  This paper introduces and investigates a regularity condition in the
asymptotic sense for optimization problems whose objective functions are
polynomial. Under this regularity condition, the normalization argument in
asymptotic analysis enables us to see the solution existence as well as the
solution stability 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, we show results on the stability
such as upper semicontinuity and local upper-H\"{o}lder stability of the
solution map of polynomial optimization problems. At the end of the paper, we
discuss the genericity of the regularity condition.
