Suppose there is a constrained optimization problem having inequality constraints. We can solve it using Karush-Kuhn-Tucker conditions. My question is how do we determine which constraints are active and which are inactive? I read it in a KKT post1, post2 that we need to try all possible combinations of active and inactive constraints.

So after reading about it I came upon the following summary for finding a local optimum satisfying some inequality constraints. If we have $n$ inequality constraints we assume $k$ of those constraints to be active and solve using KKT necessary conditions. We do this for $k$ from $0, 1, \ldots, n$. So total number of possibilities are $2^n$. Suppose among these $m$ necessary conditions are satisfied. We use KKT sufficient conditions on those $m$ cases. If $m_1$ among these are satisfied then these lead us to the local optimal solutions. Is this right?

Is this the way to theoretically approach this problem, assuming we don't need a numerical techniques to solve it?

Just a stupid query by can this problem be possibly solved by assuming that all $n$ constraints are active, and then we get values of Langrange multipliers $\lambda_i, i\in \{0,1,\ldots,n\}$. Whichever $\lambda_i$s are negative are the inactive constraints?