Optimality Conditions for functions with Lipschitzian Gradients

Question

I am looking for sufficient optimality conditions for problems of the type $$ \min_{x} \lbrace f(x) | x \in X \rbrace, $$ where $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable with Lipschitzian gradient, i.e. $f \in C^{1,1}$ and $X$ is a closed subset of $\mathbb{R}^n$. Is it possible to formulate optimality conditions in terms of generalized Hessians?

Answer 1

Short answer, yes.

Long answer, it depends.

A minimum (global or local) in a closed set X is in the open set of X (union of all open subsets of X) or in it's frontier.

If X is a closed subset then the minimums are harder to find, it can involve Lagrangian multipliers and karush kuhn tucker conditions.

On the other hand if X is an open subset then a sufficient but not necessary condition for the existance of a minimum in a point a is that the Hessian in the point a is definite positive.