# Geometric interpretation of KKT conditions

I can explain why Lagrange multipliers work for scalar functions by vector calculus. Consider optimizing $$f(\vec{x})$$ subjected to the constraint $$g(\vec{x}) = c$$.

At the optima, we can move infinitesimally within the feasible region without changing the value of $$f(\vec{x})$$. This infinitesimal movement (and any movement in the feasible region) must be along the constraint, which is one level curve of $$g$$. Since the infinitesimal movement doesn't change the value of $$f$$, the infinitesimal movement is also along the level curve of $$f$$.

As the gradient of a function at any point is perpendicular to the function's level curve, we have $$\nabla f = \lambda \nabla g$$.

Where can I find a similar explanation for KKT conditions?

• Take a look at chapters 4 and 5 of this book. The link is to the book Non-linear programming by Bazarra. Jul 3 at 14:41