使用分解技术求解不确定混合整数线性规划的鲁棒解
Robust solutions of uncertain mixed-integer linear programs using decomposition techniques
AI总结：
本文提出一种基于分解技术的鲁棒优化方法，通过将二次锥规划问题分解为线性主问题和较小规模的二次约束问题，提高大规模问题的求解效率并精确计算约束违反概率。
摘要：
鲁棒优化是一种用于处理数据不确定性的问题建模框架，近年来已成为活跃的研究领域。如果聚焦于具有不确定数据、二元决策和椭球不确定性集内硬约束的线性规划（LP）问题，本文提供了一种受分解技术启发的新鲁棒对应物（RC）解释。这种新解释允许提出一种专门的分解技术来求解RC问题，具有以下优点：i）特别提高了大规模问题的可处理性，ii）在不确定参数的概率分布完全定义时，能够提供约束违反的精确概率。我们方法的一个吸引人之处是将与鲁棒对应物相关的二次锥规划问题分解为一个线性主问题和不同规模较小的二次约束问题（QCP）。通过迭代方案中的切割平面近似来求解这些主问题和子问题，以获得最优解。此外，还给出了迭代方法收敛性的证明。
英文摘要：
Robust optimization is a framework for modeling optimization problems involving data uncertainty and during the last decades has been an area of active research. If we focus on linear programming (LP) problems with i) uncertain data, ii) binary decisions and iii) hard constraints within an ellipsoidal uncertainty set, this paper provides a different interpretation of their robust counterpart (RC) inspired from decomposition techniques. This new interpretation allows the proposal of an ad-hoc decomposition technique to solve the RC problem with the following advantages: i) it improves tractability, specially for large-scale problems, and ii) it provides the exact probability of constraint violation in case the probability distribution of uncertain parameters are completely defined by using first and second-order probability moments. An attractive aspect of our method is that it decomposes the second-order cone programming problem, associated with the robust counterpart, into a linear master problem and different quadratically constrained problems (QCP) of considerable lower size. The optimal solution is achieved through the solution of these master and subproblems within an iterative scheme based on cutting plane approximations of the second-order cone constraints. In addition, proof of convergence of the iterative method is given.
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