The Feasible Conjugate Gradient Method For Linearly Constrained Optimization Problem

In this paper,the nonlinear conjugate gradient method for solving unconstrained optimization problem is applied to solve linear equality constrained optimization problems, the global convergence of these methods is established ,numerical experiments are also given which show the efficient of the proposed methods.At first, combining the feasible direction method with the conjugate gradient method, we propose of a class of conjugate gradient method for solving linear equality constrained problem.When the method is used to solve a linear equality constrained quadratic programming, we prove that the conjugate gradient method with the exact line search terminate at the solution of the problem within the finite iterations. Further more , in this case the FR-type method, PRP-type method, CD-type method and DY-type method are equivalent. the result is extension of the subspace extension theorem for the conjugate gradient methods in the solution of unconstrained optimization problem.In chapter 2, under mild condition,we prove the global convergence of the FR-style method with exact line search for solving linear equality constrained optimization problem .In chapter 3, we propose a modified FR (MFR) style method for solving linear equality constrained optimization problem, and we also prove its global convergence in the case where inexact line search is used.Finally, we give the numerical experiments to investigate the performance of the proposed methods. The test problems are highly nonlinear, and the scale of the test problems vary from 2 dimension to 9000 dimension. The results of numerical experiments indicate that the proposed methods in this paper are effective for solving the linear equality constrained problem.