高效线性约束矩阵线性修改下的LP预热启动
Efficient LP warmstarting for linear modifications of the constraint matrix
AI总结：
本文提出三种高效算法，用于在线性约束矩阵线性修改下快速计算线性规划的最优解和目标。
摘要：
我们考虑在线性变化的线性约束下计算线性规划最优解和目标的问题。研究的问题由$\min c^t x \text{ s.t } Ax + λDx \leq b$给出，其中$λ$属于预定义的值集$Λ$。基于预先计算的基的信息，我们提出了三种高效的LP预热启动算法。每种算法要么基于特征值分解，要么基于Schur分解，或者对特征值分解进行调整，以评估这些问题的最优解和最优目标。三种算法的总体复杂度为$O(pm^2+pmn)$，其中$m$（resp. $n$）是原始问题的约束数（resp. 变量数），$p$是预处理步骤后的$Λ$中的值数。我们还提供了与最优条件相关的定理，以验证基是否仍然最优以及目标的局部界。
英文摘要：
We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by $\min c^t x \text{ s.t } Ax + λDx \leq b$ where $λ$ belongs to a set of predefined values $Λ$. Based on the information given by a precomputed basis, we present three efficient LP warm-starting algorithms. Each algorithm is either based on the eigenvalue decomposition, the Schur decomposition, or a tweaked eigenvalue decomposition to evaluate the optimal solution and optimal objective of these problems. The three algorithms have an overall complexity $O(pm^2+pmn)$ where $m$ (resp. $n$) is the number of constraints (resp. variables) of the original problem and $p$ the number of values in $Λ$ after an initial preprocessing step. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
