Mixed Algorithm For Ill-conditioned Quadratic Unconstrained Optimization

In solving ill-conditioned unconstrained optimization the steepest descent method often raises the zigzaging phenomenon and results in slow convergence and completed wrong solutions.So it is not an acceptable means for selection.But if a proper initial point in some sense is provided for it,we may improve the effect of calculation and attain the objective of solving ill-conditioned unconstrained optimization.This paper defined " a best initial point " by constructing a related auxiliary equality constraint optimization.Based on an auxiliary CHSD algorithm for roughly improving initial point and ODE method for equality constraint optimization, we got a approximate value of " best initial point " by numerical integration, which we called " acceptabile initial point ". At last we got the optimal solution by the steepest descent method.And thus we got mixed algorithm I for solving middle and small scale of ill-conditioned quadratic unconstrained optimization. However mixed algorithm 2 is formed by combining algorithm CHSD with the steepest descent method for large-scale problems. These two algorithms can not only take as new ways for solving ill-conditioned square systems of linear equations but also have significance for solving general ill-conditioned unconstrained optimization.Our preliminary numerical results obtained by executing these mixed algorithms on six test problems including a matrix of Hilbert of 1000 order as its Hessian matrix are presented in the last chapper. Numerical tests show excellent accuracy and stability of mixed algorithms and simplicity and high-speed of algorithm CHSD. All these showed that the proposed mixed algorithms have expected effective anti-illconditioned ability.