In this paper, we introduce and investigate a new regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. The normalization argument in asymptotic analysis enables us to study the existence as well as the stability of solutions of these problems. We prove a Frank–Wolfe type theorem for regular optimization problems and an Eaves type theorem for non-regular pseudoconvex optimization problems. Moreover, under the regularity condition, we show results on the stability such as upper semicontinuity and local upper-Hölder stability of the solution map of polynomial optimization problems. At the end of the paper, we discuss the genericity of the regularity condition.

在本文中，我们为目标函数为多项式的优化问题引入并研究了渐近意义上的新正则条件。渐近分析中的归一化论证使我们能够研究这些问题的解的存在性和稳定性。我们证明了正则优化问题的 Frank-Wolfe 型定理和非正则伪凸优化问题的 Eaves 型定理。此外，在正则条件下，我们展示了多项式优化问题的解图的稳定性，例如上半连续性和局部上Hölder稳定性。在论文的最后，我们讨论了正则性条件的一般性。