How bad are the BFGS and DFP methods when the objective function is quadratic?
Mathematical Programming: Series A and B
Multi-step quasi-Newton methods for optimization
ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
Using function-values in multi-step quasi-Newton methods
Proceedings of the 6th international congress on Computational and applied mathematics
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
A new structured quasi-Newton algorithm using partial information on Hessian
Journal of Computational and Applied Mathematics
Two new conjugate gradient methods based on modified secant equations
Journal of Computational and Applied Mathematics
A combined class of self-scaling and modified quasi-Newton methods
Computational Optimization and Applications
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For solving unconstrained minimization problems, quasi-Newton methods are popular iterative methods. The secant condition which employs only the gradient information is imposed on these methods. Several researchers paid attention to other secant conditions to get a better approximation of the Hessian matrix of the objective function. Recently, Zhang et al. [New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl. 102 (1999) 147-167] and Zhang and Xu [Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, J. Comput. Appl. Math. 137 (2001) 269-278] proposed the modified secant condition which uses both gradient and function value information in order to get a higher order accuracy in approximating the second curvature of the objective function. They showed the local and q-superlinear convergence property of the BFGS-like and DFP-like updates based on their proposed secant condition. In this paper, we incorporate one parameter into this secant condition to smoothly switch the standard secant condition and the secant condition of Zhang et al. We consider a modified Broyden family which includes the BFGS-like and the DFP-like updates proposed by Zhang et al. We prove the local and q-superlinear convergence of our method.