Abstract
We present algorithms to solve the problems originated from the application of the Dantzig and Wolfe decomposition principle to linear programming problems. One algorithm requires the sub-problems (slaves) completely solved at each iteration and the other uses sub-optimal solutions. All algorithms partially solve the master problem at each iteration. Also, we present a two-cut approach to save the excessive computation time in the restarting phase.
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. (1995). A cutting plane algorithm for convex programming that uses analytic centers. Mathematical Programming, (69):1–4.
Bahn, O., Merle O. du, Goffin, J. L., and Vial, J. P. (1995). A cutting plane method from analytic centers for stochastic programming. Mathematical Programming, (69):45–73.
Dantzig, G. B. and Wolfe, P. (1960). The decomposition algorithm for linear programs. Operations Research, (8):101–111.
Dantzig, G. B. and Ye, Y. (1990). A build-up interior method for linear programming: affine scaling form. Technical Report Sol 90–4, Dept. of Operations Research, Stanford, USA.
Den Hertog, D. (1992). Interior point approach to linear, quadratic and convex programming: algorithms and complexity. PhD thesis, University of Technology, Delft, The Netherlands.
Den Hertog, D., Roos, C., and Terlaky, T. (1992). A build-up variant of the path following method for 1p. Operations Research Letters, (12):181–186.
Feijoo, B., Sanchez, A., and Gonzaga, C. C. (1997). Maintaining closedness to the analytic center of a polytope by perturbing added hyperplanes. Applied Mathematics and Optimization,2(35):139–144.
Goffin, J. L.,, Gondzio, J., R., S., and Vial, J. P. (1994). Solving nonlinear multicommodity flow problems by the analytic center cutting plane method. Technical Report LOGILAB 1994. 21, GERARD, Faculty of management, McGill University.
Goffin, J. L., Haurie, A., and Vial, J. P. (1992). Decomposition and nondifferentiable optimization with the projective algorithm. Management Science, (38):284–302.
Goffin, J. L. and Vial, J. P. (1990). Cutting planes and column generation techniques with projective algorithm. Journal of Optimization Theory and Applications, (65):409–429.
Coffin, J. L. and Vial, J. P. (1997). A two-cut approach in the analytic center cutting plane method. Technical Report LOGILAB 97–20, Department of Management Studies, University of Geneva.
Goffin, J. L. and Vial, J. P. (1998). Multiple cuts in the analytic center method. Working Paper Logilab 98. 10, Department of Management Studies, University of Geneva.
Huard, P. (1967). Resolution of mathematical programming problems with nonlinear constraints by the method of centres. In Abadie, J., editor, NonLinear Programming,pages 207–21. North Holland Publishing Co. and Amsterdan.
Kelley, J. E. (1960). The cutting plane method for solving convex programming. J. Soc. Indus. Appl. Math., VIII (4): 703–712.
Kiwel, K. C. (1996). Complexity of some cutting plane methods that use analytic centers. Mathematical Programming, (74):47–54.
Mitchell, J. E. (1988). Karmakar“s algorithm and combinatorial optimization problems. PhD thesis, Cornell University.
Nesterov, Y. (1995). Complexity estimates of some cutting plane methods based on analytic barrier. Mathematical Programming, (69):149–176.
Oliveira, P. R. and Santos, M. A. d. (2000). Using analytic center and cutting planes methods for nonsmooth convex programming. In Nguyen, V., Strodiot, J., and Tossings, P., editors, Lecture Notes in Economics and Mathematical Systems-Optimization,number 481, pages 339–356. Springer-Verlag Heidelberg.
Roos, C. and Vial, J.-P. (1988). Analytic centers in linear programming. Report. 88–68, Delft University of Technology. Delft, Netherlands.
Santos, M. A. d., Lemos, A., and Oliveira, P. R. (2000). A multicut approach to the linear algebraic feasibility problem. Working paper, DCC-ICEx-UFMG, Belo Horizonte/MG, Brazil.
Santos, M. A. d. and Oliveira, P. R. (2000). Interior-point algorithms in decomposition schemes. Working paper, DCC-ICExUFMG, Belo Horizonte/MG,Brazil.
Sapatnekar, S. S., Rao, V. B., and Vaidya, P. M. (1991). A convex optimization approach to transistor sizing for cmos circuits. In Proceedings of the IEEE International Conference on Computer Aided Design, pages 482–485, Los Alamitos, CA. IEEE Computer Society Press.
Sonnevend, G. (1985). An analytical centre for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. Lecture Notes in Control and Information Sciences, (84):866–876.
Tone, K. (1991). An active-set strategy in interior point methods for linear programming. Working paper, Graduate school, Saita University, Urawa, Japan.
Ye, Y. (1992). A potential reduction algorithm allowing column generation. SIAM Journal of Optimization, (2):7–29.
Ye, Y. (1997). Complexity analysis of the analytic center cutting plane method that uses multiple cuts. Mathematical Programming, (78):85–104.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
dos Santos, M.A., Oliveira, P.R. (2001). Interior—Point Algorithms for Dantzig and Wolfe Decomposition Principle. In: Hadjisavvas, N., Pardalos, P.M. (eds) Advances in Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, vol 54. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0279-7_30
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0279-7_30
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-7923-6942-4
Online ISBN: 978-1-4613-0279-7
eBook Packages: Springer Book Archive