“創(chuàng)源”大講堂研究生學(xué)術(shù)講座

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報(bào)告人：Toh Kim-Chuan（卓金全）教授

講座地點(diǎn)：2016年12月16日下午16:00 - 17:00

講座地點(diǎn)：犀浦校區(qū)X2511

報(bào)告題目：QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming

主講人簡(jiǎn)介：Toh Kim-Chuan（卓金全），新加坡國(guó)立大學(xué)教授，博士生導(dǎo)師，新加坡國(guó)立大學(xué)數(shù)學(xué)系副主任。1990年以優(yōu)異成績(jī)本科畢業(yè)于新加坡國(guó)立大學(xué)數(shù)學(xué)系；1992年獲新加坡國(guó)立大學(xué)數(shù)學(xué)系碩士學(xué)位，1994年獲美國(guó)康奈爾大學(xué)應(yīng)用數(shù)學(xué)碩士學(xué)位；1996年獲美國(guó)康奈爾大學(xué)應(yīng)用數(shù)學(xué)博士學(xué)位（師從國(guó)際數(shù)值計(jì)算專家Lloyd N. Trefethen教授），1996年至今執(zhí)教新加坡國(guó)立大學(xué)數(shù)學(xué)系。Toh教授是國(guó)際知名數(shù)值優(yōu)化專家，主要致力于矩陣優(yōu)化、二階錐規(guī)劃、凸規(guī)劃等方面的算法設(shè)計(jì)、分析與實(shí)現(xiàn)。Toh教授及其合作者研制的軟件被學(xué)術(shù)界和工業(yè)界廣泛使用，如用于計(jì)算半定規(guī)劃、二階規(guī)劃、線性規(guī)劃的免費(fèi)軟件SDPT3，SDPNAL被廣泛使用。Toh教授在Mathematical Programming, SIAM Journal on Optimization, SIAM Journal on Matrix Analysis and Applications等國(guó)際知名期刊發(fā)表論文60余篇，Toh教授的研究結(jié)果被廣泛引用，被引4836次。Toh教授于2007年獲新加坡國(guó)立大學(xué)杰出科學(xué)家獎(jiǎng)，2010年在SIAM年會(huì)上做大會(huì)報(bào)告，并多次擔(dān)任國(guó)際重要學(xué)術(shù)會(huì)議的組織成員。Toh教授現(xiàn)任優(yōu)化著名雜志SIAM Journal on Optimization副主編，擔(dān)任Mathematical Programming Computation雜志區(qū)域主編, 擔(dān)任Optimization and Engineering, Numerical Algebra, Control and Optimization, Pacific Journal of Mathematics for Industry 等多個(gè)雜志的編委。

報(bào)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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