Forward-backward Splitting Method Research Articles

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

Incorporating history and deviations in forward–backward splitting

AbstractWe propose a variation of the forward–backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward–backward splitting scheme and an accelerated $$\mathcal {O}\left( \frac{1}{n^2}\right) $$ O 1 n 2 -convergent forward–backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization.

The forward–backward splitting method for finding the minimum like-norm solution of the mixed variational inequality problem

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set, which is by itself an optimal set of another mixed variational inequality problem in a Hilbert space. Regularized forward–backward splitting method is applied to find the minimum like-norm solution of the mixed variational inequality problem under investigation.

Modified inertial Tseng method for solving variational inclusion and fixed point problems on Hadamard manifolds

In this article, we introduce a forward–backward splitting method with a new step size rule for finding a singularity point of an inclusion problem which is defined by means of a sum of a single-valued vector field and a multi-valued vector field on a Hadamard manifold. Using a Mann, viscosity and an inertial extrapolation method, we establish a convergence result without prior knowledge of the Lipschitz constant of the underlying operator. We present some applications of our result to variational inequality problem. Finally, we present some numerical examples to demonstrate the numerical behavior of our proposed method. The result discuss in this article extends and complements many related results in the literature.

Double inertial parameters forward-backward splitting method: Applications to compressed sensing, image processing, and SCAD penalty problems

Double inertial parameters forward-backward splitting method: Applications to compressed sensing, image processing, and SCAD penalty problems

Impact Load Identification with Uniform Noise Based on Forward-backward Splitting Algorithm

In this paper, the forward-backward splitting method is introduced for the first time to solve the impact load identification problem under the condition of uniform noise in the field of structural dynamics. In the process of using this method to solve the problem of impact load identification, firstly, the load identification model is established. Then, the model is transformed into an optimization problem with infinite norm constraints by the maximum posterior estimation, and then the forward-backward splitting method is used to solve the optimization problem. The rationality and effectiveness of the forward and backward splitting method for load identification under uniform noise conditions in the field of structural dynamics are illustrated by numerical simulation examples.

Variational inequalities over the solution sets of split variational inclusion problems

We study variational inequalities over the solution sets of split variational inclusion problems with multiple output sets. In order to find a solution to such problems, we introduce a new parallel iterative algorithm by combining the inertial forward–backward splitting method with the steepest descent method. Our algorithm does not depend on the norm of the transfer mappings. Some of its applications to solving split common null point, split minimum point, and split common fixed point problems are also given.

An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application

In this paper, we propose an inertial forward–backward splitting method for solving the modified variational inclusion problem. The concept of the proposed method is based on Cholamjiak’s method. and Khuangsatung and Kangtunyakarn’s method. Cholamjiak’s inertial technique is utilized in the proposed method for increased acceleration. Moreover, we demonstrate that the proposed method strongly converges under appropriate conditions and apply the proposed method to solve the image restoration problem where the images have been subjected to various obscuring processes. In our example, we use the proposed method and Khuangsatung and Kangtunyakarn’s method to restore two medical images. To compare image quality, we also evaluate the signal-to-noise ratio (SNR) of the proposed method to that of Khuangsatung and Kangtunyakarn’s method.

Novel Algorithms with Inertial Techniques for Solving Constrained Convex Minimization Problems and Applications to Image Inpainting

In this paper, we propose two novel inertial forward–backward splitting methods for solving the constrained convex minimization of the sum of two convex functions, φ1+φ2, in Hilbert spaces and analyze their convergence behavior under some conditions. For the first method (iFBS), we use the forward–backward operator. The step size of this method depends on a constant of the Lipschitz continuity of ∇φ1, hence a weak convergence result of the proposed method is established under some conditions based on a fixed point method. With the second method (iFBS-L), we modify the step size of the first method, which is independent of the Lipschitz constant of ∇φ1 by using a line search technique introduced by Cruz and Nghia. As an application of these methods, we compare the efficiency of the proposed methods with the inertial three-operator splitting (iTOS) method by using them to solve the constrained image inpainting problem with nuclear norm regularization. Moreover, we apply our methods to solve image restoration problems by using the least absolute shrinkage and selection operator (LASSO) model, and the results are compared with those of the forward–backward splitting method with line search (FBS-L) and the fast iterative shrinkage-thresholding method (FISTA).

Modified forward-backward splitting method for split equilibrium, variational inclusion, and fixed point problems

Modified forward-backward splitting method for split equilibrium, variational inclusion, and fixed point problems

A first-order method for solving bilevel convex optimization problems in Banach space

ABSTRACT We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem in Banach space. Regularized forward–backward splitting method is applied to find the minimum like-norm solution of the minimization problem under investigation. We also introduce a gradient-based method, called the minimal like-norm gradient method, for solving this class of problems and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values.

Convergence rates of the modified forward reflected backward splitting algorithm in Banach spaces

<abstract><p>Consider the problem of minimizing the sum of two convex functions, one being smooth and the other non-smooth in Banach space. In this paper, we introduce a non-traditional forward-backward splitting method for solving such minimization problem. We establish different convergence estimates under different stepsize assumptions.</p></abstract>

Physics-informed multi-fidelity learning-driven imaging method for electrical capacitance tomography

The electrical capacitance tomography is considered to be a promising non-invasive visualization method for the measurement of multiphase flow parameters, but low-quality tomograms limit the reliability and accuracy of measurements. In order to overcome this limitation and bottleneck, the physics-informed data-dependent prior learned from the given samples is introduced and coupled with the measurement mechanism and the domain knowledge modeled by a new L1/L2 norm-based regularizer into a new optimization imaging model in this study. To improve the convergence and reduce the computational load, the built imaging model is solved by a new numerical scheme driven by the split Bregman method and the forward–backward splitting method. A new physics-informed robust sparse extreme learning machine method that not only obeys the measurement mechanism in training but also promotes the robustness and performance of the model is proposed and used to build a novel multi-fidelity learning method to infer the physics-informed data-dependent prior. The new imaging method achieves multi-source data fusion, leads to new changes in reconstruction algorithms, increases the complementarity and diversity of image priors, unifies the image reconstruction and multi-fidelity learning method, and improves robustness, numerical stability and flexibility of the model. The reconstruction results show that the new imaging method achieves more accurate reconstruction with reduced sensitivity to noise compared to important and popular imaging methods. This study leads to new changes in imaging algorithms and mining and use of image priors, and serves as a catalyst for the paradigm shift in image reconstruction.

The forward–backward splitting method for non-Lipschitz continuous minimization problems in Banach spaces

The forward–backward splitting method for non-Lipschitz continuous minimization problems in Banach spaces

Viscosity modification with parallel inertial two steps forward-backward splitting methods for inclusion problems applied to signal recovery

In this paper, we introduce a new parallel algorithm by combining viscosity modification with parallel inertial two steps forward-backward splitting methods for approximating a solution of common inclusion problems. The strongly convergent theorems are established under some suitable conditions in Hilbert spaces. The applicability and advantages of the new parallel algorithm are presented by using to solve signal recovering problem in compressed sensing. The efficiency of the algorithm is shown by comparing it with some previous parallel algorithms.

Convergence analysis and applications of the inertial algorithm solving inclusion problems

Nonlinear operator theory is an important area of nonlinear functional analysis. This area encompasses diverse nonlinear problems in many areas of mathematics, the physical sciences and engineering such as monotone operator equations, fixed point problems and more.In this work we are concern with the problem of finding a common solution of a monotone operator equation and fixed point of a nonexpansive mapping in real Hilbert spaces. Derived from dynamical systems, a simple inertial forward-backward splitting method for solving the problem is presented and analyzed under mild and standard assumptions. Some numerical examples in real-world and comparisons with related works, illustrate the theoretical advantages as well the potential applicability of the proposed scheme.

A Dynamical Splitting Method for Minimizing the Sum of Three Convex Functions

In this paper, we develop a dynamical splitting method based on a nonlinear dynamical system with two proximal operators for solving the minimum value problem of the sum of three convex functions. The method endows some existing classical splitting methods such as the Forward-Backward splitting method and the Douglas-Rachford splitting method with continuous time behaviors. We demonstrate that the proposed dynamical system converges globally to a point whose image, imaged by each of the two proximal operators, is a solution to the minimum value problem. We also shows that both the residual of the two proximal operators and the objective error enjoy the convergence rates of where t denotes time. This rate can be improved to by using an ergodic average technique. Moreover, under certain smoothness and metric subregularity conditions, we further establish the global exponential convergence of the dynamical splitting method. Finally, we apply the dynamical splitting method to deal with the problem of compressive sensing.

Stochastic generalized Nash equilibrium seeking under partial-decision information

We consider for the first time a stochastic generalized Nash equilibrium problem, i.e., with expected-value cost functions and joint feasibility constraints, under partial-decision information, meaning that the agents communicate only with some trusted neighbors. We propose several distributed algorithms for network games and aggregative games that we show being special instances of a preconditioned forward–backward splitting method. We prove that the algorithms converge to a generalized Nash equilibrium when the forward operator is restricted cocoercive by using the stochastic approximation scheme with variance reduction to estimate the expected value of the pseudogradient.

Convergence analysis of modified inertial forward–backward splitting scheme with applications

In the settings of real Hilbert spaces, this paper proposes modified forward– backward scheme with inertial extrapolation step for solving inverse problems. In our proposed method, it is possible for the inertial factor to be chosen as 1 unlike many previous inertial forward–backward splitting methods available in the literature. Weak and strong convergence of our proposed method are obtained under some standard conditions. In order to show the numerical advantage gained when the inertial factor is chosen as 1, a number of numerical implementations are performed on two different inverse problems: LASSO and SCAD penalty problems.

L1-L2 norm regularization via forward-backward splitting for fluorescence molecular tomography.

Fluorescent molecular tomography (FMT) is a highly sensitive and noninvasive imaging approach for providing three-dimensional distribution of fluorescent marker probes. However, owing to its light scattering effect and the ill-posedness of inverse problems, it is challenging to develop an efficient reconstruction algorithm that can achieve the exact location and morphology of the fluorescence source. In this study, therefore, in order to satisfy the need for early tumor detection and improve the sparsity of solution, we proposed a novel L 1-L 2 norm regularization via the forward-backward splitting method for enhancing the FMT reconstruction accuracy and the robustness. By fully considering the highly coherent nature of the system matrix of FMT, it operates by splitting the objective to be minimized into simpler functions, which are dealt with individually to obtain a sparser solution. An analytic solution of L 1-L 2 norm proximal operators and a forward-backward splitting algorithm were employed to efficiently solve the nonconvex L 1-L 2 norm minimization problem. Numerical simulations and an in-vivo glioma mouse model experiment were conducted to evaluate the performance of our algorithm. The comparative results of these experiments demonstrated that the proposed algorithm obtained superior reconstruction performance in terms of spatial location, dual-source resolution, and in-vivo practicability. It was believed that this study would promote the preclinical and clinical applications of FMT in early tumor detection.