Abstract
We analyze the convergence and the complexity of a potential reduction column generation algorithm for solving general convex or quasiconvex feasibility problems defined by a separation oracle. The oracle is called at the analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle. We show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm, provided that the feasible region has an nonempty interior. This result is based on the works of Ye [22] and Nesterov [16].
The research of the first author is supported by the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152 and by the FCAR of Quebec; the research of the second author is supported by the Natural Sciences and Engineering Research Council of Canada grant number OPG0090391; the research of the third author is supported by NSF grant DDM-9207347.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. Agmon (1954), “The Relaxation Method for Linear Inequalities,” Canadian Journal of Mathematics 6, 382–392.
D. S. Atkinson and P. M. Vaidya (1992), A Cutting Plane Algorithm that uses Analytic Centers, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA.
J. V. Burke, A. A. Goldstein, P. Tseng, and Y. Ye (1993), “Translation Cuts for Minimization,” Complexity in Numerical Optimization (Editor: P. M. Pardalos ), World Scientific, 57–73.
G. B. Dantzig and P. Wolfe (1961), “The Decomposition Algorithm for Linear Programming,” Econometrica 29, 767–778.
J. Elzinga and T. Moore (1975), “A Central Cutting Plane Algorithm for Convex Programming,” Mathematical Programming 8, 134–145.
J. L. Goffin (1980), “The Relaxation Method for Solving Systems of Linear Inequalities,” Mathematics of Operations Research 5, 388–414.
J. L. Goffin, A. Haurie, and J. P. Vial (1992), “Decomposition and Nondiffer- entiable Optimization with the Projective Algorithm,” Management Science 38, 284–302.
J. E. Kelley (1960), “The Cutting Plane Method for Solving Convex Programs,” Journal of the SIAM 8, 703 - 712.
L. G. Khacian (1980), “Polynomial Algorithms in Linear Programming,” Zh. vy- chisl. Mat. mat. Fiz, 20, No. 1 (1980) 51-68; translated in USSR Computational Mathematics and Mathematical Physics 20, No. 1, 53–72.
L. G. Khacian and M. J. Todd (1990), “On the Complexity of Approximating the Maximal Inscribed Ellipsoid for a Polytope,” Tech. Report No. 893, SORIE, Cornell University.
C. Lemaréchal, A. Nemirovskii and Y. Nesterov, “New Variants of Bundle Methods” to appear in Mathematical Programming, series B, Nondifferentiable and Large Scale Optimization, J. L. Goffin and J. P. Vial, editors.
A. Levin (1965), “An Algorithm of Minimization of Convex Functions,” Soviet Math 160, 6, 1244–1247. Doklady.
J. E. Mitchell (1988), “Karmarkar’s Algorithm and Combinatorial Optimization Problems,” Ph.D. Thesis, Department of ORIE, Cornell University, Ithaca, NY.
T. Motzkin and I. J. Schoenberg (1954), “The Relaxation Method for Linear Inequalities,” Canadian Journal of Mathematics 6, 393–404.
A. Nemirovsky and D. Yudin (1983), Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience, NY.
Y. Nesterov (1992), Cutting Plane Algorithms from Analytic Centers: Efficiency Estimates, University of Geneva, Geneva, Switzerland.
J. Renegar (1988), “A Polynomial-time Algorithm Based on Newton’s Method for Linear Programming”, Mathematical Programming 40, 59–94.
N. Z. Shor (1985), Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, Heidelberg.
G. Sonnevend (1988), “New Algorithms in Convex Programming Based on a Notion of “Centre” (for Systems of Analytic Inequalities) and on Rational Extrapolation,” in K. H. Hoffmann, J. B. Hiriat-Urruty, C. Lemarechal, and J. Zowe, editors, “Trends in Mathematical Optimization,” Proceedings of the 4th French- German Conference on Optimization in Irsee, West-Germany, April 1986, International Series of Numerical Mathematics 84, 311–327. Birkhäuser Verlag, Basel, Switzerland.
S. Tarasov, L. G. Khachiyan, I. Erlich (1988), “The Method of Inscribed Ellipsoids,” Soviet Math 37, Doklady.
P. Vaidya (1989), “A New Algorithm for Minimizing Convex Functions over Convex Sets,”to appear in Mathematical Programming.
Y. Ye (1992), “A Potential Reduction Algorithm Allowing Column Generation,” SIAM Journal on Optimization 2, 7 - 20.
Y. Ye, “Convergence of a Potential-Reduction and Column-Generation Algorithm for Convex Feasibility Problems,” Working Paper, Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242, USA.
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
Goffin, JL., Luo, ZQ., Ye, Y. (1994). On the Complexity of a Column Generation Algorithm for Convex or Quasiconvex Feasibility Problems. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds) Large Scale Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3632-7_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3632-7_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3634-1
Online ISBN: 978-1-4613-3632-7
eBook Packages: Springer Book Archive