Consider an optimization problem with sign constraints on the variables: $$ \begin{align} \min_{x\in\mathbb{R}^n}&\quad f(x) \\ \text{s.t.}&\quad g_i(x)\le 0,\quad i=1,...,m \\ &\quad h_j(x)=0,\quad j=1,....,\ell \\ &\quad x\le 0. \end{align} $$ When forming the Lagrangian for this problem, I wonder if it is necessary to assign dual variables to the constraint $x\le 0$?
To be more specific, I know for certain that I can treat $x\le 0$ as general inequality constraints, which yields the Lagrangian $$ L(x,\lambda,\mu,\nu)=f(x)+\sum_{i=1}^m\lambda_ig_i(x)+\sum_{j=1}^{\ell}\mu_jh_j(x)+\sum_{i=1}^n\nu_ix_i $$ and the dual function $$ d(\lambda,\mu,\nu)=\inf_{x\in\mathbb{R}^n}L(x,\lambda,\mu,\nu),\quad\forall\,(\lambda,\mu,\nu)\in\mathbb{R}^m_+\times\mathbb{R}^{\ell}\times\mathbb{R}^n_+, $$ so the dual problem is $$ \begin{align} (D)\quad\max&\quad d(\lambda,\mu,\nu) \\ \text{s.t.}&\quad \lambda\ge 0,\quad\nu\ge 0. \end{align} $$
What if I do not assign dual variables to the constraints $x\le 0$? That is, let $$ \widetilde{L}(x,\lambda,\mu)=f(x)+\sum_{i=1}^m\lambda_ig_i(x)+\sum_{j=1}^{\ell}\mu_jh_j(x) $$ and, when forming the dual function, perform minimization over $x\le0$ instead of $x\in\mathbb{R}^n$, which gives $$ \widetilde{d}(\lambda,\mu)=\inf_{x\le 0}L(x,\lambda,\mu),\quad\forall\,(\lambda,\mu)\in\mathbb{R}^m_+\times\mathbb{R}^{\ell}. $$ This yields $$ \begin{align} (\widetilde{D})\quad\max&\quad \widetilde{d}(\lambda,\mu) \\ \text{s.t.}&\quad \lambda\ge 0. \end{align} $$ Are $(D)$ and $(\widetilde{D})$ equivalent?
