Abstract
The notion of the central path plays an important role in the development of most primal-dual interior-point algorithms. In this work we prove that a related notion called the quasicentral path, introduced by Argáez in nonlinear programming, while being a less restrictive notion it is sufficiently strong to guide the iterates towards a solution to the problem. We use a new merit function for advancing to the quasicentral path, and weighted neighborhoods as proximity measures of this central region. We present some numerical results that demonstrate the effectiveness of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Argaez and Tapia chose the name of quasicentral path due to the fact that one of the conditions of the central path is omitted. The authors are fully aware of the fact that they use the term “quasicentral path” to denote mathematically would be known as a variety. However, we choose to retain the already established terminology originally introduced by Argaez and Tapia [1, 2].
References
Argáez, M.: Exact and inexact Newton linesearch interior-point methods for nonlinear programming. Ph.D thesis, Department of Computational and Applied Mathematics, Rice University (1997)
Argáez, M., Tapia, R.A.: On the global convergence of a modified augmented Lagrangian linesearch interior-point Newton method for nonlinear programming. J. Optimi. Theory Appl. 114(1), 1–25 (2002)
Argáez, M., Tapia, R.A., Velźquez, L.: Numerical comparisons of path-following strategies for a primal-dual interior-point method for nonlinear programming. J. of Optimi. Theory Appl. 114(2), 255–272 (2002)
Bixby, R.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50(1), 3–15 (2002)
Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)
EL-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation of the primal-dual interior point method for nonlinear programming. J. Optimi. Theory Appl. 89, 507–541 (1996)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1994)
Kojima, M., Mizuno, S., and Yoshise, A. 1989. A primal-dual interior point method for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior-Point and Related Methods, pp. 29–47. Springer, New York (1989). https://doi.org/10.1007/978-1-4613-9617-8_2
Lustig, I.J., Marsten, R., Shanno, D.F.: interior point methods for linear programming: computational state of the art. ORSA J. Comput. 6(1), 1–14 (1994)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)
Zhang, Y.: User’s guide to LIPSOL. Technical report TR95-19, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21228 (1995)
Acknowledgement
Drs. Argáez and Mendez would like to acknowledge the support of the Department of Mathematical Sciences and Dr. Velázquez the Richard Tapia Center of Excellence and Equity in Education at Rice University. The authors thank Dr. Richard A. Tapia for his reviews and helpful comments on this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Argáez, M., Mendez, O., Velázquez, L. (2022). The Notion of the Quasicentral Path in Linear Programming. In: Figueroa-García, J.C., Franco, C., Díaz-Gutierrez, Y., Hernández-Pérez, G. (eds) Applied Computer Sciences in Engineering. WEA 2022. Communications in Computer and Information Science, vol 1685. Springer, Cham. https://doi.org/10.1007/978-3-031-20611-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-031-20611-5_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20610-8
Online ISBN: 978-3-031-20611-5
eBook Packages: Computer ScienceComputer Science (R0)