约束二次规划的重叠Schwarz分解
Overlapping Schwarz Decomposition for Constrained Quadratic Programs
摘要：
提出了求解约束二次规划问题的一种重叠Schwarz分解算法。Schwarz算法传统上用于解决线性代数系统所产生的偏微分方程，但我们最近表明，他们也是有效的解决结构化最优化问题。在建议的计划中，我们认为量子点的代数结构可以表示的图形。图域被划分成重叠的子域（产生一组耦合的子问题），子问题的解决方案并行计算，并通过更新重叠区域中的原始-对偶信息来执行收敛。我们表明，收敛是有保证的，如果重叠是足够大的，并且收敛速度的重叠的大小呈指数级提高。收敛结果依赖于图结构问题的一个关键属性，即灵敏度的指数衰减。在这里，我们建立的条件下，此属性为约束QP（如网络优化和最优控制中发现的），从而扩展现有的工作，解决无约束QP。Schwarz格式的数值行为通过使用一个定义在具有9，241个节点的网络上的DC最优潮流问题来证明。
英文摘要：
We present an overlapping Schwarz decomposition algorithm for constrained quadratic programs (QPs). Schwarz algorithms have been traditionally used to solve linear algebra systems arising from partial differential equations, but we have recently shown that they are also effective at solving structured optimization problems. In the proposed scheme, we consider QPs whose algebraic structure can be represented by graphs. The graph domain is partitioned into overlapping subdomains (yielding a set of coupled subproblems), solutions for the subproblems are computed in parallel, and convergence is enforced by updating primal-dual information in the overlapping regions. We show that convergence is guaranteed if the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap. Convergence results rely on a key property of graph-structured problems that is known as exponential decay of sensitivity. Here, we establish conditions under which this property holds for constrained QPs (as those found in network optimization and optimal control), thus extending existing work that addresses unconstrained QPs. The numerical behavior of the Schwarz scheme is demonstrated by using a DC optimal power flow problem defined over a network with 9,241 nodes.
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