A Globally Convergent Hybrid FR-PRP Conjugate Gradient Method for Unconstrained Optimization Problems

In this paper, a new conjugate gradient (CG) parameter is proposed through the convex combination of the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP) CG update parameters such that the conjugacy condition of Dai-Liao is satisfied. The computational efficiency of the PRP method and the convergence profile of the FR method motivated the choice of these two CG methods. The corresponding CG algorithm satisfies the sufficient descent property and was shown to be globally convergent under the strong Wolfe line search procedure. Numerical tests on selected benchmark test functions show that the algorithm is efficient and very competitive in comparison with some existing classical methods.

A method with inertial extrapolation step for convex constrained monotone equations

In recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method.

A new class of nonlinear conjugate gradient coefficients for unconstrained optimization

The nonlinear Conjugate gradient method (CGM) is a very effective way in solving large-scale optimization problems. Zhang et al. proposed a new CG coefficient which is defined by [Formula: see text]. They proved the sufficient descent condition and the global convergence for nonconvex minimization in strong Wolfe line search. In this paper, we prove that this CG coefficient possesses sufficient descent conditions and global convergence properties under the exact line search.

A New Conjugate Gradient Method and Application to Dynamic Load Identification Problems

In this paper, a modified conjugate gradient (MCG) algorithm is proposed for solving the force reconstruction problems in practical engineering. This new method is derived from a stable regularization operator and is also strictly proved using the mathematical theory. Moreover, we also prove the sufficient descent and global convergence characteristic of the newly developed algorithm. Finally, the proposed algorithm is applied to force reconstruction for the airfoil structure and composite laminated cylindrical shell. Numerical simulations show that the proposed method is highly efficient and has robust convergence performances. Additionally, the accuracy of the proposed algorithm in identifying the expected loads is satisfactory and acceptable in practical engineering.

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

The main work of this paper focuses on identifying the heat flux in inverse problem of two-dimensional nonhomogeneous parabolic equation, which has wide application in the industrial field such as steel-making and continuous casting. Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. Furthermore, in order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard algorithm with adaptive parameter (SD-LMAP) to solve this inverse problem. At last, compared with other methods, the results of simulation experiment show that this algorithm can obviously reduce the running time and iterative number.

A NEW HYBRID PRP-MMSIS CONJUGATE GRADIENT METHOD AND ITS APPLICATION IN PORTOFOLIO SELECTION

In this paper, we propose a new hybrid coefficient of conjugate gradient method (CG) for solving unconstrained optimization model. The new coefficient is combination of part the MMSIS (Malik et.al, 2020) and PRP (Polak, Ribi'ere \& Polyak, 1969) coefficients. Under exact line search, the search direction of new method satisfies the sufficient descent condition and based on certain assumption, we establish the global convergence properties. Using some test functions, numerical results show that the proposed method is more efficient than MMSIS method. Besides, the new method can be used to solve problem in minimizing portfolio selection risk .

New hyrid conjugate gradient method as a convex combination of HZ and CD methods

In this paper, a new hybrid conjugate gradient algorithm is proposed for solving unconstrained optimization problems, the conjugate gradient parameter [Formula: see text] is computed as a convex combination of [Formula: see text] and [Formula: see text]. Under the wolfe line search, we prove the sufficient descent and the global convergence. Numerical results are reported to show the effectiveness of our procedure.