The Convergence Of Affine-scaling Interior-point Trust-region Methods For Optimization With Simple Bounds

To establish local convergence results for optimization problems with simple bounds, many papers assume strict complementarity at the solution. The main purpose of this paper is to weaken this assumption and to develop an affine-scaling interior-point trust-region algorithm that is globally and locally quadratic convergent.We first consider the quadratic model, which is established when to solve the first-order KKT condition of the problem by employing a quasi-Newton method. A trust region subproblem, which is defined by minimizing a quadratic function subject only to an ellipsoidal constraint, is solved in each iteration. We use a projection to maintain strict feasibility. Whether the trial steps are accepted depends on the objective function arid its approximation.Then we can establish global convergence to a stationary point, that is, if {xk} is the sequence generated by the affine-scaling interior-point trust-region method, then every limit point of the sequence is a stationary point for the problem.The local convergence properties of the algorithm are established. We assume that the strong second order sufficient optimality conditions at the limit point of the iterates are satisfied, but strict complementarity is not required. Our main result shows that the sequence {xk} converges to x at a quadratic rate.Finally a numerical example shows that the theoretical analysis accords with the numerical experiment result.