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wxiaoqiang
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haiting_math: 金币+4, 基金HEPI+1, ★★★很有帮助, 这是结题报告! 不过可以猜出项目的摘要! 2013-04-25 23:54:07
haiting_math: 金币+4, 基金HEPI+1, ★★★很有帮助, 这是结题报告! 不过可以猜出项目的摘要! 2013-04-25 23:54:07
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本项目期间，课题组成员主要在三个方面取得了一定的进展，发表8篇SCI期刊论文， 还有4篇SCI期刊论文已经接受发表，另有几篇论文已经完成或即将完成。发表的论文包括三方面内容， 简介如下： 1. 非负张量谱半径的各种性质及算法的收敛性以及采用凸优化方法求解最大特征值的方法； 2. 用半定松弛的方法求解一类双二次非凸优化问题，对其中一类情形，证明了半定松弛问题和原问题的最优值相等；对更一般情形， 给出了松弛问题的近似界。这些结果推广了非凸二次优化问题半定松弛的近似界结果； 3. 对凸可行性问题，采用自适应步长格式得到一类有效解法，同时对一类约束最大单调零点问题，提出了一类分裂向前向后格式及相应的松弛格式，并证明了算法的收敛性。本项目组成员参加或组织了多次学术活动，并在几个学术会议上报告了课题组的学术成果。还邀请多位知名优化专家来南开大学做报告。本项目期间，毕业博士研究生2人，硕士研究生10人，在读博士研究生5人，在读硕士研究生12人。 其中一名博士生获2011年教育部学术新人奖、2011年全国计算数学年会年会上颁发的优秀青年论文奖。2名硕士生获校级优秀硕士论文. 中文主题词： 张量特征值; Perron-Frobenius定理; 算法收敛性; 双二次优化问题;近似界 英文摘要: Our research group study three classes of projects and achieve some significant progress during performing this project. Ten research articles of ours have be published, another two papers have been accepted by the related journals and several other articles have been completed or are being prepared. In the following we introduce our three aspects' work. 1. We study the various properties of nonnegative tensors' spectral radius, in particular , Perron-Frobenius theorem on the nonnegative are developed greatly. Based on the minmax theorem on the nonnegative irreducible tensor we convert the solving the spectral radius into an equivalent convex optimization problem, so it shows that one can use the polynomial time algorithms to solve the spectral radius of the nonnegative irreducible tensor。 2. For a class of biquadratic optimization problem, we study its Semidefinite relaxation programming, and investigate the relationships between them. We show that two problems have the same optimal value in some case, while in the general situations, approximate bound of the original problem is obtained. 3. For the convex feasible problem, a self-adapt stepsize is constructed, which leads an efficient algorithm. We establish the convergence of our method. For a class of maximal monotone zero problem with constraint, we present a forward-backward scheme and its relaxed version , the convergence of the our methods are also given. Members of this project attended many important academic conferences, and presented several talks to reports the academic results of ours. We also invited some leading experts in optimization or related fields coming Nankai University to give their lectures. During the period of existing this project, two graduate students of ours obtained the PhD, ten graduate students of ours received the Master degree. Now five graduate students are studying for the PhD and twelve graduate students are studying for the Master degree. One among them is awarded the academic newer prize of State Educational Ministry of China, 2011. Two graduate students of ours were awarded the excellent master dissertation. 英文主题词： Tensors' eigenvalues; Perron-Frobenius Theorem; the convergence of algorithm; biq |
2楼2013-04-23 21:46:04