MDM 算法和希尔维斯特问题

IF 0.7 4区 数学 Q3 MATHEMATICS, APPLIED
V. N. Malozemov, N. A. Solov’eva, G. Sh. Tamasyan
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引用次数: 0

摘要

摘要在开发解决非线性最小问题的数值方法时,出现了以下辅助问题:在欧几里得空间中某个有限集合的凸壳中,找到一个具有最小规范的点。1971 年,B. Mitchell、V. Demyanov 和 V. Malozemov 提出了解决这一问题的非标准算法,后来被称为 MDM 算法(基于作者姓氏的第一个字母)。本文考虑的是一个特定的 minimax 问题:寻找包含给定有限点集的最小体积球。它被称为西尔维斯特问题,是关于集合的切比雪夫中心问题的一个特例。西尔维斯特问题与带有单纯形约束的凸二次编程问题相关联。为了解决这个问题,建议使用 MDM 算法的变体。在该算法的帮助下,可以构建一个可行解的最小化序列,使得两个连续的可行解只有两个部分不同。这些分量的指数是根据某些最优条件选择的。我们证明了所得到的可行解序列的弱收敛性,这意味着相应的向量序列在规范上收敛于西尔维斯特问题的唯一解。我们给出了平面上的四个典型例子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The MDM Algorithm and the Sylvester Problem

The MDM Algorithm and the Sylvester Problem

Abstract

When developing numerical methods for solving nonlinear minimax problems, the following auxiliary problem arose: in the convex hull of a certain finite set in Euclidean space, find a point that has the smallest norm. In 1971, B. Mitchell, V. Demyanov and V. Malozemov proposed a non-standard algorithm for solving this problem, which was later called the MDM algorithm (based on the first letters of the authors' last names). This article considers a specific minimax problem: finding the smallest volume ball containing a given finite set of points. It is called the Sylvester problem and is a special case of the problem about the Chebyshev center of a set. The Sylvester problem is associated with a convex quadratic programming problem with simplex constraints. To solve this problem, it is proposed to use a variant of the MDM algorithm. With its help, a minimizing sequence of feasible solutions is constructed such that two consecutive feasible solutions differ in only two components. The indices of these components are selected based on certain optimality conditions. We prove the weak convergence of the resulting sequence of feasible solutions that implies that the corresponding sequence of vectors converges in norm to a unique solution to the Sylvester problem. Four typical examples on a plane are given.

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来源期刊
Computational Mathematics and Mathematical Physics
Computational Mathematics and Mathematical Physics MATHEMATICS, APPLIED-PHYSICS, MATHEMATICAL
CiteScore
1.50
自引率
14.30%
发文量
125
审稿时长
4-8 weeks
期刊介绍: Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
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