Generally in the Lagrange method, you adjoin your model equations $\mathbf{g}(\mathbf{x})=\mathbf{0}$ to an Objective function $\mathcal{J}(\mathbf{x})$, to find the Lagrangian $$ \mathcal{L}(\mathbf{x},\lambda) = \mathcal{J}(\mathbf{x}) +\lambda^{\top}\mathbf{g} $$ In my setting however, I am dealing with an interative model, and if I use this model I do not get the correct equations. I did notice that when I would write the Lagrangian $$ \mathcal{L}(\mathbf{x},\lambda) = \mathbf{g}^{\top}\lambda $$ That it does give me the correct equations. However, is this mathematically correct? I can't find it in literature anywhere
Cheers, Wessel
If $\mathbf{v}$ and $\mathbf{w}$ are (column) vectors, then
$$\mathbf{w}^\top \mathbf{v} = \mathbf{v} \cdot \mathbf{w}= \mathbf{v}^\top \mathbf{w}$$
as can be checked by writing out what the terms mean in terms of components. Since they're equal, you can use them interchangeably.