Journal article
Interior-point methods for nonconvex nonlinear programming: cubic regularization
Computational optimization and applications, v 58(2), pp 323-346
01 Jun 2014
Abstract
In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization step with an appropriately chosen regularization parameter. This equivalence allows us to use the favorable theoretical results of Griewank (The modification of Newton's method for unconstrained optimization by bounding cubic terms, 1981), Nesterov and Polyak (Math. Program., Ser. A 108:177-205, 2006), Cartis et al. (Math. Program., Ser. A 127:245-295, 2011; Math. Program., Ser. A 130:295-319, 2011), but its application at every iteration of the algorithm, as proposed by these papers, is computationally expensive. We propose a hybrid method: use a Newton direction with a line search on iterations with positive definite Hessians and a cubic step, found using a sufficiently large LM perturbation to guarantee a steplength of 1, otherwise. Numerical results are provided on a large library of problems to illustrate the robustness and efficiency of the proposed approach on both unconstrained and constrained problems.
Metrics
Details
- Title
- Interior-point methods for nonconvex nonlinear programming: cubic regularization
- Creators
- Hande Y. Benson - Drexel UniversityDavid F. Shanno - Rutgers, The State University of New Jersey
- Publication Details
- Computational optimization and applications, v 58(2), pp 323-346
- Publisher
- Springer Nature
- Number of pages
- 24
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Decision Sciences (and Management Information Systems)
- Web of Science ID
- WOS:000335730500002
- Scopus ID
- 2-s2.0-84901198325
- Other Identifier
- 991019168176204721
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied
- Operations Research & Management Science