A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties

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• 作者：Xinwei Liu and Yaxiang Yuan
• 关键词：Global and local convergences ; Null ; space technique ; Primal ; dual interior ; point methods ; Nonlinear optimization with inequality and equality constraints
• 刊名：Mathematical Programming
• 出版年：2010
• 出版时间：September 2010
• 年：2010
• 卷：125
• 期：1
• 页码：163-193
• 全文大小：373.5 KB

文摘

We present a null-space primal-dual interior-point algorithm for solving nonlinear optimization problems with general inequality and equality constraints. The algorithm approximately solves a sequence of equality constrained barrier subproblems by computing a range-space step and a null-space step in every iteration. The ℓ₂ penalty function is taken as the merit function. Under very mild conditions on range-space steps and approximate Hessians, without assuming any regularity, it is proved that either every limit point of the iterate sequence is a Karush-Kuhn-Tucker point of the barrier subproblem and the penalty parameter remains bounded, or there exists a limit point that is either an infeasible stationary point of minimizing the ℓ ₂ norm of violations of constraints of the original problem, or a Fritz-John point of the original problem. In addition, we analyze the local convergence properties of the algorithm, and prove that by suitably controlling the exactness of range-space steps and selecting the barrier parameter and Hessian approximation, the algorithm generates a superlinearly or quadratically convergent step. The conditions on guaranteeing that all slack variables are still positive for a full step are presented.