Hossein Moosaei and Milan Hladík. On the optimal correction of infeasible systems of linear inequalities. J. Optim. Theory Appl., 190(1):32–55, 2021.
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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.
@article{MooHla2021a,
author = "Hossein Moosaei and Milan Hlad\'{\i}k",
title = "On the optimal correction of infeasible systems of linear inequalities",
journal = "J. Optim. Theory Appl.",
fjournal = "Journal of Optimization Theory and Applications",
volume = "190",
number = "1",
pages = "32-55",
year = "2021",
doi = "10.1007/s10957-021-01868-1",
issn = "1573-2878",
url = "https://doi.org/10.1007/s10957-021-01868-1",
bib2html_dl_html = "https://link.springer.com/article/10.1007/s10957-021-01868-1",
bib2html_dl_pdf = "https://rdcu.be/cogDk",
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.",
keywords = "Systems of linear inequalities; Infeasible problems; Fractional
programming problem; Lower and upper bounds; SQP",
}
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