近似点算法求解Hadamard流形上的多指标最优化  

Proximal point method for multicriteria optimization on Hadamard manifolds

作　　者：唐凤梅 黄平亮[1] 

机构地区：[1]上海大学理学院,上海200444

出　　处：《应用数学与计算数学学报》2017年第3期392-402,共11页Communication on Applied Mathematics and Computation

基　　金：中国博士后科学基金资助项目(2015M571536)

摘　　要：近似点算法在信号恢复和信号处理等方面有着广泛的应用.近些年,近似点算法被推广到Riemannian流形上.这种推广的意义在于:只要引入适当的Riemannian度量,可以将经典意义下的非凸问题转化为凸问题;将限制问题转化为无限制问题.为了解决Hadamard流形上的非光滑多指标最优化问题,通过引入变化的标量函数进而提出近似点算法.当目标函数是凸函数时,由这种方法产生的迭代序列收敛到弱Pareto最优点;当目标函数是强凸函数时,产生的迭代序列将收敛到Pareto最优点.The proximal point algorithm has many interesting applications such as the signal recovery, the signal processing. In recent years, the proximal point method has been extended to Riemannian manifolds. The main advantages of these extensions are that nonconvex problems in the classic sense may become convex through the introduction of an appropriate Riemannian metric, and constrained optimization problems may be seen as unconstrained ones. In this paper, we pro- pose a proximal point method for solving nonsmooth multicriteria optimization problems on Hadamard manifolds, by assuming an iterative process which uses a variable scalarization function. Under the assumption that the convexity of the objective function, the sequence generated by this method converges to a weak Pareto optimal point. If the objective function is strictly convex, then the sequence will converges to the Pareto optimal point.

关 键 词：近似点算法 Pareto最优点 Hadamard流形 Fejer收敛 多指标最优化 

分 类 号：O186.12[理学—数学]