In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a Karush–Kuhn–Tucker (KKT) point and the inner iteration (SDPTR) that calculates an approximate barrier KKT point. Algorithm SDPTR combines a commutative class of Newton-like directions with the steepest descent type direction within the framework of the trust-region strategy. We present a trust-region method in primal-dual space and prove the global convergence property of the proposed method. Some numerical experiments are given. In addition, we also present second-order approximations to the primal-dual merit function, and a trust-region method in primal space in Appendix.

本文提出了一种求解非线性半定规划问题的原始对偶内点信赖域方法。该方法由找到Karush–Kuhn-Tucker（KKT）点的外部迭代（SDPIP）和计算近似势垒KKT点的内部迭代（SDPTR）组成。算法SDPTR在信任区域策略的框架内将牛顿式方向的可交换类与最速下降的方向组合在一起。我们提出了一种原始对偶空间中的信赖域方法，并证明了该方法的全局收敛性。给出了一些数值实验。此外，我们还在附录中介绍了原始-对偶优值函数的二阶逼近，以及原始空间中的信赖域方法。