发表机构
NEC Corporation（日本电报电话株式会社）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究整数线性规划的数据驱动逆优化问题，通过将基于梯度的优化方法应用于次优性损失求解，给出投影次梯度下降实现与观测数据精确一致所需迭代次数，是样本数量等的显式函数。
AI 中文摘要
数据驱动逆优化问题（DDIOP）是估计解释观测最优解数据的目标函数参数（权重）的问题，在整数线性规划（ILP）等许多应用中出现。已知通过将基于梯度的优化方法应用于次优性损失，ILP的逆优化可在有限多次预言机迭代内精确求解，所需迭代次数以问题相关几何常数$\gamma(\ell_{\mathrm{sub}})$表示为$T=O(1/\gamma(\ell_{\mathrm{sub}})^2)$。但此前无法根据问题规模从下方界定$\gamma(\ell_{\mathrm{sub}})$，因此迭代次数不能作为问题规模的显式函数给出。本文给出当正向问题为ILP时，应用于次优性损失的投影次梯度下降实现与观测数据精确一致所需的迭代次数，它是样本数量、特征维度、特征范围和约束系数矩阵结构的完全显式函数，直至基本常数（权重集直径、步长参数和次优性损失的利普希茨常数）中的多项式因子。
英文摘要
A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as $T=O(1/γ(\ell_{\mathrm{sub}})^2)$ in terms of a problem-dependent geometric constant $γ(\ell_{\mathrm{sub}})$. However, no means of bounding $γ(\ell_{\mathrm{sub}})$ from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).
Comments34 page. This paper was splited from arXiv:2405.14273v7
