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.

我们通过使用Frobenius范数在系数矩阵和右侧向量中进行最小的更改来研究不可行的线性不等式系统的最佳校正。这导致了一个特殊的结构化无约束非线性和非凸问题，可以将其重新构造为一维参数最小化问题，以便每个目标函数都对应一个信任区域子问题。我们表明，在某些假设下，参数函数是可微的且严格为单峰的。我们提出了最佳条件，提出了最佳值的上限和下限，并讨论了最佳值的可达到性。为了解决原始问题，我们提出了一种二元搜索方法，并辅以一种Newton-Lagrange方法来解决子问题。