球约束加权极大极小离差问题的SDP松弛的注记  

A Note on Semi-Definite Programming Relaxations of Ball-Constrained Weighted Maximin Dispersion Problems

作　　者：张思颖 罗洪林 ZHANG Siying;LUO Honglin(College of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《重庆师范大学学报（自然科学版）》2020年第1期107-112,共6页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.11601050,No.11771064);重庆市科委(No.cstc2016jcyjA0116);研究生创新基金(No.YKC19003)。

摘　　要：【目的】研究利用CVX软件有效求解球约束下的加权极大极小离差问题的SDP松弛模型。【方法】应用半定规划的强对偶定理和Gershgorin圆盘定理。【结果】证明了Haines等人给出的球形约束下离差问题的SDP松弛的解的存在性;同时提出了另一个球形约束下的离差问题,并给出了它的SDP松弛模型的解的存在性证明。【结论】提出的新的证明方法为CVX中嵌入的SeDuMi和SDPT3这两种内点算法提供了有效求解SDP松弛模型的理论依据。[Purposes]Lay the theoretical foundation of the effectiveness for solving the SDP relaxation of the problem(1)by the software CVX.[Methods]Based on strong duality theorem and Gershgorin circle theorem.[Findings]Provide a new proof of the existence of solution for the semi-definite programming(SDP)relaxation of(Pball)proposed by Haines,Loeppky,Tseng and Wang.A new SDP relaxation is proposed for another ball-constrained weighted maximin dispersion problem which is equivalent to(Pball)in sense that they share the same optimal solution.[Conclusions]The new proof of the existence of solution for the semi-definite programming(SDP)lay the theoretical foundation of the effectiveness for solving the SDP relaxation by the software CVX.

关 键 词：半定规划松弛 球约束加权极大极小离差问题 Gershgorin圆盘定理 强对偶定理 

分 类 号：O221.6[理学—运筹学与控制论]