realistic application scenario and obtain the statistical transition probabilities. The forward-backward algorithm is implemented to carry out the Bayesian inference. The algorithm performance is illustrated using the channel models generated by a ray launching simulator. The background log-likelihood is suggested to adapt the algorithm to the instant channel state change in a multipath environment.

关键词： Angle-of-arrival (AoA) estimation Bayesian inference forward-backward algorithm human blockage terahertz (THz) communication

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Approximate forward-backward algorithm for a switching linear Gaussian model

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2011年第1期55卷 154-167页

作者： Hammer, Hugo Tjelmeland, Hakon Norwegian Univ Sci & Technol Dept Math Sci N-7034 Trondheim Norway

A hidden Markov model with two hidden layers is considered. The bottom layer is a Markov chain and given this the variables in the second hidden layer are assumed conditionally independent and Gaussian distributed. The observation process is Gaussian with mean values that are linear functions of the second hidden layer. The forward-backward algorithm is not directly feasible for this model as the recursions result in a mixture of Gaussian densities where the number of terms grows exponentially with the length of the Markov chain. By dropping the less important Gaussian terms an approximate forward-backward algorithm is defined. Thereby one gets a computationally feasible algorithm that generates samples from an approximation to the conditional distribution of the unobserved layers given the data. The approximate algorithm is also used as a proposal distribution in a Metropolis-Hastings setting, and this gives high acceptance rates and good convergence and mixing properties. The model considered is related to what is known as switching linear dynamical systems. The proposed algorithm can in principle also be used for these models and the potential use of the algorithm is therefore large. In simulation examples the algorithm is used for the problem of seismic inversion. The simulations demonstrate the effectiveness and quality of the proposed approximate algorithm. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Approximation forward-backward algorithm Hidden Markov model Metropolis-Hastings algorithm Seismic inversion

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Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space

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OPTIMIZATION 2021年第1期70卷 169-190页

作者： Wang, Yamin Wang, Fenghui Zhang, Haixia Henan Normal Univ Coll Math & Informat Sci Xinxiang Henan Peoples R China Henan Normal Univ Coll Math & Informat Sci Henan Engn Lab Big Data Stat Anal & Optimal Contr Xinxiang Henan Peoples R China Luoyang Normal Univ Dept Math Luoyang Peoples R China

In recent years, the forward-backward algorithm (FBA) received much attention due to its various applications in image recovery, signal processing, and machine learning. In this paper, we consider the FBA in the setting of Banach spaces that are uniformly convex and q-uniformly smooth. We introduce two viscosity FBA, and one of them with weakly contractive mapping, which generalizes many previous results on the viscosity approximation method with fixed contraction. Moreover, we establish their strong convergences under more general conditions.

关键词： forward-backward algorithm accretive operator Banach space strong convergence

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A forward-backward algorithm for Nested Hidden semi-Markov Model and Application to Network Traffic

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COMPUTER JOURNAL 2013年第2期56卷 229-238页

作者： Xie, Yi Hu, J. Tang, S. Huang, X. Sun Yat Sen Univ Sch Informat Sci & Technol Guangzhou 510275 Guangdong Peoples R China Univ New S Wales Sch Engn & & Informat Technol Australian Def Force Acad UNSW ADFA Canberra ACT 2600 Australia Missouri Western State Univ Dept Engn Technol St Joseph MO 64507 USA Sun Yat Sen Univ Network & Informat Technol Ctr Guangzhou 510275 Guangdong Peoples R China

Doubly hidden Markov models (DHMMs) have been widely used to analyze a type of time process whose driving factors are hierarchical and hierarchically correlated. A common issue of these models is that they implicitly assume that the dwell time of any system state is constant or exponentially distributed. This property comes from the standard hidden Markov models and causes the DHMM to limitations in some actual application environment, where an application has latent temporal structure and does not follow the exponential distribution but has the period-like or variable-period feature. Such problems are frequently encountered in practice, e.g. network traffic. In this paper, we remove this limitation by a new structural discrete approach named nested hidden semi-Markov model. The proposed model includes a nested latent semi-Markov chain and one observable discrete stochastic process. The bottom latent semi-Markov chain is the core layer and controls the second-layer semi-Markov chain that generates the observable process. The state duration of both the semi-Markov chains can be variable or explicit. The model makes no assumptions on the distribution of the state-duration and the observable processes. An efficient forward and backward recursion procedure is developed for estimating the generator of the proposed model and inferring the underlying state processes for a given observation sequence. To evaluate the performance of the proposed model, we apply the model to the arrival process of network traffic and compare its simulation traffic and the real traffic. The performance evaluation in the experiments includes time dynamic process, auto-correlation, cross-correlation, statistical distribution and self-similarity.

关键词： forward-backward algorithm nested HsMM network traffic

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Extended forward-backward algorithm

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2013年第1期403卷 167-172页

作者： Lassonde, Marc Nagesseur, Ludovic Univ Antilles Guyane LAMIA F-97159 Pointe A Pitre Guadeloupe France

We propose an extended forward backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators. We establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward backward algorithms. This provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Maximal monotone operator epsilon-enlargement Extended sum forward-backward algorithm Projected subgradient algorithm Splitting algorithm Convergence in average

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On convergence and complexity analysis of an accelerated forward-backward algorithm with linesearch technique for convex minimization problems and applications to data prediction and classification

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-20页

作者： Sarnmeta, Panitarn Inthakon, Warunun Chumpungam, Dawan Suantai, Suthep Chang Mai Univ Fac Sci Data Sci Res Ctr Dept Math Chang Mai 56200 Thailand Chang Mai Univ Fac Sci Dept Math Res Ctr Math & Appl Math Chang Mai 56200 Thailand

In this work, we introduce a new accelerated algorithm using a linesearch technique for solving convex minimization problems in the form of a summation of two lower semicontinuous convex functions. A weak convergence of the proposed algorithm is given without assuming the Lipschitz continuity on the gradient of the objective function. Moreover, the convexity of this algorithm is also analyzed. Some numerical experiments in machine learning are also discussed, namely regression and classification problems. Furthermore, in our experiments, we evaluate the convergent behavior of this new algorithm, then compare it with various algorithms mentioned in the literature. It is found that our algorithm performs better than the others.

关键词： Convex minimization problems Machine learning forward-backward algorithm Linesearch Accelerated algorithm Data classification

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Variable Metric forward-backward algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第1期162卷 107-132页

作者： Chouzenoux, Emilie Pesquet, Jean-Christophe Repetti, Audrey Univ Paris Est Marne la Vallee Lab Informat Gaspard Monge F-77454 Champs Sur Marne Marne La Vallee France Univ Paris Est Marne la Vallee CNRS UMR 8049 F-77454 Champs Sur Marne Marne La Vallee France

We consider the minimization of a function G defined on , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka-Aojasiewicz property. Such a problem can be solved with the forward-backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize-Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric forward-backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.

关键词： Nonconvex optimization Nonsmooth optimization Majorize-Minimize algorithms forward-backward algorithm Image reconstruction Proximity operator

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An inertial based forward-backward algorithm for monotone inclusion problems and split mixed equilibrium problems in Hilbert spaces

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ADVANCES IN DIFFERENCE EQUATIONS 2020年第1期2020卷 1-25页

作者： Arfat, Yasir Kumam, Poom Ngiamsunthorn, Parinya Sa Khan, Muhammad Aqeel Ahmad King Mongkuts Univ Technol Thonburi KMUTT KMUTT Fixed Point Res Lab KMUTT Fixed Point Theory & Applicat Res Grp Dept MathFac Sci 126 Pracha Uthit Rd Bangkok 10140 Thailand King Mongkuts Univ Technol Thonburi KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan King Mongkuts Univ Technol Thonburi KMUTT Dept Math Fac Sci 126 Pracha Uthit Rd Bangkok 10140 Thailand COMSATS Univ Islamabad Dept Math Lahore Campus Lahore 54000 Pakistan

Iterative algorithms are widely applied to solve convex optimization problems under a suitable set of constraints. In this paper, we develop an iterative algorithm whose architecture comprises a modified version of the forward-backward splitting algorithm and the hybrid shrinking projection algorithm. We provide theoretical results concerning weak and strong convergence of the proposed algorithm towards a common solution of the monotone inclusion problem and the split mixed equilibrium problem in Hilbert spaces. Moreover, numerical experiments compare favorably the efficiency of the proposed algorithm with the existing algorithms. As a consequence, our results improve various existing results in the current literature.

关键词： Mixed split equilibrium problem Inertial method Inclusion problem forward-backward algorithm Shrinking projection method 47H04 47H05 47H10 47J22

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