Is it necessary to add the non-active constraints within the Lagrangian for solving a convex optimization problem?

Question

Suppose we have a convex optimization problem as below $$\min~~ f_0(x)$$ $$\text{s.t. }f_1(x)\leq 0\\ f_2(x)\leq 0\\ \vdots \\f_{m-1}(x)\leq 0\\ f_m(x)\leq 0.$$ Suppose we know, through some way, that the last constraint is not active in the optimal solution. Then, is it ok to write the Lagrangian of the above problem as follows $$L(\lambda_1,\lambda_2,\cdots \lambda_{m-1},x)=f_0(x)+\lambda_1f_1(x)+\cdots \lambda_{m-1}f_{m-1}(x).$$ That is, can we ignore that constraint which we know is inactive in the optimal solution while writing the Lagrangian of the problem? Thanks in advance.

Answer 1

Yes, you may drop any constraint that you know does not bind. This follows from the complementary slackness conditions in the KKT theorem, which read $$\lambda_i \ge 0, \quad \lambda_i f_i(x) =0 $$ for all $i=1,\ldots,m$, where $\lambda_i$ is a Lagrange multiplier for the $i$-th constraint. Let $x^*$ be a solution to your minimisation problem. If you know that at $x^*$, the $m$-th constraint does not bind, then you must have that $f_m(x^*) < 0$. Complementary slackness then requires that $\lambda_m = 0$. This means that this constraint effectively does not feature in your optimisation problem.

To help convince yourself, it might be instructive to consider a simple example. Suppose you wished to minimise $x^2$ subject to the constraint $x \le 1$. Obviously, the constraint will not bind, and you are free to ignore it. If you try to solve this problem by considering the Lagrangian, you will find that the Lagrange multiplier for the constraint has to be zero, which is the same as saying you can effectively ignore the constraint, as it does not bind.