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QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming

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“创源”大讲堂研究生学术讲座

海  报

报告人:Toh Kim-Chuan(卓金全)教授

讲座地点:20161216日下午16:00 - 17:00

讲座地点:犀浦校区X2511

报告题目:QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming

主讲人概况:Toh Kim-Chuan(卓金全),新加坡国立大学教授,博士生导师,新加坡国立大学数学系副主任。1990年以优异成绩本科毕业于新加坡国立大学数学系;1992年获新加坡国立大学数学系硕士学位,1994年获美国康奈尔大学应用数学硕士学位;1996年获美国康奈尔大学应用数学博士学位(师从国际数值计算专家Lloyd N. Trefethen教授),1996年至今执教新加坡国立大学数学系。Toh教授是国际知名数值优化专家,主要致力于矩阵优化、二阶锥规划、凸规划等方面的算法设计、分析与实现。Toh教授及其合编辑研制的App被学术界和工业界广泛使用,如用于计算半定规划、二阶规划、线性规划的免费AppSDPT3SDPNAL被广泛使用。Toh教授在Mathematical Programming, SIAM Journal on Optimization, SIAM Journal on Matrix Analysis and Applications等国际知名期刊发表论文60余篇,Toh教授的研究结果被广泛引用,被引4836次。Toh教授于2007年获新加坡国立大学杰出科学家奖,2010年在SIAM年会上做大会报告,并多次担任国际重要学术会议的组织成员。Toh教授现任优化著名杂志SIAM Journal on Optimization副主编,担任Mathematical Programming Computation杂志区域主编, 担任Optimization and Engineering, Numerical Algebra, Control and Optimization, Pacific Journal of Mathematics for Industry 等多个杂志的编委。

报告摘要:we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality, inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite {cone} constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) where the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are provided for the ALM. Moreover, under mild conditions, we are able to analyze the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence rate of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and innovative shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase framework is not only fast but also robust in obtaining accurate solutions.

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