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We provide sharp worst-case evaluation complexity bounds for nonconvexminimization problems with general inexpensive constraints, i.e.\ problemswhere the cost of evaluating/enforcing of the (possibly nonconvex or evendisconnected) constraints, if any, is negligible compared to that of evaluatingthe objective function. These bounds unify, extend or improve all known upperand lower complexity bounds for unconstrained and convexly-constrainedproblems. It is shown that, given an accuracy level $\epsilon$, a degree ofhighest available Lipschitz continuous derivatives $p$ and a desired optimalityorder $q$ between one and $p$, a conceptual regularization algorithm requiresno more than $O(\epsilon^{-\frac{p+1}{p-q+1}})$ evaluations of the objectivefunction and its derivatives to compute a suitably approximate $q$-th orderminimizer. With an appropriate choice of the regularization, a similar resultalso holds if the $p$-th derivative is merely H\"older rather than Lipschitzcontinuous. We provide an example that shows that the above complexity bound issharp for unconstrained and a wide class of constrained problems, we also givereasons for the optimality of regularization methods from a worst-casecomplexity point of view, within a large class of algorithms that use the samederivative information.

Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints