Consider the following problem: $$ \min \mathbf{c}^T\mathbf{x}, \mathbf{x} \in \mathbf{R}^N\\ \mathbf{A}\mathbf{x}=\mathbf{b} $$ where $\mathbf{A}\in\mathbf{R}^{D \times N}, \mathbf{b}\in\mathbf{R}^D$.
I am trying to find the Lagrangian and then the dual. I wrote down: $$ L(\mathbf{x},\mathbf{\lambda},\mathbf{\nu})=\mathbf{c}^T\mathbf{x} + \sum \lambda_i(\mathbf{a}_i^T\mathbf{x}-b_i) $$ where $\mathbf{a}_i$ are the rows of the matrix $\mathbf{A}$. Then the derivative would be: $$ \mathbf{c}+\sum\lambda_i\mathbf{a}_i $$
But this does not depend on $\mathbf{x}$. I am almost sure I made a mistake somewhere in the derivative. How can I find the dual if I cannot replace it in the Lagrangian?