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.
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Notes
Recall Example C from Sect. 1.2: we are to find a payoff matrix \(A \in \varTheta \) to achieve a prescribed game value. Assume that in addition, a fixed matrix \(A_0\) is given. For \(A, A' \in \varTheta ^*\), define \(A \le ^* A'\) iff \(\Vert A - A_0\Vert \ge \Vert A' - A_0\Vert \), where \(\Vert \cdot \Vert \) is a matrix norm. This problem can be read as follows: find a payoff matrix such to achieve the prescribed game value, and if the solution is not unique, then find the solution which differs from the reference game \(A_0\) as little as possible.
This leads us to computational geometry, where the process or replacement of an intricate set A by another simpler object approximating A in some well defined sense is referred to as “geometric rounding” of A.
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The work of both authors was supported by the Czech Science Foundation under Grant P403/12/1947.
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Černý, M., Hladík, M. Inverse optimization: towards the optimal parameter set of inverse LP with interval coefficients. Cent Eur J Oper Res 24, 747–762 (2016). https://doi.org/10.1007/s10100-015-0402-y
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DOI: https://doi.org/10.1007/s10100-015-0402-y