混合整数 recourse 的凸近似的残差中心化与误差界
Residual Centering and Error Bounds for Convex Approximation of Mixed-Integer Recourse
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中文总结 AI 辅助
该研究针对两阶段随机规划的混合整数 recourse 函数凸近似问题，通过残差条件分解残差，建立误差界，为第一阶段决策提供质量保证，还扩展了相关估计与方法。
中文摘要 AI 辅助
我们研究两阶段随机规划中混合整数 recourse 函数的凸近似。对于第一阶段决策，相关近似误差是凸近似与整数 recourse 值函数之间的带符号期望残差。在有界不确定箱上，我们确定了能给出该期望误差确定性界的残差条件。核心条件是坐标切片中心化，在此条件下分部积分可得到各向异性全变差界和混合导数界，其中全坐标情形通过密度的 Vitali 变差表达。然而，即使对于全幺模上限 recourse，精确中心化也可能与凸性不兼容。因此，我们使用可交换坐标投影器将任意残差分解为中心化分量和显式切片均值缺陷。所得的缺陷调整界提供了可选的误差证书和统一的第一阶段决策质量保证。我们进一步确定了精确常数，将混合导数估计扩展到非光滑密度，并开发了该框架的张量积和几何扩展，以及 max-affine 凸拟合与审计方法。
英文摘要
We study convex approximations of mixed-integer recourse functions in two-stage stochastic programming. For first-stage decisions, the relevant approximation error is the signed expected residual between the convex approximation and the integer-recourse value function. On bounded uncertainty boxes, we identify residual conditions that yield deterministic bounds for this expectation error. The central condition is coordinate slice-centering, under which integration by parts gives an anisotropic total-variation bound and a mixed-derivative bound whose all-coordinate case is expressed through Vitali variation of the density. Exact centering, however, may be incompatible with convexity, even for totally unimodular ceiling recourse. We therefore use commuting coordinate projectors to decompose an arbitrary residual into a centered component and an explicit slice-mean defect. The resulting defect-adjusted bounds provide selectable error certificates and uniform first-stage decision-quality guarantees. We further establish sharp constants, extend the mixed-derivative estimate to nonsmooth densities, and develop tensor-product and geometric extensions of the framework, alongside a max-affine convex fitting and audit methodology.
