  In this paper, we focus on a class of constrained nonlinear optimization
problems (NLP), where some of its equality constraints define a closed embedded
submanifold $\mathcal{M}$ in $\mathbb{R}^n$. Although NLP can be solved
directly by various existing approaches for constrained optimization in
Euclidean space, these approaches usually fail to recognize the manifold
structure of $\mathcal{M}$. To achieve better efficiency by utilizing the
manifold structure of $\mathcal{M}$ in directly applying these existing
optimization approaches, we propose a partial penalty function approach for
NLP. In our proposed penalty function approach, we transform NLP into the
corresponding constraint dissolving problem (CDP) in the Euclidean space, where
the constraints that define $\mathcal{M}$ are eliminated through exact
penalization. We establish the relationships on the constraint qualifications
between NLP and CDP, and prove that NLP and CDP have the same stationary points
and KKT points in a neighborhood of the feasible region under mild conditions.
Therefore, various existing optimization approaches developed for constrained
optimization in the Euclidean space can be directly applied to solve NLP
through CDP. Preliminary numerical experiments demonstrate that by dissolving
the constraints that define $\mathcal{M}$, CDP gains superior computational
efficiency when compared to directly applying existing optimization approaches
to solve NLP, especially in high dimensional scenarios.
