论文标题
本地Lipschitz连续性下单调夹杂物的原始二重推外方法
Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity
论文作者
论文摘要
在本文中，我们考虑了一类单调包含（MI）的问题，即找到两个单调算子的总和中的零，其中一个操作员是最大单调的，而另一个操作员是{\ es {\ it locally lipschitz}连续的。我们提出了使用点和操作员推断技术来解决它们的原始双重外推方法，其参数是通过回溯线搜索方案选择的。提出的方法享有$ {\ cal o}（\logε^{ - 1}）$和$ {\ cal o}（ε^{ - 1} \logε^{ - 1}）$的操作复杂性$ \ varepsilon $ - 分别对强烈和非巧妙的MI问题的解决方案。后一种复杂性显着改善了以前最佳的操作复杂度$ {\ cal o}（\ varepsilon^{ - 2}）$。作为副产品，还可以获得原始二偶推迟方法的复杂性结果，以找到$ \ varepsilon $ -KKT或$ \ varepsilon $ - 均衡的解决方案，用于凸圆锥优化，锥形约束鞍点，锥形约束的鞍点，和各种不平等问题。我们提供初步的数值结果，以证明所提出的方法的性能。
In this paper we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone while the other is {\it locally Lipschitz} continuous. We propose primal-dual extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of ${\cal O}(\log ε^{-1})$ and ${\cal O}(ε^{-1}\log ε^{-1})$, measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an $\varepsilon$-residual solution of strongly and non-strongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity ${\cal O}(\varepsilon^{-2})$. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an $\varepsilon$-KKT or $\varepsilon$-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under {\it local Lipschitz} continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods.
