两种求解单调变分不等式的投影收缩算法的开题报告

两种求解单调变分不等式的投影收缩算法的开题报
告
题目：两种求解单调变分不等式的投影收缩算法
摘要：
单调变分不等式在优化理论与应用中有着广泛的应用。

本文将介绍
两种基于收缩操作的投影算法（Projection algorithm with contraction operation），用于求解单调变分不等式。

第一种算法是基于倍数调整的投影收缩算法（Projection and Contraction Type I algorithm）。

该算法基于将当前的变量值乘以一个可调整的倍数，然后将其投影回可行域内来实现收缩操作。

该算法的收敛
速度快，但不一定能够保证全局收敛性。

第二种算法是基于可行域调整的投影收缩算法（Projection and Contraction Type II algorithm）。

该算法基于将可行域进行扩张或缩小，然后将当前的变量值投影到扩张或缩小后的可行域内来实现收缩操作。

该算法能够保证全局收敛性，但收敛速度相对较慢。

本文将首先介绍单调变分不等式及其应用，然后详细介绍两种投影
收缩算法及其收敛性分析，最后通过数值实验验证算法的有效性。

关键词：单调变分不等式，投影算法，收缩操作，可行域调整，倍
数调整，全局收敛性
Abstract:
Monotone variational inequalities have been widely used in optimization theory and applications. In this paper, we introduce two projection algorithms based on contraction operations for solving
monotone variational inequalities.
The first algorithm is the Projection and Contraction Type I algorithm based on scaling adjustment. This algorithm multiplies the current variable value by an adjustable scale and then projects it back
into the feasible domain to achieve contraction operation. The algorithm has a fast convergence speed but may not guarantee global
convergence.
The second algorithm is the Projection and Contraction Type II
algorithm based on feasible domain adjustment. This algorithm adjusts
the feasible domain by expanding or contracting it, and then projects the current variable value into the expanded or contracted feasible
domain to achieve contraction operation. The algorithm can guarantee
global convergence, but the convergence speed is relatively slower.
This paper first introduces monotone variational inequalities and their applications, then details the two projection algorithms and their convergence analysis, and finally verifies the effectiveness of the algorithm through numerical experiments.
Keywords: Monotone variational inequality, projection algorithm, contraction operation, feasible domain adjustment, scaling adjustment, global convergence.。