Does the $\max_x \min_\lambda L(x,\lambda)$ concept still work for inequality constrained optimization?

Question

When we want to maximize a function $f(x)$ with respect to $x$, and w.r.t a constraint $g(x)=0$ we can formulate the Lagrangian $$L(x, \lambda) = f(x) + \lambda h(x)$$

And then maximize $L(x,\lambda^*)$ with respect to $x$, where $\lambda^*$ is the value that minimizes $L$ with respect to $\lambda$.

That is, The above equality constraint problem is equivalent to the problem: $$\max_x\min_\lambda L(x,\lambda)$$

My question is, does an analogous condition also exist for inequality constraints? My intuition is no, because with the inequality constraint, we have the additional "complementary slackness" requirement, and I'm not sure how to fit that in.

Answer 1

My understanding is "Yes". You can still apply LP theory with inequality constraints. See Equality and Inequality Constraints or Note.