Handling an inequality constraint by introducing a slack variable and then using Lagrange multipliers

Question

According to the Wikipedia page, the Lagrangian method can only be used with equality constraints. If I have an inequality constraint, can I introduce a slack variable to convert the inequality constraint into an equality constraint and then apply the same Lagrangian method? If I can do so, what is the difference between KKT condition and this?

Answer 1

From my personal understandings(mostly from prof. Stephen Boyd's book), Lagrange multipliers can be used to both equality constrained problems and inequality constrained problems. The KKT conditions are in fact, I am not sure whether it's proper to say though, derived from the lagrangian under strong duality conditions.

Back to your question, the answer is yes, you can do that, but be careful about the sign of the multiplier(for equality constrained problems, the sign of the Lagrange multiplier doesn't matter, but not for inequality constrained ones. It's positive when inequality are defined under 'less or equal', negative on the other case).

But whether you formulate the Lagrangian using slack variables, or directly by inequalities(they are essentially the same), you can't change the fact that you are dealing with an inequality constrained problem, so "applying the same lagrange method" will not work. Some iterative method is necessary, consider interior point method.