关于复合迫近束方法对偶问题的研究

Research on the Dual Problem of Composite Proximal Bundle Method

对于具有复合形式目标函数的优化问题,复合迫近束方法展示了很好的数值结果,因此,对于该类问题的研究越来越受到人们的关注.本文中,c:Rn→Rm是光滑映射,h:Rm→R是正齐次凸函数.我们将复合无约束问题min x∈Rn(hc)(x)的研究转化成一系列二次规划问题min d∈Rn h∨l(c k(d))+12|d|2k,l的求解.本文利用文献[1]中惩罚束方法的研究方式,采用对偶空间思想,对惩罚子问题展开研究,刻画了原问题与对偶问题之间的关系.

For optimization problems with the composite form objective function, composite proximal bundle method shows excellent numerical results. Therefore, Much more attention had been paid to the research on this problem. In this paper, is a smooth map,is a positively homogeneous convex function. We convert the study of the composite unconstrained problems into a series of quadratic programming problems . This paper utilizes the approach of the penalized bundle method in [ 1] and the dual space ideas, studies the penalized subproblems, describes the relationship between the original problem and the dual problem, and finally gets relevant properties of solutions to these two problems.