Abstract
For linear programs, we consider schemes for the formation of a generalized central path, which arise under the simultaneous use of interior and exterior penalty terms in the traditional Lagrange function and the minimax problems generated by it. The advantage of the new schemes is that they do not require a priori knowledge of feasible interior points in the primal or dual problem. Moreover, when applied to problems with inconsistent constraints, the schemes automatically lead to some of their generalized solutions, which have an important applied content. Descriptions of the algorithms, their justification, and results of numerical experiments are presented.
Similar content being viewed by others
REFERENCES
C. Roos, T. Terlaky, and J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach (Wiley, Chichester, 1997).
I. I. Eremin, Theory of Linear Optimization (VSP, Utrecht, 2002).
I. I. Eremin, Vl. D. Mazurov, and N. N. Astaf’ev, Improper Problems of Linear and Convex Programming (Nauka, Moscow, 1983) [in Russian].
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Wiley, New York, 1981).
F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
I. I. Eremin and N. N. Astaf’ev, Introduction to the Theory of Linear and Convex Programming (Nauka, Moscow, 1976) [in Russian].
L. D. Popov, “Use of barrier functions for optimal correction of improper problems of linear programming of the 1st kind,” Autom. Remote Control 73 (3), 417–424 (2012). doi 10.1134/S0005117912030010
L. D. Popov, “Dual approach to the application of barrier functions for the optimal correction of improper linear programming problems of the first kind,” Proc. Steklov Inst. Math. 288 (Suppl. 1), S173–S179 (2015). doi 10.1134/S0081543815020170
Funding
This work was supported by the Russian Foundation for Basic Research (project no. 16-07-00266).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Tselishcheva
Rights and permissions
About this article
Cite this article
Popov, L.D. Interior Point Methods Adapted to Improper Linear Programs. Proc. Steklov Inst. Math. 309 (Suppl 1), S116–S124 (2020). https://doi.org/10.1134/S0081543820040148
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543820040148