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On the Optimal Correction of Infeasible Systems of Linear Inequalities

Author

Listed:
  • Hossein Moosaei

    (University of Bojnord
    Charles University)

  • Milan Hladík

    (Charles University)

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.

Suggested Citation

  • Hossein Moosaei & Milan Hladík, 2021. "On the Optimal Correction of Infeasible Systems of Linear Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 32-55, July.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:1:d:10.1007_s10957-021-01868-1
    DOI: 10.1007/s10957-021-01868-1
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    References listed on IDEAS

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    1. Saeed Ketabchi & Hossein Moosaei, 2012. "Optimal Error Correction and Methods of Feasible Directions," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 209-216, July.
    2. C. Kanzow & H. Qi & L. Qi, 2003. "On the Minimum Norm Solution of Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 116(2), pages 333-345, February.
    3. O. L. Mangasarian, 2004. "A Newton Method for Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 1-18, April.
    4. Paula Amaral & Luís Fernandes & Joaquim Júdice & Hanif Sherali, 2009. "On optimal zero-preserving corrections for inconsistent linear systems," Computational Optimization and Applications, Springer, vol. 45(4), pages 645-666, December.
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