AI 中文总结
该研究系统分类静态、两阶段及K-适应性三类鲁棒优化问题的计算复杂性，证明其归属于多项式层次、实数存在理论或不可判定问题，还给出双层线性优化属NP类的另一证明及K-适应性的相关归约结论。
AI 中文摘要
我们研究三类经典鲁棒优化问题的决策版本的计算复杂性：静态鲁棒优化、两阶段（可调整）鲁棒优化以及K-适应性。我们考虑可行性集合、不确定性集合和追索集合为多面体或整数规划可表示集合，不确定性为决策独立或决策依赖，可行区域为有界或无界。尽管这些问题在现有文献中已得到充分研究，我们仍给出系统分类，将所得问题归为P类、NP类、coNP类及多项式层次的更高层级（Σ₂^p和Σ₃^p）；当决策依赖不确定性引入二次约束后，问题归为实数存在理论及其层次（∃ℝ、Σ₂ℝ和Σ₃ℝ）或不可判定问题。除了难解性归约，我们特别关注成员资格证明，即便所考虑集合通常包含编码长度为指数的向量，仍建立了多项式大小的证书。作为副产品，我们利用双层线性优化与决策依赖鲁棒优化的关联，给出双层线性优化属于NP类的另一种证明。对于K-适应性，我们将该问题与其静态和两阶段对应问题关联，表明难解性随K单调增长，且对于完全离散的决策依赖实例，K-适应性可归约回静态问题。
英文摘要
We study the computational complexity of the decision versions of three classic robust optimization problems: static robust optimization, two-stage (adjustable) robust optimization, and $K$-adaptability. We consider that the feasibility, uncertainty, and recourse sets are polyhedra or integer-programming-representable sets, the uncertainty is decision-independent or decision-dependent, and the feasible regions are bounded or unbounded. While these problems are well-established in the current literature, we give a systematic classification that places the resulting problems in $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{coNP}$, and higher levels of the polynomial hierarchy ($Σ_2^p$ and $Σ_3^p$), and, once decision-dependent uncertainty introduces quadratic constraints, in the existential theory of the reals and its hierarchy ($\exists\mathbb{R}$, $Σ_2\mathbb{R}$, and $Σ_3\mathbb{R}$) or among the undecidable problems. Beyond hardness reductions, we pay particular attention to membership proofs, establishing polynomial-size certificates even though the sets considered generally contain vectors of exponential encoding length. As a by-product, we give an alternative proof that bilevel linear optimization lies in $\mathsf{NP}$, exploiting its connection to decision-dependent robust optimization. For $K$-adaptability, we relate the problem to its static and two-stage counterparts, showing that hardness grows monotonically with $K$ and that, for fully discrete decision-dependent instances, $K$-adaptability reduces back to the static problem.
