非凸函数束方法模型构造及其对偶问题

Construction of Model of the Bundle Method for Nonconvex Functions and Its Dual Problem

对于非光滑凸优化问题,迫近束方法展示出较高的有效性,我们试图通过改变相应的参数将其推广至非凸非精确优化问题中.我们给出求解一类已知目标函数近似值的非凸非光滑优化问题的迫近束方法,利用函数的近似信息构造一种切平面模型,给出的参数选取方式不仅可以保证线性化误差非负,还可以通过求解惩罚子问题得到下一个迭代点.此外,我们还研究了惩罚子问题的对偶问题,讨论了惩罚子问题解的表达形式及相应次微分的归属关系.

Proximal bundle methods have been shown to be highly efficient for nonsmooth convex optimization problems. We attempt to extend convex cases to nonconvex and inexact cases by modifying the corresponding parameter. In this paper,we give a proximal bundle method for solving a class of nonconvex nonsmooth optimization problems with the approximate values of objective function. We employ the approximate information of the function to build a kind of cutting- planes model. The way given to choose the parameters can not only enforce the nonnegative property of linearization errors,but also can obtain the next iteration point by solving a penalized subproblem. In addition,we study the dual problem of the penalized subproblem,and discuss the solution expression of the penalized subproblem and the ownership of the corresponding subdifferential.