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Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

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Abstract

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. In particular, the proposed algorithm runs in polynomial time for interval linear programs with no equality constraints.

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Acknowledgements

The work of M. Rada was supported by the Czech Science Foundation Grant P403/17-13086S. The work of M. Hladík and E. Garajová was supported by the Czech Science Foundation Grant P403/18-04735S. The work of E. Garajová was also supported by Grant No. 156317 of Grant Agency of Charles University.

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Authors and Affiliations

  1. Department of Financial Accounting and Auditing, Faculty of Finance and Accounting, University of Economics, W. Churchill’s Sq. 4, 130 67, Prague, Czech Republic

    Miroslav Rada

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík & Elif Garajová

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  1. Miroslav Rada
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  2. Milan Hladík
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  3. Elif Garajová
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Correspondence to Miroslav Rada.

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Rada, M., Hladík, M. & Garajová, E. Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases. Optim Lett 13, 875–890 (2019). https://doi.org/10.1007/s11590-018-1289-z

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