Can I use Lagrange Multipliers with inequality constraints?

Question

Suppose I had a problem: Maximize $f(\bf{x}) $ subject to the contraints $g_i(\bf{x})< b$

Can I still use Lagrange multipliers? My text says that the constraints need to be equalities.

Answer 1

You do want $\le$, not $<$, otherwise there is likely to be no maximum. The generalization of Lagrange multipliers to allow inequality constraints is the Karush-Kuhn-Tucker conditions.