Milan Hladík. Linear programming sensitivity measured by the optimal value worst-case analysis. Optim. Methods Softw., 2024. in press
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This paper introduces the concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the worst case optimal value of an interval LP problem when the nominal data are inflated to intervals according to given perturbation patterns. By definition, the derivative expresses how the optimal value can worsen when the data are subject to variations. In addition, it also gives a certain sensitivity measure or condition number of an LP problem. If the LP problem is nondegenerate, the derivatives are easy to calculate from the computed primal and dual optimal solutions. For degenerate problems, the computation is more difficult. We propose an upper bound and some kind of characterization, but there are many open problems remaining. We carried out numerical experiments with specific LP problems and with real LP data from Netlib repository. They show that the derivatives give a suitable sensitivity measure of LP problems. It remains an open problem how to efficiently and rigorously handle degenerate problems.
@article{Hla2023uc,
author ="Milan Hlad\'{\i}k",
title = "Linear programming sensitivity measured by the optimal value worst-case analysis",
journal = "Optim. Methods Softw.",
fjournal = "Optimization Methods and Software",
note = "in press",
year = "2024",
doi = "10.1080/10556788.2024.2329590",
issn = "1029-4937",
url = "https://www.tandfonline.com/doi/full/10.1080/10556788.2024.2329590",
bib2html_dl_html = "https://doi.org/10.1080/10556788.2024.2329590",
abstract = "This paper introduces the concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the worst case optimal value of an interval LP problem when the nominal data are inflated to intervals according to given perturbation patterns. By definition, the derivative expresses how the optimal value can worsen when the data are subject to variations. In addition, it also gives a certain sensitivity measure or condition number of an LP problem. If the LP problem is nondegenerate, the derivatives are easy to calculate from the computed primal and dual optimal solutions. For degenerate problems, the computation is more difficult. We propose an upper bound and some kind of characterization, but there are many open problems remaining. We carried out numerical experiments with specific LP problems and with real LP data from Netlib repository. They show that the derivatives give a suitable sensitivity measure of LP problems. It remains an open problem how to efficiently and rigorously handle degenerate problems.",
keywords = "Linear programming; Sensitivity analysis; Interval analysis; Condition number",
}
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