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On the Optimal Correction of Infeasible Systems of Linear Inequalities

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Abstract

We study the optimum correction of infeasible systems of linear inequalities through making minimal changes in the coefficient matrix and the right-hand side vector by using the Frobenius norm. It leads to a special structured unconstrained nonlinear and nonconvex problem, which can be reformulated as a one-dimensional parametric minimization problem such that each objective function corresponds to a trust region subproblem. We show that, under some assumptions, the parametric function is differentiable and strictly unimodal. We present optimally conditions, propose lower and upper bounds on the optimal value and discuss attainability of the optimal value. To solve the original problem, we propose a binary search method accompanied by a type of Newton–Lagrange method for solving the subproblem. The numerical results illustrate the effectiveness of the suggested method.

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Acknowledgements

This work was supported by the Czech Science Foundation Grant P403-18-04735S.

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Authors and Affiliations

  1. Department of Mathematics, University of Bojnord, Bojnord, Iran

    Hossein Moosaei

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Hossein Moosaei & Milan Hladík

Authors
  1. Hossein Moosaei
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  2. Milan Hladík
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Correspondence to Hossein Moosaei.

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Communicated by Florian Potra.

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Moosaei, H., Hladík, M. On the Optimal Correction of Infeasible Systems of Linear Inequalities. J Optim Theory Appl 190, 32–55 (2021). https://doi.org/10.1007/s10957-021-01868-1

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