On the convergence of the forward–backward splitting method with linesearches

In optimization theory, convex minimization problems have been intensively investigated in the current literature due to its wide range in applications. A major and effective tool for solving such problem is the forward‐backward splitting algorithm. However, to guarantee the convergence, it is usually assumed that the gradient of functions is Lipschitz continuous and the stepsize depends on the Lipschitz constant, which is not an easy task in practice. In this work, we propose the modified forward‐backward splitting method using new linesearches for choosing suitable stepsizes and discuss the convergence analysis including its complexity without any Lipschitz continuity assumption on the gradient. Finally, we provide numerical experiments in signal recovery to demonstrate the computational performance of our algorithm in comparison to some well‐known methods. Our reports show that the proposed algorithm has a good convergence behavior and can outperform the compared methods.

Section: Resultsmentioning

confidence: 99%

“…It is well‐known that the proximal operator is single‐valued with full domain. Furthermore, note that

\frac{z - {prox}_{α g} false(z false)}{α} \in \partial g false({prox}_{α g} false(z false) false) for all z \in H, α > 0 .

…”

Section: Definitions and Lemmasmentioning

confidence: 99%

“…Following Bello Cruz and Nghia, we assume that two below conditions hold: (A1)

f, g : H \to double-struckR \cup false{+ \infty false}

are two proper, lower semi‐continuous, and convex functions with dom g ⊆dom f . (A2)The function f is Fr

\overset{e}{true}

chet differentiable on an open set containing dom g . The gradient ∇ f is uniformly continuous on any bounded subset of dom g and maps any bounded subset of dom g to a bounded set in H . …”

Section: Definitions and Lemmasmentioning

confidence: 99%

“…Following Bello Cruz and Nghia, 9 we assume that two below conditions hold: Remark 2.6. The second part of (A2) still holds when ∇f is Lipschitz continuous on domg.…”

Section: Definitions and Lemmasmentioning

confidence: 99%

“…Bello Cruz and Nghia proposed the following forward‐backward method with linesearch which dose not depend on the Lipschitz constant for computing.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On convergence and complexity of the modified forward‐backward method involving new linesearches for convex minimization

Kankam

Pholasa

Cholamjiak

2019

Modified proximal gradient methods involving double inertial extrapolations for monotone inclusion

Inkrong,

Cholamjiak

2024

In this work, we propose a novel class of forward‐backward‐forward algorithms for solving monotone inclusion problems. Our approach incorporates a self‐adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions and utilizes two‐step inertial extrapolations to enhance the convergence rate of the algorithm. We establish a weak convergence theorem under mild assumptions. Furthermore, we conduct numerical tests on image deblurring and data classification as practical applications. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature which shows its superior performance and effectiveness.

A modified forward‐backward splitting methods for the sum of two monotone operators with applications to breast cancer prediction

Peeyada

Dutta

Shiangjen

et al. 2022

This work proposes a modified forward-backward splitting algorithm combining an inertial technique for solving the monotone variational inclusion problem.The weak convergence theorem is established under some suitable conditions in Hilbert space, and a new step size is presented for our algorithm to speed up the convergence. We give an example and numerical results for supporting our main theorem in infinite dimensional spaces. We also provide an application to predict breast cancer by using our proposed algorithm for updating the optimal weight in machine learning. Moreover, we use the Wisconsin original breast cancer data set as a training set to show efficiency comparing with the other three algorithms in terms of three key parameters, namely, accuracy, recall, and precision.