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Solving systems of nonlinear equations has evolved into an active research field, with numerous iterative methods being proposed. Notably, iterative methods characterized by fast convergence remain of interest. In this paper, based on the modified line search scheme by Ou and Li, we introduce a derivative-free algorithm with a relaxed-inertial technique for approximating solutions of nonlinear systems involving pseudo-monotone mappings in Euclidean space. The global convergence of the proposed algorithm is established without Lipschitz continuity of the underlying mapping. Moreover, our approach allows flexibility in selecting the inertial extrapolation step length within the interval [0, 1]. To show the efficiency of the proposed method, we embed a derivative-free search direction into the scheme. Numerical experiments are given to illustrate the efficiency of the proposed algorithm for large-scale systems and sparse signal reconstruction. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view. [ABSTRACT FROM AUTHOR]

We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions and exploits a new strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from the CUTEst package. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient-like algorithm is proposed to solve uncon-strained optimization problems. The step directions generated by the new algorithm satisfy sufficient descent condition independent of the line search. The global convergence of the new algorithm, with the Armijo backtracking line search, is proved. Numerical experiments indicate the efficiency and robustness of the new algorithm. [ABSTRACT FROM AUTHOR]

Optimization using radial basis functions as an interpolation tool in trust region (ORBIT) is a derivative-free framework based on fully linear radial basis function (RBF) models. In this paper, an improved version of ORBIT algorithm based on two novel ideas is proposed. The accuracy and stability of RBFs depend on a so-called shape parameter, so it is more appropriate to determine the shape parameter according to the optimization problem. While ORBIT in all problems uses a fixed value as a shape parameter, our new version, Hybrid-ORBIT, uses a statistical technique to select an appropriate shape parameter. In addition, ORBIT uses some stored points to build a fully linear RBF model without considering their function values, while in the Hybrid-ORBIT algorithm the stored points are sorted based on their function values and the RBF model is built using the points with lower function values, and the best point in the sense of function value is defined as the trust-region center. Numerical results indicate the efficiency of the improved version compared with the original version. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid derivative-free conjugate gradient method that inherits the structures of two conjugate gradient methods is introduced to recover sparse signal in compressive sensing by solving the nonlinear convex constrained equations. The global convergence of the proposed method is proved, under some appropriate assumptions. Numerical experiments and comparisons suggest that the proposed algorithm is an efficient approach for sparse signal and image reconstruction in compressive sensing. [ABSTRACT FROM AUTHOR]