The goal of this work is to propose a new type of constraint for linear programs: inequalities having infinite, finite, and infinitesimal values in the right-hand side. Because of the nature of such constraints, the feasible region polyhedron becomes more complex, since its vertices can be represented by non-purely finite coordinates, and so is the optimum of the problem. The introduction of such constraints enlarges the class of linear programs, where those described by finite values only become a special case. To tackle optimization problems over such polyhedra, there is a need for an ad-hoc solving routine: this work proposes a generalization of the Simplex algorithm, which is able to solve common linear programs as corner cases. Finally, the study presents three relevant applications that can benefit from the use of these novel constraints, making the use of the extended Simplex algorithm essential. For each application, an exemplifying benchmark is solved, showing the effectiveness of the proposed routine.

中文翻译：

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右侧显示无穷大、有限值和无穷小值的线性规划


这项工作的目标是为线性规划提出一种新型的约束：在右侧具有无限、有限和无穷小值的不等式。由于这种约束的性质，可行区域多面体变得更加复杂，因为它的顶点可以用非纯有限坐标表示，因此问题的最优值也是如此。这种约束的引入扩大了线性规划的类，其中由有限值描述的线性规划仅成为一种特殊情况。为了解决这种多面体上的优化问题，需要一个临时求解例程：这项工作提出了单纯形算法的泛化，它能够将常见的线性规划作为极端情况进行求解。最后，该研究提出了三个相关的应用程序，这些应用程序可以从使用这些新约束中受益，这使得扩展单纯形算法的使用变得至关重要。对于每个应用程序，都会解决一个示例基准，显示建议的例程的有效性。