關於變分不等式的輔助問題原理

Abstract

輔助問題原理允許我們藉由解決輔助問題的一個數列去尋找最佳化的問題(例如：最小化問題，鞍點問題，變分不等式問題，...等）的解。 根據 Cohen 的輔助問題原理，我們介紹並分析一個演算法來解決一般性的變分不等式 VI（T，C）問題。 為了解決關於一般的非單調算子在自反的巴那赫空間中多值的變分不等式問題，所以在這篇文章裡，近似方法的觀念被介紹而且一個收斂的演算法也被提出。而我們文章的目標就是為了輔助問題原理去建立類似的連結。事實上，這篇論文的要旨有兩層： (1）一般化單調算子的條件之下，以輔助問題原理為基礎， 我們處理演算法的收斂性，例如: pseudo-Dunn property，強偽單調性，$alpha$-強偽單調性，...等。 (2）我們提出一個修改的演算法，在一個缺乏強單調性質的輔助函數條件之下，來解決變分不等式的解之收斂性。

The auxiliary problem principle allows us to find the solution of an optimization problem (minimization problem, saddle-point problem, variational inequality problem, etc.) by solving a sequence of auxiliary problem. Following the auxiliary problem principle of Cohen, we introduce and analyze an algorithm to solve the usual variational inequality VI(T,C). In this paper, the concept of proximal method is introduced and a convergent algorithm is proposed for solving set-valued variational inequalities involving nonmonotone operators in reflexive Banach spaces. The aim of our work is to establish similar links for the auxiliary problem principle. In fact, the purpose of this paper has two folds : (1) We first deal with the convergence of algorithm based on the auxiliary problem principle under generalized monotonicity, such as, pseudo-Dunn property, strong pseudomonotonicity, $alpha$-strong pseudomonotonicity, etc. (2) We present a modified algorithm for solving our variational inequalities under a weaker condition on the auxiliary function without strong monotonicity.