Representation of positive polynomials on a generalized strip and its application to polynomial optimization

IF 1.3 4区数学 Q2 MATHEMATICS, APPLIED

Optimization Letters Pub Date : 2024-01-11 DOI:10.1007/s11590-023-02087-5

引用次数: 0

Abstract

We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are called generalized strips). This class includes the strip \([a,b] \times {\mathbb {R}}\) which was studied by Marshall in (Proc Am Math Soc 138(5):1559–1567, 2010). A denominator-free Nichtnegativstellensätz holds true on a generalized strip when the width of the generalized strip is constant and fails otherwise. As a consequence, we confirm that the standard hierarchy of semidefinite programming relaxations defined for the compact case can indeed be adapted to the generalized strip with constant width. For polynomial optimization problems on the generalized strip with non-constant width, we follow Ha-Pham’s work: Solving polynomial optimization problems via the truncated tangency variety and sums of squares.

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正多项式在广义条带上的表示及其在多项式优化中的应用

摘要我们研究了两变量非负多项式在某类无界封闭基本半代数集合（称为广义条带）上的表示。这一类包括 Marshall 在 (Proc Am Math Soc 138(5):1559-1567, 2010) 中研究过的条（[a,b] \times {\mathbb {R}}）。当广义条带的宽度恒定时，广义条带上的无分母 Nichtnegativstellensätz 成立，反之则不成立。因此，我们证实了在紧凑情况下定义的半定式编程松弛的标准层次确实可以适用于宽度恒定的广义条带。对于宽度非恒定的广义条带上的多项式优化问题，我们遵循 Ha-Pham 的工作方法：通过截切线和平方和解决多项式优化问题。

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

Optimization Letters 管理科学-应用数学

CiteScore

3.40

自引率

6.20%

发文量

116

审稿时长

9 months

期刊介绍： Optimization Letters is an international journal covering all aspects of optimization, including theory, algorithms, computational studies, and applications, and providing an outlet for rapid publication of short communications in the field. Originality, significance, quality and clarity are the essential criteria for choosing the material to be published. Optimization Letters has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time one of the most striking trends in optimization is the constantly increasing interdisciplinary nature of the field. Optimization Letters aims to communicate in a timely fashion all recent developments in optimization with concise short articles (limited to a total of ten journal pages). Such concise articles will be easily accessible by readers working in any aspects of optimization and wish to be informed of recent developments.