In this work, motivated by the effect of inertial step in accelerating algorithms for solving nonlinear problems such as equilibrium problems, we propose two hybrid spectral algorithms with inertial effect for solving system of nonlinear equations with convex constraints. The search directions in these algorithms use the convex combination of the modified Barzilai and Borwein spectral parameters (IMA journal of numerical analysis, vol. 8, no. 1, pp. 141-148, 1988) and their geometric mean proposed by Dai et al. (In Numerical Analysis and Optimization, pp. 59-75, Springer, 2015). The incorporation of the inertial-step aids the proposed algorithms in producing more efficient results in comparison with three existing spectral algorithms. Under the assumption that the function under consideration is monotone and satisfies Lipschitz continuity, we prove the global convergence of the proposed algorithms. In addition, we also show the application of the proposed algorithms in motion control of two-joint planar robotic manipulator.

Many problems in science and engineering can be formulated as nonlinear least-squares (NLS) problems. Thus, the need for efficient algorithms to solve these problems can not be overemphasized. In that sense, we introduce a generalized structured-based diagonal Hessian algorithm for solving NLS problems. The formulation associated with this algorithm is essentially a generalization of a similar result in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021). However, in this work, the structured diagonal Hessian update is derived under a weighted Frobenius norm; this allows other choices of the weighted matrix analogous to the Davidon-Fletcher-Powell (DFP) method. Moreover, to theoretically fill the gap in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021), we have shown that the proposed algorithm is R-linearly convergent under some standard conditions devoid of any safeguarding strategy. Furthermore, we experimentally tested the proposed scheme on some standard benchmark problems in the literature. Finally, we applied this algorithm to solve robotic motion control problem consisting of 3DOF (degrees of freedom).

In this paper, we propose a modified conjugate descent (CD) projection algorithm for solving system of nonlinear monotone equations with convex constraints. The search direction in this algorithm use a convex combination of the steepest descent algorithm and the well-known CD method. The algorithm proves to be quite efficient for solving large scale monotone nonlinear equations, as it has low storage requirement and does not need the computation of Jacobian matrix. We prove the convergence of the algorithm using some conditions, and perform numerical experiments on some test problems. In order to show the efficiency of our proposed algorithm, the numerical performance is compared with some existing algorithms. Finally, by reformulating ℓ1 regularized problem into monotone equation, we successfully apply the algorithm to restore some blurred images. The numerical results obtained prove that the algorithm can be used as an efficient and qualitative solver for image restoration problems.