Two-Phase Semi-Asynchronous Parallel Iterative Methods for the Symmetric Linear Complementarity Problems

Abstract

在本文中，我們提出了一個在求解對稱線性互補問題上的二階段半非同步平行演算法。此法的求解程序可分成一連串交互的兩個演算階段。在第一階段中，我們進行完全非同步的演算，亦即，無論是各處理器在更新疊代解的次數上，或是各處理器之間的通訊次序上，都完全沒有任何的限制；而在第二階段的演算中，我們僅執行一個簡單的線性搜索步驟。此演算法大大地減少了因同步化及通訊上所產的冗額成本，因此增進了演算法的效率；而且由於此法在更新疊代解的演算中，擷取了大部份最新的資訊，也因而加快了收斂速度。至於在建立此法的收斂理論上，我們僅須對線性搜索範圍給予一微弱的限制，即可保證此半非同步演算過程，終將收斂到線性互補問題的解。我們將此法在IBM 3090的機器上模擬執行，並應用於求解一個由二次規劃所導出之對稱線性互補問題，其數值結果亦顯示出此法優於一些同步的平行疊代法。

In this paper, we propose a two-phase semi-asynchronous parallel iterative method for solving the symmetric linear complementarity problem. The solution process is divided into 1st-phase and 2nd-phase alternatingly. In the 1st- phase, we allow the process to be completely asynchronous. That is. the computation for new iterate on separate processors and communication in between them could be absolutely tree and chaotic. We impose essentially no restrictions on the ordering and the number of times of such iterate updating and information exchanging among processors. In the 2nd-phase. we simply perform a line search. The method dramatically reduces the overhead arising from synchronization and communication. Moreover, the process has always used and taken advantage of the most recent information available, when updating the iterate. Convergence of the overall scheme is established under a mild restriction on the range of the line search. The method is simulated and implemented on the IBM 3090 machine for solving the symmetric linear complementarity problem arising from the constrained convex quadratic program. Numerical results are also reported.