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On the optimal solution set in interval linear programming

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Abstract

Determining the set of all optimal solutions of a linear program with interval data is one of the most challenging problems discussed in interval optimization. In this paper, we study the topological and geometric properties of the optimal set and examine sufficient conditions for its closedness, boundedness, connectedness and convexity. We also prove that testing boundedness is co-NP-hard for inequality-constrained problems with free variables. Furthermore, we prove that computing the exact interval hull of the optimal set is NP-hard for linear programs with an interval right-hand-side vector. We then propose a new decomposition method for approximating the optimal solution set based on complementary slackness and show that the method provides the exact description of the optimal set for problems with a fixed coefficient matrix. Finally, we conduct computational experiments to compare our method with the existing orthant decomposition method.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 11800, Prague, Czech Republic

    Elif Garajová & Milan Hladík

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  1. Elif Garajová
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  2. Milan Hladík
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Correspondence to Elif Garajová.

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This paper was presented at the 12th International Conference on Applied Mathematical Programming and Modelling (APMOD 2016) held in Brno, Czech Republic.

The authors were supported by the Czech Science Foundation Grant P403-18-04735S and by the Charles University, Project GA UK No. 156317.

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Garajová, E., Hladík, M. On the optimal solution set in interval linear programming. Comput Optim Appl 72, 269–292 (2019). https://doi.org/10.1007/s10589-018-0029-8

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