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In matrix analysis, the scaling technique reduces the chances of an ill-conditioning of the matrix. This article proposes a one-parameter scaling memoryless Davidon–Fletcher–Powell (DFP) algorithm for solving a system of monotone nonlinear equations with convex constraints. The measure function that involves all the eigenvalues of the memoryless DFP matrix is minimized to obtain the scaling parameter's optimal value. The resulting algorithm is matrix and derivative-free with low memory requirements and is globally convergent under some mild conditions. A numerical comparison showed that the algorithm is efficient in terms of the number of iterations, function evaluations, and CPU time. The performance of the algorithm is further illustrated by solving problems arising from image restoration. [ABSTRACT FROM AUTHOR]

A new accelerated common fixed point algorithm is introduced and analyzed for a countable family of nonexpansive mappings and then we apply it to solve some convex bilevel optimization problems. Then, under some suitable conditions, we prove a strong convergence result of the proposed algorithm. As an application, we employ the proposed algorithm for regression and classification problems. Moreover, we compare the performance of our algorithm with others. By numerical experiments, we found that our algorithm has a better performance than the others. [ABSTRACT FROM AUTHOR]

Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction of the proposed method is derived based on a modified conjugate gradient method, in such a way that it is sufficiently descent. It is worth noting that, unlike many existing methods that require a monotonicity assumption to prove the convergence result, our new method needs the underlying function to be pseudomonotone, which is a weaker assumption. The performance of the proposed algorithm is demonstrated on a set of some test problems and applications arising from compressive sensing. The obtained results confirm that the proposed method is effective compared to some existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

Two new inertial-type extragradient methods are proposed to find a numerical common solution to the variational inequality problem involving a pseudomonotone and Lipschitz continuous operator, as well as the fixed point problem in real Hilbert spaces with a ρ -demicontractive mapping. These inertial-type iterative methods use self-adaptive step size rules that do not require previous knowledge of the Lipschitz constant. We also show that the proposed methods strongly converge to a solution of the variational inequality and fixed point problems under appropriate standard test conditions. Finally, we present several numerical examples to show the effectiveness and validation of the proposed methods. [ABSTRACT FROM AUTHOR]