I'm currently taking a convex optimization course at my school using the textbook *Convex Optimization (Boyd & Vandenberghe)* and had a question regarding KKT conditions and their connection to Lagrange multipliers. I've taken a look at other questions regarding the two topics, such as:

- Lagrange multipliers and KKT conditions - what do we gain?
- Connection between method of Lagrange multipliers and KKT conditions?
- Lagrange multiplier vs KKT

and have read the Wikipedia article on KKT conditions as well but haven't been able to figure out my question.

I'm reading that "Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints," and "when there are no inequality constraints, the KKT conditions turn into the Lagrange multipliers."

I'm confused about this concept because when I studied about Lagrange multipliers earlier in the chapter, I studied that **"the Lagrange multiplier $\lambda_i$ is associated with the $i$th inequality constraint $f_i(x)\le 0$."**

What does it mean when it says that Lagrange multiplier allow only equality constraints, when there seems to clearly be inequality constraints included as well?

Thank you.