Abstract
In this paper, weak optimal inverse problems of interval linear programming (IvLP) are studied based on KKT conditions. Firstly, the problem is precisely defined. Specifically, by adjusting the minimum change of the current cost coefficient, a given weak solution can become optimal. Then, an equivalent characterization of weak optimal inverse IvLP problems is obtained. Finally, the problem is simplified without adjusting the cost coefficient of null variable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S Ahmed, Y Guan. The inverse optimal value problem, Mathematical Programming, 2005, 102(1): 91–110.
R K Ahuja, J B Orlin. Inverse optimization, Operations Research, 2001, 49(5): 771–783.
M S Bazaraa, J J Jarvis. Linear programming and network flows, J Wiley, New York, 1977.
D Burton, P L Toint. On the use of an inverse shortest paths algorithm for recovering linearly correlated costs, Mathematical Programming, 1994, 63(1): 1–22.
M Fiedler, J Nedoma, J Ramík, J Rohn, K Zimmermann. Linear optimization problems within exact data, Springer-Verlag, New York, 2006.
C Finn, S Levine, P Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization, JMLR: W&CP, 2016, 48: 49–58.
W Gerlach. Zur lösung linearer ungleichungssysteme bei störimg der rechten seite und der koeffizientenmatrix, Mathematische Operationsforschung und Statistik, Series Optimization, 1981, 12(1): 41–43.
X Guan, P M Pardalos, B Zhang. Inverse max+sum spanning tree problem under weighted l1norm by modifying the sum-cost vector, Optimization Letters, 2018, 12(5): 1065–1077.
M Hladík. Interval linear programming: A survey, Chapter 2, In: Mann ZA (ed) Linear Programming New Frontiers in Theory and Applications, Nova Science, New York, 2012, 85–120.
M Hladík. Weak and strong solvability of interval linear systems of equations and inequalities, Linear Algebra and Its Applications, 2013, 438(11): 4156–4165.
D S Hochbaum. Efficient Algorithms for the Inverse Spanning-Tree Problem, Operations Research, 2003, 51(5): 785–797.
H Ishibuchi, H Tanaka. Multiobjective programming in optimization of the interval objective function, European Journal of Operational Research, 1990, 48(2): 219–225.
C Jansson, S M Rump. Rigorous solution of linear programming problems with uncertain data, Zeitschrift für Operations Research, 1991, 35(2): 87–111.
D Li, Y Leung, W Wu. Multiobjective interval linear programming in admissible-order vector space, Information Sciences, 2019, 486: 1–19.
W Li, X Liu, H Li. Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions, Optimization Methods and Software, 2015, 30(3): 516–530.
W Li, J Luo, C Deng. Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming, Linear Algebra and Its Applications, 2013, 439(10): 3241–3255.
X Liu, W Li, P Liu. Construction method of constraint matrices corresponded by an optimal solution, Applied Mechanics and Materials, 2014, 513–517: 1617–1620.
J Luo, W Li. Strong optimal solutions of interval linear programming, Linear Algebra and Its Applications, 2013, 439(8): 2479–2493.
A Mostafaee, M Hladík, M Černỳ. Inverse linear programming with interval coefficients, Journal of Computational and Applied Mathematics, 2016, 292: 591–608.
M Mohammadi, M Gentili. Bounds on the worst optimal value in interval linear programming, Soft Computing, 2019, 23(21): 11055–11061.
J Novotná, M Hladík, T Masařík. Duality gap in interval linear programming, Journal of Optimization Theory and Applications, 2020, 184(2): 565–580.
W Oettli, W Prager. Compatibility of approximate solution of linear equations with given error bounds for coefficients and right hand sides, Numerische Mathematik, 1964, 6(1): 405–409.
M Rada, M Hladík, E Garajová. Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases, Optimization Letters, 2019, 13(4): 875–890.
J Rohn. Strong solvability of interval linear programming problems, Computing, 1981, 26(1): 79–82.
R E Steuer. Algorithms for linear programming problems with interval objective function coefficients, Mathematics of Operations Research, 1981, 6(3): 333–348.
H Zhang, T Feng, G Yang. Distributed cooperative optimal control for multiagent systems on directed graphs: An inverse optimal approach, IEEE Transactions on Cybernetics, 2015, 45(7): 1315–1326.
J Zhang, Z Liu. Calculating some inverse linear programming problems, Journal of Computational and Applied Mathematics, 1996, 72(2): 261–273.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China(11971433), First Class Discipline of Zhejiang — A (Zhejiang Gongshang University- Statistics, 1020JYN4120004G-091), Graduate Scientific Research and Innovation Foundation of Zhejiang Gongshang University.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Liu, X., Jiang, T. & Li, Hh. Weak optimal inverse problems of interval linear programming based on KKT conditions. Appl. Math. J. Chin. Univ. 36, 462–474 (2021). https://doi.org/10.1007/s11766-021-4324-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-021-4324-2