Position Paper: Post-Solve Robustness in Decision Engines: Feasible Regions and Smoothness Under Perturbations

立场论文：决策引擎中的求解后鲁棒性：扰动下的可行域与平滑度

Abstract: Mixed-Integer Linear Programming (MILP) decision engines routinely output nominally optimal plans for high-stakes industrial systems. Yet deployment rarely matches solve-time assumptions: small perturbations in costs, demands, or resource availability can invalidate feasibility or trigger discontinuous shifts to qualitatively different solutions.

摘要： 混合整数线性规划（MILP）决策引擎通常为高风险工业系统输出名义上的最优计划。然而，实际部署情况往往与求解时的假设不符：成本、需求或资源可用性的微小扰动可能会导致可行性失效，或引发向性质截然不同的解的不连续跳变。

We argue that this post-solve robustness gap is a missing layer in today’s optimization pipelines and a missing evaluation dimension for learning-enabled decision systems. Rather than replacing robust optimization or stochastic programming, the proposed layer audits a solved incumbent and returns solver-backed evidence about how far that solution can be trusted.

我们认为，这种“求解后鲁棒性差距”是当前优化流程中缺失的一环，也是学习型决策系统评估维度中的空白。所提出的这一层并非要取代鲁棒优化或随机规划，而是旨在对已求解的现行方案进行审计，并提供基于求解器的证据，以证明该方案的可信度边界。

We formalize two central objects: (i) an $\epsilon$-near-optimal feasible neighborhood in parameter space, capturing when an incumbent remains feasible and near-optimal under perturbations, and (ii) solution smoothness in decision space, capturing whether nearby alternatives with small combinatorial edits remain competitive.

我们形式化了两个核心对象：(i) 参数空间中的 $\epsilon$-近优可行邻域，用于捕捉现行方案在扰动下保持可行且近优的条件；(ii) 决策空间中的解平滑度，用于捕捉经过微小组合编辑后的邻近替代方案是否仍具竞争力。

We then synthesize the most relevant partial answers from sensitivity and stability analysis, robust optimization, neighborhood search, adversarial testing, and learning-based enhancements, and articulate an agenda for a unified post-solve robustness layer. Concretely, we call for certified inner approximations around the incumbent, probabilistic robustness estimation with calibrated uncertainty, adversarial robustness margins, and learning-based prediction and explanation aligned with solver-backed verification.

随后，我们综合了来自灵敏度与稳定性分析、鲁棒优化、邻域搜索、对抗性测试以及基于学习的增强技术中最相关的部分答案，并阐述了构建统一的“求解后鲁棒性层”的议程。具体而言，我们呼吁实现围绕现行方案的认证内近似、带有校准不确定性的概率鲁棒性估计、对抗性鲁棒性边界，以及与求解器验证相一致的基于学习的预测与解释。

We conclude with a compact reporting template and evaluation protocol that would make robustness a first-class output of decision engines.

最后，我们提供了一个简洁的报告模板和评估协议，旨在使鲁棒性成为决策引擎的一项核心输出指标。