Adaptive Cubic Regularisation Quasi-Newton Algorithm And Numerical Experiments

Through getting the Quadratic Taylor expansions of the objective function, we can get an approximate module, which forms the subproblem of the optimization problem. When we calculate the exact Hessian matrix of the objective function, we often replace it with an approximate matrix. Usually we take the BFGS correction formulae to up-date the exact Hessian matrix of the objective function. The other method is that we assume the second derivative is Lipschitz continuous, the Lipschitz constant is L, using the continuity of the second derivation, we get an approximate module with a three items, the approximate module which forms the sub-problem of the initial problem has a better accuracy and adaptivity.This paper is organized as follows:In the second chapter, by taking the cubic overestimation of the objective function f, we get a new quasi-Newton equation, according to the new quasi-Newton equation, we get two new quasi-Newton updating formulaes, which are the adaptive symmetric rank-1updating formulae and the adaptive symmetric rank-2updating formulae. Besides, we analysis the difference between the adaptive symmetric updating formulae and the classical updating formulae.In the third chapter, we have implemented the adaptive cubic regularisation quasi-Newton algorithm, where Hessian is taken from the adaptive symmetric rank-2updating formulae. Preliminary numerical experiments with small-scale test problems form An Unconstrained Optimization Test Functions Collection, which shows encouraging perfor-mance of the adaptive cubic regularisation quasi-Newton algorithm when compared to an adaptive cubic regularisation algorithm implementation.