Nonlinear Monotone Equations Research Articles

Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations

AbstractThe one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.

An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

In this work, for unconstrained optimization, we introduce an improved Dai‐Liao‐style hybrid conjugate gradient method based on the hybridization‐based self‐adaptive technique, and the search direction generated fulfills the sufficient descent and trust region properties regardless of any line search. The global convergence is established under standard Wolfe line search and common assumptions. Then, combining the hyperplane projection technique and a new self‐adaptive line search, we extend the proposed conjugate gradient method and obtain an improved Dai‐Liao‐style hybrid conjugate gradient projection method to solve constrained nonlinear monotone equations. Under mild conditions, we obtain its global convergence without Lipschitz continuity. In addition, the convergence rates for the two proposed methods are analyzed, respectively. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

A Solution Method for Nonlinear Monotone Equations via Hybrid Spectral Conjugate Gradient and Signal Recovery Problems

A Solution Method for Nonlinear Monotone Equations via Hybrid Spectral Conjugate Gradient and Signal Recovery Problems

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

Iterative methods for solving constraint nonlinear monotone equations have been developed and improved by many researchers. The aim of this research is to present a modified three-term conjugate descent (TTCD) derivative-free method for constrained nonlinear monotone equations. The proposed algorithm requires low storage memory; therefore, it has the capability to solve large-scale nonlinear equations. The algorithm generates a descent and bounded search direction dk at every iteration independent of the line search. The method is shown to be globally convergent under monotonicity and Lipschitz continuity conditions. Numerical results show that the suggested method can serve as an alternative to find the approximate solutions of nonlinear monotone equations. Furthermore, the method is promising for the reconstruction of sparse signal problems.

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

Modification of a conjugate gradient approach for convex constrained nonlinear monotone equations with applications in signal recovery and image restoration

Conjugate gradient (CG )methods have received widespread use in recent years for recovering damaged signals in compressive sensing. This study attempts to create a CG method that is more effective than the recent existing scheme for recovering disrupted signals. The newly created method is a modification of the method proposed by Halilu et al. (2021). Because the authors’ CG parameter can be undefined while solving general functions. As a result, to accomplish the desired result, a new CG method combined with the projection scheme of Solodov and Svaiter (1999) is presented. In addition, under mild conditions, the proposed method converges globally. Numerical experiments conducted with recent methods show the effectiveness of the proposed method. Furthermore, the proposed method is applied efficiently to address the signal recovery and image restoration problems, yielding a better result than some existing methods in the literature.

Stabilized BB projection algorithm for large-scale convex constrained nonlinear monotone equations to signal and image processing problems

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP.

An efficient inertial subspace minimization CG algorithm with convergence rate analysis for constrained nonlinear monotone equations

Conjugate gradient (CG) methods are efficient algorithms for unconstrained optimization. An inertial step strategy is usually used to accelerate iterative methods. In 1995, Yuan and Stoer (1995) introduced a subspace study on CG algorithms and presented some subspace minimization CG (SMCG) methods. Motivated by these approaches, we incorporate a new inertial step strategy in SMCG methods and present an efficient inertial SMCG method for solving constrained nonlinear monotone equations by combining with the projection technique. The proposed inertial step strategy makes use of information from recent iterations. It can be considered as an extension of the known inertial step strategy. The resulting direction always satisfies the sufficient descent property and that of trust region. Indeed, these properties do not depend on line search rules. The global convergence and Q-linear convergence rate of the proposed method are established under mild assumptions, without any Lipschitz continuity. Numerical results indicate that the proposed method is very promising.

An Efficient Subspace Minimization Conjugate Gradient Method for Solving Nonlinear Monotone Equations with Convex Constraints

The subspace minimization conjugate gradient (SMCG) methods proposed by Yuan and Store are efficient iterative methods for unconstrained optimization, where the search directions are generated by minimizing the quadratic approximate models of the objective function at the current iterative point. Although the SMCG methods have illustrated excellent numerical performance, they are only used to solve unconstrained optimization problems at present. In this paper, we extend the SMCG methods and present an efficient SMCG method for solving nonlinear monotone equations with convex constraints by combining it with the projection technique, where the search direction is sufficiently descent.Under mild conditions, we establish the global convergence and R-linear convergence rate of the proposed method. The numerical experiment indicates that the proposed method is very promising.

An effective inertial-relaxed CGPM for nonlinear monotone equations

An effective inertial-relaxed CGPM for nonlinear monotone equations

A class of derivative free three-term descent Hestenes-Stiefel conjugate gradient algorithms for constrained nonlinear problems

This paper presents a novel method for solving large-scale nonlinear monotone equations with convex constraints. The method builds upon the conjugate gradient approach and incorporates a search direction of three terms, complemented by a modified line search. The proposed method generates a search direction that guarantees sufficient descent property, irrespective of the line search technique used. We establish the global convergence of the method under mild conditions and conduct numerical experiments to compare it with existing algorithms from the literature. The results demonstrate the method’s effectiveness and robustness, highlighting its potential for practical applications.

An efficient modified HS conjugate gradient algorithm in machine learning

<p>The Hestenes-Stiefe (HS) conjugate gradient method is very effective in resolving larger-scale sophisticated smoothing optimization tasks due to its low computational requirements and high computational efficiency. Additionally, the algorithm has been employed in practical applications to address image restoration and machine learning issues. In this paper, the authors proposed an improved Hestenes-Stiefe conjugate gradient algorithm having characteristics like: ⅰ) The algorithm depicts the decreasing features and trust region properties free of conditionalities. ⅱ) The algorithm satisfies global convergence. ⅲ) The algorithm can be applied to tackle the image restoration problem, monotone nonlinear equations, and machine learning problems. Numerical results revealed that the proffered technique is a competitive method.</p>

An Adaptive Projection Algorithm for Solving Nonlinear Monotone Equations with Convex Constraints

An Adaptive Projection Algorithm for Solving Nonlinear Monotone Equations with Convex Constraints

An improved inertial projection method for solving convex constrained monotone nonlinear equations with applications

An improved inertial projection method for solving convex constrained monotone nonlinear equations with applications

An efficient projection algorithm for large-scale system of monotone nonlinear equations with applications in signal recovery

An efficient projection algorithm for large-scale system of monotone nonlinear equations with applications in signal recovery

A new hybrid CGPM-based algorithm for constrained nonlinear monotone equations with applications

A new hybrid CGPM-based algorithm for constrained nonlinear monotone equations with applications

A three-term projection method based on spectral secant equation for nonlinear monotone equations

A three-term projection method based on spectral secant equation for nonlinear monotone equations

A projection method for solving monotone nonlinear equations with application

Nonlinear equations are one of the most trending research areas due to their vast applications in the sciences, social sciences, and engineering. However, the conjugate gradient method (CG) is one of the most rapidly developing iterative techniques for solving nonlinear monotone problems. Recently, much work has been done on using the CG method to solve monotone nonlinear equations. This paper discusses a new variant for solving constrained monotone nonlinear equations. The method satisfies the sufficient descent condition and proves global convergence and R-linear convergence with the help of some reasonable assumptions. In addition, two sets of numerical tests were conducted. The first experiment shows the good performance of the proposed method compared to existing methods, while the second experiment displays the performance of the proposed algorithm in compressive sensing.

A sufficient descent LS-PRP-BFGS-like method for solving nonlinear monotone equations with application to image restoration

A sufficient descent LS-PRP-BFGS-like method for solving nonlinear monotone equations with application to image restoration

A framework for convex-constrained monotone nonlinear equations and its special cases

A framework for convex-constrained monotone nonlinear equations and its special cases