We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth, so hard to solve. In this paper, we study fundamental properties on the topology and the geometric shape of the solution set, and also conditions for convexity, connectedness, boundedness and integrality of the vertices. Further, we address various complexity issues, showing that many basic questions are NP-hard to solve. We show that the feasible set is a (nonconvex) polyhedral set and, more importantly, every nonconvex polyhedral set can be described by means of absolute value constraints. We also provide a necessary and sufficient condition when a KKT point of a nonconvex quadratic programming reformulation solves the original problem.

翻译：本文研究涉及绝对值形式的线性规划问题，此类问题无法用标准线性规划表达。绝对值的存在导致问题非凸且非光滑，求解困难。本文探讨了解集的拓扑与几何形态的基本性质，以及顶点凸性、连通性、有界性和整数性的条件。此外，我们讨论了各类复杂性难题，证明许多基本问题是NP-难解的。研究表明可行域是一个（非凸）多面体集，更重要的是，每个非凸多面体集均可通过绝对值约束描述。我们还给出了非凸二次规划重构的KKT点求解原问题的充要条件。