Consider an optimization problem $$ \begin{align} \max_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad & g_i(x) \le0,\,i=1,\ldots,m\\ \quad & h_j(x)=0,\,j=1,\ldots,l\\ \end{align} $$ The (complete) Lagrangian for the above problem is $$ \mathcal{L}(x,\mu,\lambda) = f(x) + \sum_{i=1}^m\mu_ig_i(x) + \sum_{j=1}^l\lambda_jh_j(x) $$ The set of KKT conditions are
1) Stationarity: $\nabla_x\mathcal{L}(x,\mu,\lambda)=0\equiv\nabla f(x) + \sum_{i=1}^m\mu_i\nabla g_i(x) + \sum_{j=1}^l\lambda_j\nabla h_j(x) = 0$
2) Complementary slackness: $\mu_ig_i(x) = 0$ for all $i$
3) Dual feasibility: $\mu_i\ge0$ for all $i$
4) Primal feasibility: $g_i(x)\le0$ and $h_j(x) = 0$ for all $i,j$
BUT we can also form a partial Lagrangian, say $$\mathcal{L}(x,\mu) = f(x) + \sum_{i=1}^m\mu_ig_i(x)$$ by just dualizing the inequality constraints. The partial Lagrangian would have a different set of KKT conditions, right?
Does this mean that we can have many different KKT conditions for the same optimization problem?
Is this the correct way of thinking about KKT conditions? Do they correspond to the optimization problem or the Lagrangian?