Linear Constrained Optimization Problem Of Indefinite Dogleg Path Methods

Optimization, which makes research on how to find the optimal solution among many feasible plans, is widely applied in many fields such as finance, trade, management and scientific research.Both line search and trust region algorithm are well-accepted methods in the optimization to assure global convergence. The modern trust region concept minimizes an appropriate quadratic model that approximates the objective function in a region centered at current iterate point. By correcting the trust region radius, we can pick acceptable step. Coleman and Li [4] presented a trust region affine scaling interior point algorithm for the minimization problem subject only to linear inequality constraints. By using affine scaling to formulate an appropriate quadratic function and trust region subproblem, difficulties imposed by constraints are overcome. For unconstrained minimization, Nocedal and Yuan [12] suggested a combination of the trust region and line search method which is called backtracking. Zhu [17] studied unconstrained optimal problem by combining optimal path and modified path with nonmonotonic trust region methods. Besides, Zhang and Xu [16] proposed a class of indefinite dogleg path methods.In this paper, we construct a relevant affine scaling matrix and introduce eigenvalue correcting value and negative curvilinear direction which are based on dogleg path, to solve the problem whose Hesse matrix is not definite. If the iterate direction doesn't satisfy acceptable rules, we can get new step which can decrease the function value by using both line search and interior point backtracking technique . This indefinite dogleg algorithm has less computation and fast convergence rate. Numerical results indicate that the algorithm is useful and effective in practice.