Linear Programming on the Stiefel Manifold

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-02-15 DOI:10.1137/23m1552243

Mengmeng Song, Yong Xia

引用次数: 0

Abstract

SIAM Journal on Optimization, Volume 34, Issue 1, Page 718-741, March 2024.
Abstract. Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all [math]-tuples of orthonormal vectors in [math] satisfying [math] additional linear constraints. Despite the classical polynomial-time solvable case [math], general (LPS) is NP-hard. According to the Shapiro–Barvinok–Pataki theorem, (LPS) admits an exact semidefinite programming relaxation when [math], which is tight when [math]. Surprisingly, we can greatly strengthen this sufficient exactness condition to [math], which covers the classical case [math] and [math]. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order local necessary optimality conditions are sufficient for global optimality when [math].

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Stiefel Manifold 上的线性规划

SIAM 优化期刊》，第 34 卷，第 1 期，第 718-741 页，2024 年 3 月。摘要首次研究了 Stiefel 流形（LPS）上的线性规划。它旨在最小化[math]中所有[math]正交向量的[math]元组集合上满足[math]附加线性约束的线性目标函数。尽管有经典的多项式时间可解情况[math]，但一般（LPS）是 NP 难的。根据 Shapiro-Barvinok-Pataki 定理，当[math]时，（LPS）允许精确的半定式编程松弛，而当[math]时，（LPS）是紧密的。令人惊奇的是，我们可以将这一充分精确性条件大大强化为[math]，它涵盖了经典情况[math]和[math]。将（LPS）视为平稳非线性编程问题，我们揭示了一个很好的性质，即在线性独立约束条件下，当[math]时，标准的一阶和二阶局部必要最优条件对全局最优是充分的。

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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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.