分块大规模优化的ADMM-SQP型算法理论与应用

The alternating direction methods of multiplier (ADMM) and the sequence quadratic programming (SQP) type algorithms are two important methods and subject foreland for researching and designing effective algorithms for solving constrained optimization. However, each of the two types of algorithms has its advantages and disadvantages. ADMM needs to solve minimization subproblems exactly at each iteration. For large-scale optimization, SQP type algorithms need to solve the same scale quadratic programming (QP) subproblems at each iteration. Both of the two above are difficult and costly. In this project, we will embed the idea of ADMM in solving the QP subproblems of SQP type algorithms, and propose ADMM-SQP type algorithms and theory for block large-scale optimization. (a) With the help of the idea of ADMM and the block structure of the discussed optimization, the direction finding subproblems of SQP methods, stabilized SQP (sSQP) methods and sequential quadratic constrained quadratic programming (SQCQP) methods will be decomposed into several small-scale QPs. (b) Based on the Augmented Lagrange function, the associated appropriate and effective merit functions will be constructed, and then some valid ADMM-SQP, ADMM-sSQP, ADMM-SQCQP and other ADMM-SQP type algorithms will be presented. (c) The theoretical characteristics will be analyzed, and the numerical efficiency of the proposed algorithms will be tested. (d) The algorithms will be applied to study and solve the optimal power flow, unit combination, economic dispatch and other practical problems.

乘子交替方向法(ADMM))和序列二次规划(SQP)型算法是研究和设计约束优化有效算法的重要方法及学科前沿, 然而其各有利弊。ADMM迭代中需精确求解交替子问题；对于大规模优化，SQP型算法迭代需求解同等规模的二次规划(QP)子问题等，这些都是困难和高耗费的工作。本项将在SQP型算法QP子问题的求解中植入ADMM思想，创新分块大规模优化的ADMM-SQP型算法与理论。(a)借助ADMM思想和分块结构，将SQP、稳定SQP(sSQP)、序列二次约束二次规划(SQCQP)等SQP型算法的搜索方向子问题分解为若干小规模QP。(b)基于增广Lagrane函数，创新合适的效益函数,进而设计出有效的ADMM-SQP、ADMM-sSQP、ADMM-SQCQP等ADMM-SQP型算法。(c)分析ADMM-SQP型算法的理论特征，测试其数值效果。(d)将算法用于研究最优潮流和机组组合等实际问题。

分块大规模优化的ADMM-SQP型算法理论与应用项目始终按原计划展开研究工作，已取得一批有特色、有影响的成果，正式发表学术论文32篇，其中SCI收录27篇（T1期刊3篇、T2期刊5篇和T3期刊7篇），中文核心5篇（均为T3期刊），以及授权发明专利1项。成果的主要贡献和创新有：1、约束分块非凸光滑问题的分裂SQP算法、新型ADMM-SQP算法、基于二次约束二次规划的分裂SQP算法和线性约束的非凸多分块部分对称正则化ADMM以及改进的SQP型算法；2、机组组合问题的分层ADMM和两阶段全分布式方法、动态经济调度问题的新型分布式方法、不确定的直流安全约束最优潮流的并行方法；3、大规模优化问题的优化算法，如：极大极小问题QP-free算法和邻近投影部分束方法、共轭梯度法、凸约束方程组共轭梯度投影法。. 在项目经费的资助下，课题组参加了30多人次学术会议、邀请近20位专家来校为课题组作学术报告、项目负责人简金宝教授受邀到10多所高校进行学术交流；课题组成员中晋升副教授和讲师各1人；培养了博士4人，硕士16人。