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Numerous attempts have been made to develop efficient methods for solving the system of constrained nonlinear equations due to its widespread use in diverse engineering applications. In this article, we present a family of inertial‐based derivative‐free projection methods with a correction step for solving such system, in which the selection of the derivative‐free search direction is flexible. This family does not require the computation of corresponding Jacobian matrix or approximate matrix at every iteration and possess the following theoretical properties: (i) the inertial‐based corrected direction framework always automatically satisfies the sufficient descent and trust region properties without specific search directions, and is independent of any line search; (ii) the global convergence of the proposed family is proven under a weaker monotonicity condition on the mapping E$$ E $$, without the typical monotonicity or pseudo‐monotonicity assumption; (iii) the results about convergence rate of the proposed family are established under slightly stronger assumptions. Furthermore, we propose two effective inertial‐based derivative‐free projection methods, each embedding a specific search direction into the proposed family. We present preliminary numerical experiments on certain test problems to demonstrate the effectiveness and superiority of the proposed methods in comparison with existing ones. Additionally, we utilize these methods for solving sparse signal restorations and image restorations in compressive sensing applications. [ABSTRACT FROM AUTHOR]

Al-Baali et al. (Comput. Optim. Appl. 60:89–110, 2015) have proposed a three-term conjugate gradient method which satisfies a sufficient descent condition and global convergence. In this paper, we extend this method to a family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial versions for constrained nonlinear pseudo-monotone equations. The accelerated gradient-descent method MSM and the relaxed-inertial strategy are incorporated into the proposed methods to obtain better computational performance. The global convergence of the extended methods are established theoretically without the Lipschitz continuity of the underlying mapping. Numerical results on constrained nonlinear equations show that the extended methods with different settings are efficient. The applicability and efficiency of the extended methods in the regularized decentralized logistic regression and sparse signal restoration problems are also presented and verified. [ABSTRACT FROM AUTHOR]

Combining the derivative-free projection with inertial technique, we propose a hybrid inertial spectral conjugate gradient projection method for solving constrained nonlinear monotone equations. The conjugate parameter is a hybrid modification based on the memoryless BFGS update. The spectral parameter is obtained from quasi-Newton equations and double-truncated to ensure the sufficient descent. The search direction with a restart procedure satisfies sufficient descent condition and the trust region property at each iteration, independent of the choice of line search. We also investigate the theoretical properties, such as the global convergence and linear convergence rate, of the inertial projection method under normal assumptions. Numerical performances indicate the superiority of the proposed method in solving large-scale equations and restoring the blurred images contaminated by the Gaussian noise. [ABSTRACT FROM AUTHOR]

In this work, by considering the hyperplane projection and hybrid techniques, three scaled three‐term conjugate gradient methods are extended to solve the system of constrained monotone nonlinear equations, and the developed methods have the advantages of low storage and only using function values. The new methods satisfy the sufficient descent condition independent of any line search criterion. It has been proved that three new methods converge globally under some mild conditions. The numerical experiments for constrained monotone nonlinear equations and image de‐blurring problems illustrate that the proposed methods are numerically effective and efficient. [ABSTRACT FROM AUTHOR]

The conjugate gradient projection method is one of the most effective methods for solving large-scale monotone nonlinear equations with convex constraints. In this paper, a new conjugate parameter is designed to generate the search direction, and an adaptive line search strategy is improved to yield the step size, and then, a new conjugate gradient projection method is proposed for large-scale monotone nonlinear equations with convex constraints. Under mild conditions, the proposed method is proved to be globally convergent. A large number of numerical experiments for the presented method and its comparisons are executed, which indicates that the presented method is very promising. Finally, the proposed method is applied to deal with the recovery of sparse signals. [ABSTRACT FROM AUTHOR]

This paper discusses a class of two-block nonconvex smooth optimization problems with nonlinear constraints. Based on a quadratically constrained quadratic programming (QCQP) approximation, an augmented Lagrangian function (ALF), and a Lagrangian splitting technique into small-scale subproblems, we propose a novel sequential quadratic programming (SQP) algorithm. First, inspired by the augmented Lagrangian method (ALM), we penalize the quadratic equality constraints associated with the QCQP approximation subproblem in its objective by means of the ALF, and then split the resulting subproblem into two small-scale ones, but both of them are not quadratic programming (QP) due to the square of the quadratic equality constraints in the objective. Second, by ignoring the three-order infinitesimal arising from the squared term, the two small-scale subproblems are reduced to two standard QP subproblems, which can yield an improved search direction. Third, taking the ALF of the discussed problem as a merit function, the next iterate point is generated by the Armijo line search. As a result, a new SQP method, called QCQP-based splitting SQP method, is proposed. Under suitable conditions, the global convergence, strong convergence, iteration complexity and convergence rate of the proposed method are analyzed and obtained. Finally, preliminary numerical experiments and applications were carried out, and these show that the proposed method is promising. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid version of the Rivaie–Mustafa–Ismail–Leong, RMIL for short, conjugate gradient method is proposed for unconstrained optimization problem. The search direction fulfills the sufficient descent condition and trust region property at each iteration, independent of any line search. We establish the global convergence of the proposed algorithm under normal assumptions and Wolfe line search. To move forward, incorporating with derivative-free projection technique, the proposed conjugate coefficient is extended to deal with nonlinear monotone equations. Without Lipschitz continuity assumption, the global convergence is proved with mild conditions. The numerical results for both unconstrained optimization instances and large-scale nonlinear equations prove the efficiency of the proposed methods with less computation cost. In particular, we also explore the practical applications on signal recovery and image denoising problems. [ABSTRACT FROM AUTHOR]