We consider a class of optimization problems defined by a system of linear equations with min and max operators. This class of optimization problems has been studied under restrictive conditions, such as, (C1) the halting or stability condition; (C2) the non-negative coefficients condition; (C3) the sum up to 1 condition; and (C4) the only min or only max oerator condition. Several seminal results in the literature focus on special cases. For example, turn-based stochastic games correspond to conditions C2 and C3; and Markov decision process to conditions C2, C3, and C4. However, the systematic computational complexity study of all the cases has not been explored, which we address in this work. Some highlights of our results are: with conditions C2 and C4, and with conditions C3 and C4, the problem is NP-complete, whereas with condition C1 only, the problem is in UP intersects coUP. Finally, we establish the computational complexity of the decision problem of checking the respective conditions.

翻译：本文研究一类由带有最小和最大算子的线性方程组定义的优化问题。该类优化问题此前仅在限制性条件下被研究，例如：(C1) 停机或稳定性条件；(C2) 非负系数条件；(C3) 和为1条件；(C4) 仅含最小或仅含最大算子条件。文献中的若干奠基性成果聚焦于特殊情形：例如回合制随机博弈对应条件C2与C3；马尔可夫决策过程对应条件C2、C3与C4。然而，针对所有情形的系统性计算复杂性研究尚未展开，这正是本工作的研究目标。我们成果的若干亮点包括：在条件C2与C4下，以及在条件C3与C4下，该问题为NP完全问题；而仅满足条件C1时，该问题属于UP与coUP的交集。最后，我们建立了检验各条件本身的判定问题的计算复杂性。