| This article mainly deals with some latest developments of interior-point methods in the past few years. It consists of two sections. In the first section, the stable interior-point methods are discussed. The definition of qusi-definte matrix is introduced as well as its properties. We state that such kind of matrix is well-applied in the interior-point methods. In particular, this section deals with quadratic programming with strict equation restriction. The stable interior-point methods are modified so that the quasi-definite matrix can be applied in computing the direction from the system arising in the interior-point methods. At the end of this section, the improved algorithm and an example are provided. In the second section, we discuss the affine-scaling interior-point methods. The history as well as the latest developments of these methods are introduced. The optimal conditions of affine-scaling methods for general nonlinear problems are systematically considered, including the first order and second order optimal conditions. The newton methods basing on the affine-scaling optimal conditions are introduced. we discuss, furthermore, the situation where the strict complementary condition is violated. The affine-scaling matrix is modified and the computing of newton step is simplified without changing the convergence rate. |
