Abstract
This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint functions of the problem are analytic, we also characterize the limiting behavior of these paths as they approach the set of optimal solutions. A duality relationship between a certain pair of logarithmic barrier problems is also discussed.
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I. Adler and R. D. C. Monteiro, “Limiting behavior of the affine scaling continuous trajectories for linear programming problems,” Mathematical Programming, vol. 50, pp. 29-51, 1991.
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons: New York, 1968. Reprint: Volume 4 of “SIAM Classics in Applied Mathematics,” SIAM Publications, Philadelphia, PA 19104-2688, USA, 1990.
A. M. Geoffrion, “Duality in nonlinear programming: a simplified applications-oriented development,” SIAM Review, vol. 13, pp. 1-37, 1971.
O. Güler, “Limiting behavior of the weighted central paths in linear programming,” Mathematical Programming, vol. 65, pp. 347-363, 1994.
O. Güler and C. Roos and T. Terlaky and J. P. 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.
J.-B. Hiriart-Urruty and C. Lemaréchal, “Convex Analysis and Minimization Algorithms I,” volume 305 of Comprehensive Study in Mathematics. Springer-Verlag: New York, 1993.
M. Kojima, S. Mizuno and T. Noma, “Limiting behavior of trajectories by a continuation method for monotone complementarity problems,” Mathematics of Operations Research, vol. 15, pp. 662-675, 1990.
L. McLinden, “An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem,” Pacific Journal of Mathematics, vol. 88, pp. 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, pp. 131-158. Springer Verlag: New York, 1989. Identical version in Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, 1986, pp. 1-35.
N. Megiddo and M. Shub, “Boundary behavior of interior point algorithms in linear programming,” Mathematics of Operations Research, vol. 14, pp. 97-114, 1989.
R. D. C. Monteiro, “Convergence and boundary behavior of the projective scaling trajectories for linear programming,” Mathematics of Operations Research, vol. 16, pp. 842-858, 1991.
R. D. C. Monteiro and J.-S. Pang, “Properties of an interior-point mapping for mixed complementarity problems,” Mathematics of Operations Research, vol. 21, 629-654, 1996.
R. D. C. Monteiro and T. Tsuchiya, “Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem,” Mathematics of Operations Research, vol. 21, pp. 793-814, 1996.
R. T. Rockafellar, Convex Analysis, Princeton University Press: Princeton, NJ, 1970.
C. Witzgall, P. T. Boggs and P. D. Domich, “On the convergence behavior of trajectories for linear programming,” In J. C. Lagarias and M. J. Todd, editors, Mathematical Developments Arising from Linear Programming: Proceedings of a Joint Summer Research Conference held at Bowdoin College, Brunswick, Maine, USA, June/July 1988, volume 114 of Contemporary Mathematics, pp. 161-187, American Mathematical Society: Providence, Rhode Island, USA, 1990.
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Monteiro, R.D., Zou, F. On the Existence and Convergence of the Central Path for Convex Programming and Some Duality Results. Computational Optimization and Applications 10, 51–77 (1998). https://doi.org/10.1023/A:1018339901042
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DOI: https://doi.org/10.1023/A:1018339901042