混合整数多项式电网优化的低秩与提升半定规划
Low-Rank and Lifted Semidefinite Programming for Mixed-Integer Polynomial Power Grid Optimization
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中文总结 AI 辅助
本文提出将低秩半定规划与基于矩的松弛收紧相结合，用于求解交流最优输电切换问题，并通过原始-对偶目标一致性和原始可行性验证，获得全局最优解，可作为分支定界法的可扩展替代方案。
中文摘要 AI 辅助
一个局部求解器能否返回非凸混合整数多项式电网优化问题的保证全局最优解？通过将低秩半定规划（SDP）与Lasserre层级中的基于矩的提升相结合，本文提供了支持肯定答案的实例证据。具体而言，我们迭代地收紧基于矩的交流最优输电切换（AC-OTS）问题的松弛，包括基于Lasserre层级的更高阶矩约束，直到该松弛变得紧（即原始可行）。然后，我们使用局部低秩SDP求解器（Knitro）重新求解该问题，并将相关的对偶变量代入一个有界的拉格朗日对偶函数中。通过展示原始和对偶目标值的一致性以及原始可行性，我们确认该解为精确的全局解。从这些初步结果中，我们假设将（i）低秩（即高度可扩展）SDP方法与（ii）基于矩的松弛收紧相结合，可能是基于全局分支定界方法的可扩展替代方案。
英文摘要
Can a local solver return the guaranteed globally optimal solution to a nonconvex mixed-integer polynomial power grid optimization problem? By mixing low-rank semidefinite programming (SDP) and moment-based lifting in the Lasserre hierarchy, this paper provides anecdotal evidence in the affirmative. Specifically, we iteratively tighten a moment-based relaxation of the AC Optimal Transmission Switching (AC-OTS) problem, including higher order moment-based constraints, rooted in the Lasserre hierarchy, until it is tight (i.e., primal feasible). We then re-solve this problem using a local, low-rank SDP solver (Knitro), plugging the associated dual variables into a bounding Lagrange dual function. By demonstrating primal and dual objective agreement and primal feasibility, we confirm the solution as the exact, global solution. From these initial results, we hypothesize that mixing (i) low-rank (i.e., highly scalable) SDP methods with (ii) moment-based relaxation tightening could be a scalable alternative to global branch-and-bound-based methods.
发表机构
- University of Vermont（佛蒙特大学）
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