非凸半定规划的鞍点存在性研究

The Study of the Existence of Saddle Point for Nonconvex Semidefinite Programming Problems

主要利用矩阵分析的谱分解、Frobenius内积及其相关性质,凸分析的凸集分离定理来研究非凸半定规划问题的鞍点的存在性,通过3种不同的方式给出并证明了鞍点存在的一些充分、必要以及充分必要条件。首先,利用一个不等式系统给出了与文献[1]中的对偶定理等价的一个鞍点存在的充分必要条件。然后,给出了广义的KKT条件,并在不变凸性的假设下,证明了广义KKT条件是鞍点存在的一个充分条件;若x∈int C,则广义KKT条件是鞍点存在的一个必要条件。最后,定义了一个扰动函数ν,并在非凸半定规划问题的最优解存在的假设下,利用此扰动函数给出了鞍点存在的一个充分必要条件:若非凸半定规划问题的最优解存在,则对偶可达且无对偶间隙等价于扰动函数ν的上图在点(0,ν(0))处存在支撑超平面。

In this paper,we devote to study the existence of the saddle point of nonconvex semidefinite programming problems by means of Spectral decomposition,Inner product and correlative properties of Matrix Analysis and Separation theorem of convex set of Convex Analysis.In this case,some necessary and/or sufficient conditions for existence for the saddle point are derived and proved in three different ways.First,we present a sufficient and necessary condition which is equivalent to dual theorem in ref.[1]by utilizing an inequality system.Then,we give a generalized Karush-Kuhn-Tucker condition and prove this condition is sufficient conditions for existence of the saddle point under invex convexity assumption.In addition,if x∈int C,this sufficient condition is also necessary condition.Finally,we define a perturbation function which is used to deduce a sufficient and necessary condition for existence of the saddle point：dual attainment and the absence of a duality gap is equivalent to the existence of a supporting hyperplane for the epigraph ofνat the point（0,ν（0））.