Abstract
We analyze the effect of adding and deleting a constraint of the linear program on the position of the central point, the “distance” to the path, and the change in the barrier function value. Based on these results we propose a column generation and deletion variant of the logarithmic barrier method for linear programming. The algorithm starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to or violates one of the constraints, which is in turn added to the current system. It also has the possibility of deleting constraints which are likely to be nonbinding in the optimal solutions. A complexity analysis for this algorithm will be given.
This work is completed with the support of a research grant from SHELL.
On leave from the Eötvös University, Budapest, and partially supported by OTKA No. 2116.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Atkinson, D.S., and Vaidya, P.M.: 1992, ‘A Cutting Plane Algorithm that Uses Analytic Centers’, Working Paper, Department of Mathematics, University of Illinois at Urbana-Champaign.
Bahn, O., Goffin, J.L., Vial, J.-Ph., and Du Merle, O.: 1991, ‘Implementation and Behavior of an Interior Point Cutting Plane Algorithm for Convex Programming: an Application to Geometric Programming’, Working Paper, Université de Genève, Genève, Switzerland. To Appear in Annals of Discrete Mathematics.
Dantzig, G.B., and Ye, Y.: 1990, ‘A Build-Up Interior Method for Linear Programming: Affine Scaling Form’, Technical Report SOL 90 -4, Systems Optimzation Laboratory, Department of Operations Research, Stanford University, Stanford, California.
Den Hertog, D., Roos, C., and Terlaky, T.: 1992, ‘A Build-Up Variant of the Path-Following Method for LP’, Operations Research Letters Vol. no. 12, pp. 181–186.
Den Hertog, D., Roos, C., and Vial, J.-Ph.: 1992, ‘A Complexity Reduction for the Long-Step Path-Following Algorithm for Linear Programming’, SIAM Journal on Optimization Vol. no. 2, pp. 71–87.
Den Hertog, D., Kaliski, J.A., Roos, C., and Terlaky, T.: 1992, ‘A Logarithmic Barrier Cutting Plane Method for Convex Programming’, Report No. 93–43, Faculty of Mathematics and Informatics, Delft University of Technology, Delft, Holland.
Goffin, J.L., and Vial, J.-Ph.: 1990, ‘Cutting Planes and Column Generation Techniques with the Projective Algorithm’, Journal of Optimization Theory and Applications Vol. no. 65, pp. 409–429.
Kaliski, J.A., and Ye, Y.: 1991, ‘A Decomposition Variant of the Potential Reduction Algorithm for Linear Programming’, Working Paper 91 — 11, Department of Management Sciences, The University of Iowa, Iowa City, Iowa.
Karmarkar, N.: 1984, ‘A New Polynomial-Time Algorithm for Linear Programming’, Combinatorica Vol. no. 4, pp. 373–395.
Mitchell, J.E.: 1988, ‘Karmarkar’s Algorithm and Combinatorial Optimization Problems’, Ph.D. Thesis, Cornell University, Ithaca.
Roos, C. and Vial, J.-Ph.: 1992, ‘A Polynomial Method of Approximate Centers for Linear Programming’, Mathematical Programming Vol. no. 54, pp. 295–305.
Tone, K.: 1991, ‘An Active-Set Strategy in Interior Point Method for Linear Programming’, Working Paper, Graduate School of Policy Science, Saitama University, Urawa, Saitama 338, Japan.
Vaidya, P.M.: 1989, ‘A New Algorithm for Minimizing Convex Functions over Convex Sets’, Proceedings of 30th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, California, 338–343. To Appear in Mathematical Programming.
Ye, Y.: 1989, ‘Eliminating Columns and Rows in Potential Reduction and Path-Following Algorithms for Linear Programming’, Working Paper Series No. 89 -7, Department of Management Sciences, The University of Iowa, Iowa City, Iowa.
Ye, Y.: 1990, ‘The “Build-Down” Scheme for Linear Programming’, Mathematical Programming Vol. no. 46, pp. 61–72.
Ye, Y.: 1991, ‘An O(n3L) Potential Reduction Algorithm for Linear Programming’, Mathematical Programming Vol. no. 50, pp. 239-258.
Ye, Y.: 1992, ‘A Potential Reduction Algorithm Allowing Column Generation’, SIAM Journal on Optimization Vol. no. 2, pp. 7–20.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Kluwer Academic Publishers
About this chapter
Cite this chapter
Den Hertog, D., Roos, C., Terlaky, T. (1994). Adding and Deleting Constraints in the Logarithmic Barrier Method for LP. In: Du, DZ., Sun, J. (eds) Advances in Optimization and Approximation. Nonconvex Optimization and Its Applications, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3629-7_8
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3629-7_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3631-0
Online ISBN: 978-1-4613-3629-7
eBook Packages: Springer Book Archive