Abstract
We discuss the basic concepts of interior point methods for linear programming, viz., duality, the existence of a strictly complementary solution, analytic centers and the central path with its properties. To solve the initialization problem we give an embedding of the primal and the dual problem in a skew-symmetric self-dual reformulation that has an obvious initial interior point. Finally, we consider the topic of interior point based sensitivity analysis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. Adler and R.D.C. Monteiro. A geometric view of parametric linear programming.Algorithmica, 8: 161–176, 1992.
M. Akgül. A note on shadow prices in linear programming.J. Opl. Res. Soc., 35: 425–431, 1984.
E.D. Andersen and Y. Ye. Combining interior-point and pivoting algorithms for linear programming. Technical Report, Department of Management Sciences, University of Iowa, Iowa City, USA, 1994
D.C. Aucamp and D.I. Steinberg. The computation of shadow prices in linear programming.J. Opl. Res. Soc., 33: 557–565, 1982.
J.F. Benders. Partitioning procedures for solving mixed variables programming problems.Numerische Mathematik, 4: 238–252, 1962.
J.R. Evans and N.R. Baker. Degeneracy and the (mis)interpretation of sensitivity analysis in linear programming.Decision Sciences, 13: 348–354, 1982.
A.V. Fiacco and G.P. McCormick.Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley & Sons, New York, 1968. (Reprint: Volume 4 ofSIAM Classics in Applied Mathematics, SI AM Publications, Philadelphia, USA, 1990 ).
T. Gal.Postoptimal analyses, parametric programming and related topics. Mac-Graw Hill Inc., New York/Berlin, 1979.
T. Gal. Shadow prices and sensitivity analysis in linear programming under degeneracy, state-of-the-art-survey.OR Spektrum, 8: 59–71, 1986.
T. Gal. Weakly redundant constraints and their impact on postoptimal analyses in LP. Diskussionsbeitrag 151, FernUniversität Hagen, Hagen, Germany, 1990.
J. Gauvin. Quelques precisions sur les prix marginaux en programmation lineaire.INFOR, 18:68-73, 1980. (In French).
A.J. Goldman and A.W. Tucker. Theory of linear programming. In H.W. Kuhn and A.W. Tucker, editors,Linear Inequalities and Related Systems, Annals of Mathematical Studies, No.38, pages 53–97. Princeton University Press, Princeton, New Jersey, 1956.
J. Gondzio and T. Terlaky. A computational view of interior-point methods for linear programming. In J. Beasley, editor,Advances in linear and integer programming. Oxford University Press, Oxford, UK, 1995.
H.J. Greenberg. An analysis of degeneracy.Naval Research Logistics Quarterly, 33: 635–655, 1986.
H.J. Greenberg. The use of the optimal partition in a linear programming solution for postoptimal analysis.Operations Research Letters, 15: 179–186, 1994.
O. Güler, C. Roos, T. Terlaky, and J.-Ph. Vial. Interior point approach to the theory of linear programming. Cahiers de Recherche 1992.3, Faculte des Sciences Economique et Sociales, Universite de Geneve, Geneve, Switzerland, 1992. ( To appear in Management Science).
O. Güler and Y. Ye. Convergence behavior of interior-point algorithms.Mathematical Programming, 60: 215–228, 1993.
B. Jansen, C. Roos, and T. Terlaky. An interior point approach to postoptimal and parametric analysis in linear programming. Technical Report 92–21, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands, 1992.
J.J. de Jong. A computational study of recent approaches to sensitivity analysis in linear programming. Optimal basis, optimal partition and optimal value approach. Master’s thesis, Delft University of Technology, Delft, The Netherlands, 1993.
G. Knolmayer. The effects of degeneracy on cost-coefficient ranges and an algorithm to resolve interpretation problems.Decision Sciences, 15: 14–21, 1984.
M. Kojima, S. Mizuno, and A. Yoshise. A primal-dual interior point algorithm for linear programming. In N. Megiddo, editor,Progress in Mathematical Programming: Interior Point and Related Methods, pages 29–47. Springer Verlag, New York, 1989.
T.L. Magnanti and J.B. Orlin. Parametric linear programming and anti-cycling pivoting rules.Mathematical Programming, 41: 317–325, 1988.
T.L. Magnanti and R.T. Wong. Accelerating Benders decomposition: algorithmic enhancement and model selection criteria.Operations Research, 29: 464–484, 1981.
L. McLinden. The analogue of Moreau’s proximation theorem, with applications to the nonlinear complementarity problem.Pacific Journal of Mathematics, 88: 101–161, 1980.
N. Megiddo. Pathways to the optimal set in linear programming. In N. Megiddo, editor,Progress in Mathematical Programming: Interior Point and Related Methods, pages 131–158. Springer Verlag, New York, 1989.
S. Mehrotra and R.D.C. Monteiro. Parametric and range analysis for interior point methods. Technical Report, Dept. of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, USA, 1992
S. Mehrotra and Y. Ye. Finding an interior point in the optimal face of linear programs.Mathematical Programming, 62: 497–515, 1993.
R.D.C. Monteiro and I. Adler. Interior path following primal-dual algorithms: Part I: Linear programming.Mathematical Programming, 44: 27–41, 1989
R.T. Rockafellar.Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970.
D.S. Rubin and H.M. Wagner. Shadow prices: tips and traps for managers and instructors.Interfaces, 20: 150–157, 1990.
R. Sharda. Linear programming software for personal computers: 1992 survey.OR/MS Today, pages 44–60, June 1992.
G. Sonnevend. An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In A. Prekopa, J. Szelezsan, and B. Strazicky, editors,System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, volume 84 ofLecture Notes in Control and Information Sciences, pages 866–876. Springer Verlag, Berlin, Germany, 1986.
T. Terlaky. Onℓp-programming.European Journal of Operational Research, 22: 70–100, 1985.
T. Terlaky and S. Zhang. Pivot rules for linear programming: a survey on recent theoretical developments.Annals of Operations Research, 46: 203–233, 1993.
A.W. Tucker. Dual systems of homogeneous linear relations. In H.W. Kuhn and A.W. Tucker, editors,Linear Inequalities and Related Systems,Annals of Mathematical Studies, No.38, pages 3–18. Princeton University Press, Princeton, New Jersey, 1956.
J.E. Ward and R.E. Wendell. Approaches to sensitivity analysis in linear programmingAnnals of Operations Research, 27: 3–38, 1990.
X. Xu, P.F. Hung, and Y. Ye. A simplified homogeneous and self-dual linear programming algorithm and its implementation. Technical Report, Department of Mathematics, University of Iowa, Iowa City, Iowa, USA, 1994.
Y. Ye, M.J. Todd, and S. Mizuno. An O$$\[\left({\sqrt n L} \right)$$-iteration homogeneous and self-dual linear programming algorithm.Mathematics of Operations Research, 19: 53–67, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Jansen, B., Roos, C., Terlaky, T. (1996). Introduction to the Theory of Interior Point Methods. In: Terlaky, T. (eds) Interior Point Methods of Mathematical Programming. Applied Optimization, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3449-1_1
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3449-1_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3451-4
Online ISBN: 978-1-4613-3449-1
eBook Packages: Springer Book Archive