An improved quasi-Newton equation on the quasi-Newton methods for unconstrained optimizations

<span><span>Quasi-Newton methods are a class of numerical methods for </span>solving the problem of unconstrained optimization. To improve the overall efficiency of resulting algorithms, we use the quasi-Newton methods which is interesting for quasi-Newton equation. In this manuscript, we present a modified BFGS update formula based on the new quasi-Newton equation, which give a new search direction for solving unconstrained optimizations proplems. We analyse the convergence rate of quasi-Newton method under some mild condition. Numerical experiments are conducted to demonstrate the efficiency of new methods using some test problems. The results indicates that the proposed method is competitive compared to the BFGS methods as it yielded fewer iteration and fewer function evaluations.</span>

Modified BFGS Update (H-Version) Based on the Determinant Property of Inverse of Hessian Matrix for Unconstrained Optimization

The study presents the modification of the Broyden-Flecher-Goldfarb-Shanno (BFGS) update (H-Version) based on the determinant property of inverse of Hessian matrix (second derivative of the objective function), via updating of the vector s ( the difference between the next solution and the current solution), such that the determinant of the next inverse of Hessian matrix is equal to the determinant of the current inverse of Hessian matrix at every iteration. Moreover, the sequence of inverse of Hessian matrix generated by the method would never approach a near-singular matrix, such that the program would never break before the minimum value of the objective function is obtained. Moreover, the new modification of BFGS update (H-version) preserves the symmetric property and the positive definite property without any condition.

Global Convergence of a Modified Two-Parameter Scaled BFGS Method with Yuan-Wei-Lu Line Search for Unconstrained Optimization

The BFGS method is one of the most efficient quasi-Newton methods for solving small- and medium-size unconstrained optimization problems. For the sake of exploring its more interesting properties, a modified two-parameter scaled BFGS method is stated in this paper. The intention of the modified scaled BFGS method is to improve the eigenvalues structure of the BFGS update. In this method, the first two terms and the last term of the standard BFGS update formula are scaled with two different positive parameters, and the new value of yk is given. Meanwhile, Yuan-Wei-Lu line search is also proposed. Under the mentioned line search, the modified two-parameter scaled BFGS method is globally convergent for nonconvex functions. The extensive numerical experiments show that this form of the scaled BFGS method outperforms the standard BFGS method or some similar scaled methods.