无穷维广义析取规划的求解方法
Solution Methods for Infinite-Dimensional Generalized Disjunctive Programming
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中文总结 AI 辅助
该研究将四种GDP求解方法推广到无穷维优化，提出新型高斯过程变体MBM-GP，经动态与随机优化案例验证，其性能优于基准方法且重构成本更低。
中文摘要 AI 辅助
广义析取规划（GDP）通过布尔指示变量和析取式表达混合离散-连续决策，可通过文献中提出的一系列方法系统求解。近期的InfiniteGDP抽象将该建模层拓展至时间、空间、不确定性等连续域，但仅松弛谱两端的大M法和凸包重构法被推广到无穷维场景。本研究通过将另外四种GDP求解方法推广到无穷维优化，填补了这一空白：多大M重构法、P-分裂重构法、割平面重构法以及基于逻辑的外层近似算法。此外，本研究提出了MBM-GP，这是多大M法的一种新型高斯过程变体，通过少量子问题求解从无穷域中学习大M函数。上述方法均在Julia包中实现，并在动态优化和随机优化的案例研究中进行了基准测试。结果表明，这些推广后的求解方法可优于大M法和凸包法，且MBM-GP在保留多大M法紧度的同时，仅需其一小部分的重构成本。
英文摘要
Generalized disjunctive programming (GDP) expresses mixed discrete-continuous decisions through Boolean indicators and disjunctions, and can be systematically solved via a library of methods proposed in the literature. The recent InfiniteGDP abstraction lifts this modeling layer to continuous domains such as time, space, and uncertainty, but only the big-M and hull reformulations, the two endpoints of the relaxation spectrum, have been generalized to the infinite setting. This work closes this gap by generalizing four other GDP solution methods to infinite-dimensional optimization: the multiple big-M reformulation, P-split reformulation, cutting plane reformulation, and the logic-based outer approximation algorithm. It further proposes MBM-GP, a novel Gaussian-process variant of multiple big-M that learns the big-M function over the infinite domain from a small subset of the subproblem solves. Moreover, these approaches are implemented in the Julia package InfiniteDisjunctiveProgramming.jl. The methods are benchmarked on case studies arising in dynamic and stochastic optimization. The results demonstrate how the generalized solution methods can outperform big-M and hull, with MBM-GP retaining the tightness of multiple big-M at a fraction of its reformulation cost.
