Michal Černý and Milan Hladík. Inverse optimization: towards the optimal parameter set of inverse LP with interval coefficients. Cent. Eur. J. Oper. Res., 24(3):747–762, 2016.
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We study the inverse optimization problem in the following formulation: given a family of parametrized optimization problems and a real number called demand, determine for which values of parameters the optimal value of the objective function equals to the demand. We formulate general questions and problems about the optimal parameter set and the optimal value function. Then we turn our attention to the case of linear programming, when parameters can be selected from given intervals (“inverse interval LP”). We prove that the problem is NP-hard not only in general, but even in a very special case. We inspect three special cases---the case when parameters appear in the right-hand sides, the case when parameters appear in the objective function, and the case when parameters appear in both the right-hand sides and the objective function. We design a technique based on parametric programming, which allows us to inspect the optimal parameter set. We illustrate the theory by examples.
@article{CerHla2016a,
author = "Michal {\v{C}}ern\'{y} and Milan Hlad\'{\i}k",
title = "Inverse optimization: towards the optimal parameter set of inverse {LP} with interval coefficients",
journal = "Cent. Eur. J. Oper. Res.",
fjournal = "Central European Journal of Operations Research",
volume = "24",
number = "3",
pages = "747-762",
year = "2016",
doi = "10.1007/s10100-015-0402-y",
issn = "1613-9178",
bib2html_dl_html = "http://dx.doi.org/10.1007/s10100-015-0402-y",
bib2html_dl_pdf = "https://rdcu.be/7t5I",
abstract = "We study the inverse optimization problem in the following formulation: given a family of parametrized optimization problems and a real number called demand, determine for which values of parameters the optimal value of the objective function equals to the demand. We formulate general questions and problems about the optimal parameter set and the optimal value function. Then we turn our attention to the case of linear programming, when parameters can be selected from given intervals (``inverse interval LP''). We prove that the problem is NP-hard not only in general, but even in a very special case. We inspect three special cases---the case when parameters appear in the right-hand sides, the case when parameters appear in the objective function, and the case when parameters appear in both the right-hand sides and the objective function. We design a technique based on parametric programming, which allows us to inspect the optimal parameter set. We illustrate the theory by examples.",
keywords = "Interval optimization; Inverse optimization; Linear programming; Parametric programming",
}
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