Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

Chee-Khian Sim ()
Additional contact information
Chee-Khian Sim: The Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2011, vol. 148, issue 1, No 6, 79-106

Abstract: Abstract An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. $\sqrt{\mu}$ and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.

Keywords: Semidefinite linear complementarity problem; Interior point methods; NT direction; Local convergence; Ordinary differential equations (search for similar items in EconPapers)
Date: 2011
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10957-010-9746-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:148:y:2011:i:1:d:10.1007_s10957-010-9746-6

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-010-9746-6

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().