Suppose we have an optimization problem $P$ :
$$
\begin{aligned}
&\min\limits_{\theta \in S} \space L(\theta)\\
\end{aligned}
$$
with
$S=\{\theta| \forall k_1,k_2 \in \{1,\ldots, K\}, s_{k_i}(\theta) = s_{k_j}(\theta)\}$ for an array of real-valued constraint functions $s_{1},...,s_{K}$.
S can be rewritten as
$$ \begin{aligned} S = &\{\theta|\forall k \in \{1,\ldots,K-1\}, s_k(\theta)=s_{k+1}(\theta)\}\\ = &\{\theta|\exists c\in\mathbb{R}, \forall k=\{1,\ldots,K\}, s_k(\theta) = c\}\\ \end{aligned} $$
Now I consider the consider for a fixed value $c\in\mathbb{R}$ the optimization problem $P_{c}$
$$
\begin{aligned}
&\min\limits_{\theta \in S_c} \space L(\theta)\\
\end{aligned}
$$
where $$S_c=\{\theta| \forall k \in \{1,\ldots, K\}, s_{k}(\theta) = c\}$$
If I'm correct, we have that $\forall c, S_c\subset S$, and that $S=\cup_{c\in\mathbb{R}} S_c$.
What can be said about the relationship between the solutions of $P$ and the solutions of $P_c$, supposing that they exist?