The evaluation complexity of finding high-order minimizers of nonconvex optimization

Abstract

We introduce the concept of strong high-order approximate minimizers of nonconvex optimization problems. These apply in both standard smooth and composite nonsmooth 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, the evaluation complexity of this algorithm is investigated. The bounds obtained not only provide, to the best of our knowledge, the first known result 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.