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.

中文翻译：

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广义带上正多项式的表示及其在多项式优化中的应用

摘要

我们研究一类无界闭基本半代数集（称为广义带）上两个变量的非负多项式的表示。该类包括马歇尔在 (Proc Am Math Soc 138(5):1559–1567, 2010) 中研究的条带\([a,b] \times {\mathbb {R}}\) 。当广义带的宽度恒定时，无分母 Nichtnegativstellensätz 在广义带上成立，否则失败。因此，我们确认为紧凑情况定义的半定规划松弛的标准层次结构确实可以适应具有恒定宽度的广义条带。对于具有非恒定宽度的广义带上的多项式优化问题，我们遵循 Ha-Pham 的工作：通过截断切线变化和平方和求解多项式优化问题。