A Modified Class of Conjugate Gradient Algorithms Based on Quadratic Model for Nonlinear Unconstrained Optimization

Basim Hassan,Hameed Sadiq

doi:10.33899/csmj.2014.163729

Abstract

In this paper, we have investigated a new class of conjugate gradient algorithms for unconstrained non-linear optimization which are based on the quadratic model. Some theoretical results are investigated which are sufficient descent and ensure the local convergence of the new proposed algorithms. Numerical results show that the proposed algorithms are effective by comparing with the Polak and Ribiere algorithm.

Highlights

Our problem is to minimize a function of n variable minimize f (x), x R n....................(1)where, f is smooth and its gradient g(x) f (x) is available
Several formulas of k where considered, which are equivalent for strictly convex quadratic objective function .Within the framework of linear conjugate gradient method, the conjugacy condition is defined by d
The quadratic model is obtained from Taylor expansion of the function upto the second order terms, which can be written f

Summary

Introduction

Several formulas of k where considered, which are equivalent for strictly convex quadratic objective function .Within the framework of linear conjugate gradient method, the conjugacy condition is defined by d. = 0, i j for search directions, and this condition guarantees the finite termination of linear conjugate gradient methods. On the other hand, (2) and (3) are called the non-linear conjugate gradient method for general unconstrained optimization problem. Within the framework of nonlinear conjugate gradient methods, the conjugacy condition is replaced by d. Condition (7) means that the search directions dk+1 and d k are mutually conjugate with respect to Hessian matrix 2 f (x) at some point. The extension of conjugacy condition was studied by Perry [4] He tried to accelerate the conjugate gradient method by incorporating the second-order information into it. Section (5) establishes some numerical results to show the effectiveness of the proposed CG-method and Section (6) gives brief conclusions and discussions

New Formulae for k

The Descent Property and Descent Algorithm

Numerical Results

Conclusions and Discussions