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

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Abstract

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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Hebei University of Technology, Tianjin, China

    Xinwei Liu

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Zhong Guan Cun Donglu 55, 100190, Beijing, China

    Yaxiang Yuan

Authors
  1. Xinwei Liu
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  2. Yaxiang Yuan
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Correspondence to Yaxiang Yuan.

Additional information

This work was partially supported by Grants 10571039, 10231060, 10831006 of National Natural Science Foundation of China, and a grant from Hebei University of Technology.

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Liu, X., Yuan, Y. A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties. Math. Program. 125, 163–193 (2010). https://doi.org/10.1007/s10107-009-0272-y

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  • DOI: https://doi.org/10.1007/s10107-009-0272-y

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