A Bicomposition of Conical Projections

Pub Date : 2024-02-12 DOI:10.1134/s0081543823060160
E. A. Nurminski
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Abstract

We consider a decomposition approach to the problem of finding the orthogonal projection of a given point onto a convex polyhedral cone represented by a finite set of its generators. The reducibility of an arbitrary linear optimization problem to such projection problem potentially makes this approach one of the possible new ways to solve large-scale linear programming problems. Such an approach can be based on the idea of a recurrent dichotomy that splits the original large-scale problem into a binary tree of conical projections corresponding to a successive decomposition of the initial cone into the sum of lesser subcones. The key operation of this approach consists in solving the problem of projecting a certain point onto a cone represented as the sum of two subcones with the smallest possible modification of these subcones and their arbitrary selection. Three iterative algorithms implementing this basic operation are proposed, their convergence is proved, and numerical experiments demonstrating both the computational efficiency of the algorithms and certain challenges in their application are performed.

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圆锥投影的二重组合
我们考虑用分解法来解决寻找给定点在凸多面体圆锥上的正交投影的问题,凸多面体圆锥由其生成器的有限集合表示。任意线性优化问题都可以还原成这样的投影问题,这使得这种方法可能成为解决大规模线性规划问题的新方法之一。这种方法可以基于循环二分法的思想,将原始的大规模问题分割成二叉圆锥投影树,对应于将初始圆锥连续分解为较小子圆锥的总和。这种方法的关键操作在于解决将某个点投影到由两个子圆锥之和表示的圆锥上的问题,并对这些子圆锥进行尽可能小的修改和任意选择。本文提出了实现这一基本操作的三种迭代算法,证明了它们的收敛性,并通过数值实验证明了算法的计算效率及其应用中的某些挑战。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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