Let $f:\mathbb{C}^n \rightarrow \mathbb{R}$ be an analytic function and let $\{h_i:\mathbb{C}^n \rightarrow \mathbb{C} \}$ $i=1,\cdots,n$ be a set of analytic functions. Now consider the optimization problem $$ \min f(z) $$ subjected to $$ h_i(z) = u_i, i=1,\cdots,n $$ where $z \in \mathbb{C}^n$ and $u_i \in \mathbb{C}$.
1) Is it possible to formulate Lagrange multipliers for a local optimum $v \in \mathbb{C}^n$ as follows $$ \frac{\partial f}{\partial z_i}\big{|}_{z=v} = \sum_{j=1}^n \lambda_j \frac{\partial h_j}{\partial z_i}\big{|}_{z=v} $$ where $\lambda_j \in \mathbb{C}$. or
2) Is it possible to formulate Lagrange multipliers considering that $f$ and $h_i$ have Real and Imaginary parts from $z_i$? $$ \frac{\partial Re(f)}{\partial Re(z_i)}\big{|}_{z=v} = \sum_{j=1}^n \lambda_j \frac{\partial Re(h_j)}{\partial Re(z_i)}\big{|}_{z=v} $$ and $$ \frac{\partial Re(f)}{\partial Im(z_i)}\big{|}_{z=v} = \sum_{j=1}^n \lambda_j \frac{\partial Re(h_j)}{\partial Im(z_i)}\big{|}_{z=v} $$ and $$ \frac{\partial Im(f)}{\partial Re(z_i)}\big{|}_{z=v} = 0 = \sum_{j=1}^n \lambda_j \frac{\partial Im(h_j)}{\partial Re(z_i)}\big{|}_{z=v} $$ and $$ \frac{\partial Im(f)}{\partial Im(z_i)}\big{|}_{z=v} = 0 = \sum_{j=1}^n \lambda_j \frac{\partial Im(h_j)}{\partial Im(z_i)}\big{|}_{z=v} $$ where $\lambda_j \in \mathbb{R}$.
Thanks in advance.