Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A globally convergent primal-dual interior-point relaxation method for nonlinear programs


Authors: Xin-Wei Liu and Yu-Hong Dai
Journal: Math. Comp. 89 (2020), 1301-1329
MSC (2010): Primary 90C30, 90C51; Secondary 90C26
DOI: https://doi.org/10.1090/mcom/3487
Published electronically: December 6, 2019
MathSciNet review: 4063319
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the classic logarithmic barrier problem is equivalent to a particular logarithmic barrier positive relaxation problem with barrier and scaling parameters. Based on the equivalence, a line-search primal-dual interior-point relaxation method for nonlinear programs is presented. Our method does not require any primal or dual iterates to be interior-points, which has similarity to some warmstarting interior-point methods and is different from most of the globally convergent interior-point methods in the literature. A new logarithmic barrier penalty function dependent on both primal and dual variables is used to prompt the global convergence of the method, where the penalty parameter is adaptively updated. Without assuming any regularity condition, it is proved that our method will either terminate at an approximate KKT point of the original problem, an approximate infeasible stationary point, or an approximate singular stationary point of the original problem. Some preliminary numerical results are reported, including the results for a well-posed problem for which many line-search interior-point methods were demonstrated not to be globally convergent, a feasible problem for which the LICQ and the MFCQ fail to hold at the solution and an infeasible problem, and for some standard test problems of the CUTE collection. Correspondingly, for comparison we also report the numerical results obtained by the interior-point solver IPOPT. These results show that our algorithm is not only efficient for well-posed feasible problems, but is also applicable for some feasible problems without LICQ or MFCQ and some infeasible problems.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 90C30, 90C51, 90C26

Retrieve articles in all journals with MSC (2010): 90C30, 90C51, 90C26


Additional Information

Xin-Wei Liu
Affiliation: Institute of Mathematics, Hebei University of Technology, Tianjin 300401, People’s Republic of China
Email: mathlxw@hebut.edu.cn

Yu-Hong Dai
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Email: dyh@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/mcom/3487
Keywords: Nonlinear programming, constrained optimization, interior-point method, logarithmic barrier problem, global convergence
Received by editor(s): December 1, 2017
Received by editor(s) in revised form: August 15, 2018, April 7, 2019, May 29, 2019, and June 20, 2019
Published electronically: December 6, 2019
Additional Notes: The first author is the corresponding author
The research was supported in part by the Chinese NSF grants (nos. 11671116 and 11271107), the Major Research Plan of the National Natural Science Foundation of China (no. 91630202), and a grant (no. A2015202365) from the Hebei Natural Science Foundation of China
The second author was supported in part by the Chinese NSF grants (nos. 11631013, 11331012 and 81173633) and the National Key Basic Research Program of China (no. 2015CB856000)
Article copyright: © Copyright 2019 American Mathematical Society