Levenberg-Marquardt method for absolute value equation associated with second-order cone

<p style='text-indent:20px;'>In this paper, we suggest the Levenberg-Marquardt method with Armijo line search for solving absolute value equations associated with the second-order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. We analyze the convergence of the proposed algorithm. For numerical reports, we not only show the efficiency of the proposed method, but also present numerical comparison with smoothing Newton method. It indicates that the proposed algorithm could also be a good choice for solving the SOCAVE.</p>

Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery

In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.

A New Inexact Line Search Method for Convex Optimization Problems

In general one can say that line search procedure for the steplength and search direction are two important elements of a line search algorithm. The line search procedure requires much attention because of its far implications on the robustness and efficiency of the algorithm. The purpose of this paper is to propose a simple yet effective line search strategy in solving unconstrained convex optimization problems. This line search procedure does not require the evaluation of the objective function. Instead, it forces reduction in gradient norm on each direction. Hence it is suitable for problems when function evaluation is very costly. To illustrate the effectiveness of our line search procedure, we employ this procedure together with the symmetric rank one quasi-Newton update and test it against the same quasi-Newton method with the well-known Armijo line search. Numerical results on a set of standard unconstrained optimization problems showed that the proposed procedure is superior to the Armijo line search.

An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

A Modified Nonlinear Conjugate Gradient Method with the Armijo Line Search and Its Application

In this article, a modified Polak-Ribière-Polyak (PRP) conjugate gradient method is proposed for image restoration. The presented method can generate sufficient descent directions without any line search conditions. Under some mild conditions, this method is globally convergent with the Armijo line search. Moreover, the linear convergence rate of the modified PRP method is established. The experimental results of unconstrained optimization, image restoration, and compressive sensing show that the proposed method is promising and competitive with other conjugate gradient methods.

A Simulated Annealing-Based Barzilai–Borwein Gradient Method for Unconstrained Optimization Problems

In this paper, we propose a simulated annealing-based Barzilai–Borwein (SABB) gradient method for unconstrained optimization problems. The SABB method accepts the Barzilai–Borwein (BB) step by a simulated annealing rule. If the BB step cannot be accepted, the Armijo line search is used. The global convergence of the SABB method is established under some mild conditions. Numerical experiments indicate that, compared to some existing BB methods using nonmonotone line search technique, the SABB method performs well with high efficiency.

A Novel Algorithm for Structural Reliability Analysis Based on Finite Step Length and Armijo Line Search

This paper presents a novel algorithm for structural reliability analysis based on the finite step length and Armijo line search to remove the drawbacks of the Hasofer–Lind and Rakwitz–Fiessler (HL-RF) algorithm that may be subjected to non-convergence in the first-order reliability method (FORM). Initially, the sensitivity factor with finite step length is introduced for preventing the iterative process of the algorithm from entering a periodic loop. Subsequently, an optimization method based on the sufficient descent condition with the Armijo line search technique is proposed. With that, the initial step length and adjusting coefficient are optimized to enhance the applicability of the algorithm emphatically for highly nonlinear functions. A comparison analysis is carried out between the proposed algorithm and existing FORM-based algorithms to validate the robustness and efficiency of the proposed algorithm. The results of this demonstrate that the proposed algorithm is superior to the HL-RF algorithm in terms of robustness and surpass the other existing FORM-based algorithms in connection to efficiency.

Modified Three-Term Conjugate Gradient Method and Its Applications

We propose a modified three-term conjugate gradient method with the Armijo line search for solving unconstrained optimization problems. The proposed method possesses the sufficient descent property. Under mild assumptions, the global convergence property of the proposed method with the Armijo line search is proved. Due to simplicity, low storage, and nice convergence properties, the proposed method is used to solve M-tensor systems and a kind of nonsmooth optimization problems with l1-norm. Finally, the given numerical experiments show the efficiency of the proposed method.