# Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is shown that using a model of degree p, this algorithm will find a strong approximate q-th-order minimizer in at most O(\max_{1 \leq j \leq q}}\epsilon_j^{-(p+1)/(p-j+1)}) evaluations of the problemâ€™s functions and their derivatives, where $\epsilon_j$ is the j-th order accuracy tolerance; this bound applies when either q = 1 or the problem is not composite with q \leq 2. For general non-composite problems, even when the feasible set is nonconvex, the bound becomes O(\max_{1\leq j\leq q}\epsilon_j^{-q(p+1)/p}) evaluations. If the problem is composite, and either q > 1 or the feasible set is not convex, the bound is then O(\max_{1\leq j\leq q}\epsilon_j^{-(q+1)}) evaluations. These results not only provide, to our knowledge, the first known bound for (unconstrained or inexpensively-constrained) composite problems for optimality orders exceeding one, but also give the first sharp bounds for high-order strong approximate q-th order minimizers of standard (unconstrained and inexpensively constrained) smooth problems, thereby complementing known results for weak minimizers.